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LOUDON, B.A. DKMONSIRATOR IN PHYSICS IN Till- UNIVERSITY OV TORONTO V » *' fxyjhcu ayav Xclu |[Jork MACMILLAN AND CO. AND LONDON 1896 All rights reserved Copyright, 1895, Bv MACMILLAN AND CO. NorhjooO 53rf3B J. S. CushiiiK it Co. -- Berwick & Smith Norwood Muss. U.S.A. t ti t; tl Ci n k: C( i PREFACE. This elementary treatise on Rij;id Dynamics has arisen out of a course of lectures delivered by me, during the past few years, to advanced classes in the University. It is intended as a text-book for those who, having already mastered the elements of the Calculus and acquired some familiarity with the methods of Particle Dynamics, wish to become acquainted with the ]:)rinciples underlying the equations of motion of a solid body. Although indebted to the exhaustive works of Routh and Price for many suggestions and problems, I believe that the arrangement of the work, method of treatment, and more par- ticularly the illustrations, are entirely new and original ; and that they will not only aid beginners in appreciating fundamen- tal truths, but will also point out to them the road along which they must travel in order to become intimate with those higher complex motions of a material system which have their culmi- nating point in the region of Physical Astronomy. My thanks are due Mr. J. C. Glashan of Ottawa, who has kindly read the proofshects and supplied me with a large collection of miscellaneous problems. W. J. LOUUOxN. University ok Toronto, Aug. 19, 1895. ^ 1 f CONTENTS. l\romcnts of Inertia Illustrative Examples CHAPTER I. PACK I II CHAPTER II. Ellipsoids of Inertia Illustrative Examples . ••••*» i6 18 Equimomental Systems Principal Axes 20 . . . 22 Illustrative Examples . CHAPTER III. . 26 D'Alembert's Principle . • 31 • 34 • 35 • 1>1 • 39 Impulsive Equations of Motion Illustrative Examples The Principle of Energy Illustrative Examples CHAPTER IV Motion about a Fixed Axis. Finite Forces The Pendulum .... Illustrative Examples . Determination of .^ by the Pendulum Pressure on Fixed Axis Illustrative Examples . 45 49 51 53 58 62 Vll vUi CONTliNTS. CIIAITKK V. Motion about a Fixed \xis. IiniJiiisivc Korcrs Centre of I'crciission .... Illustrative Examples .... Initial Motions. Changes of Constraint liinstrativc I')xanii)Ies .... Tiic liallislic Pendulum PAOB 70 72 74 78 78 8S CHAI'IKR VI. Motion aljout a Fixed Point. Finite Forces Anjj;ular VY'locity ....... Ciuneral Equation.s of Motion .... Equations of Motion referred to Axes fixed in Space luiler'.s Equations of Motion ..... Aufjular Coordinate-, of the IJody .... Pressure on the Fixed Point ..... Illustrative Examples ...... Top spinning on a Rough Horizontal Plane . Top spinning with Great Velocity on a Rough Horizontal Plane The Ciyroscope moving in a Horizontal Plane about a Fixed Point 88 88 98 100 lOI IDS 109 III 112 120 128 CHAPTER VII. Motion about a Fixed Point. Impulsive Forces Illustrative Examples 134 137 CHAPTER VIII. Motion about a Fixed Point. No Forces acting 140 CHAPTER IX. Motion of a Free Body Illustrative Examples Impulsive Actions Illustrative Examples 145 148 156 158 CDNTKNTS. IX PAfJK 70 72 74 78 78 85 CHAl'TKK X. The (iyroscope ....... To prove the Rotation of t lie Karth upon its Axis Hopkins' i:irxtrical Gyroscope . . . , Fcssel's (iyroscft>- CHAPTER I. MOMENTS OF INERTIA. 1. In attempting to solve the equations of motion of a Rigid Body in a manner similar to that employed for a single particle, it will be found that certain new quantities appear, v hich depend on the extent and shape of the body, on iis density, and on the way in which it may be moving in respect of some particular line or system of coordinate axes. 2. These quantities are called Moments of Inertia and Products of Inertia. A moment of inertia of a body about any line is defined to be the sum of the products of all the material elements of the body.by the squares of their perpendicular dis- tances from this line. It may be denoted in general by the letter /, and when / is expressed in the form AIK^, where M is the mass of the body, K is called the radius of gyratioti. When the body is referred to three codrdinate rectangular axes, the moments of inertia about the three axes will evidently be A^^m^f^z^), B=-.tm{a'^+x'^), C=tm{x^+f), ni being the mass of any element at the point (,r, y, s), and the summation being taken throughout the body. A product of inertia is defined with reference to two planes at right angles to one another and is found by multiplying the elements by the products of their distances from these coordi- In 2 RIGID DYNAMICS. nate planes, and summing them throughout the body. ProcKicts of inertia exist in sets of three, and for three rectangular axes D=—in}>a, E=^inax, F=1nixy. 3. It is evident that when the law of ;// is known and the shajie of the body is given, the finding of a moment or of a jjrcnluct of inertia involves an integration ; and the following examples will serve to show how the process of integration may be uf'id for this purpose. Further on, several propositions will be given by which the method may be usually much simplified. 4. Illustrations of finding Moments of Inertia by Integration, (a) A uniform rod of small cross-section about a line perpen- dicular to it at one end. Here, if the length of the rod be 2 a, and the density p, X2a px^d: '^dx--=M 4a^ {b) A circular arc of uniform density about an axis through its midpoint perpendicular to its plane. In Fig. I, let OA = r, OCA = 6, OCB^n; then the n.oment of inertia of the arc BOD about an axis through O perpen- dicular to the plane of the paper is 2 Ipds • >^, where ds is an element of the arc at A. 1 ■.')v'^"^ ■^$. MOMENTS OF INERTIA. kment Irpen- is an 1=8 pa^ sin2-,/0 = 4pr;3 p(i -cos 0),/e = 2Jl/(i -''-— V. (c) An elliptic plate, of small thickness and uniform density, Fig. 2. In Fig. 2, divide the plate into strips, and then we have /about 0V=4 Cpx^fdx=4p C - x^^d^-,vdx=-.M--. c/" Jo a 4 Similarly, /about OX=M — And / about a line through O perpendicular to the plate will evidently be M For a circular plate a = b. (d) A rectangular plate, sides 2 a, 7 b. By dividing the plate into strips of mass m it will be seen that ./ 4 /A 4<^ /about side 2a^\\m \ = M 3 ' 3 / 3' Also, / about a line through a corner perpendicular to the plate is M^irr-^b^). For a square plate a = b. and /aboutside2^ = :iC;/^if^')==J/i^ 4 RIGID DYNAMICS. {e) A triangular plate. Let the triangle be ABC, and choosing C as origin of coordi- nates, let CA, CB be the axes. Then, dividing the triangle into strips parallel to AC^ an elemental mass at (,r, y) is equal to pdxdy sin C, p being the density. The distances of this ele- ment from AC, BC, and the point C are xsinC, ^ sin C, and V,f '^ +y^ +2xycosC. Hence / about AC= ( | px^ sin^ Cd.rdy, °(a-x) / about BC= I i py^ sin^ Cdxdy, and /about a line through C perpendicular to the triangle "(« X) Xa /*a I p sin ^ (,r2 +7^ + 2 -ij cos C)dxdy. These integrals can easily be evaluated, and the moments of inertia expressed in terms of the two sides and included angle. (/) A sphere about a diameter. Dividing the sphere up into small circular plates of thickness a. I m Fig. 3. MOMENTS OF INERTIA. dv, as in Fi?^. 3, wc have I about a diameter = 2 i^p • irfdx • - =p7r i (^i2_ ^-2^2^^^= j/ 2 ^2 1-2 (^r) A right circular cone, about its axis. Fig. 4. Dividing the cone up into circular strips, perpendicular to its axis, as in Fig. 4, we have, if a be its height, I=^P'jTy^ • dx — , and y= --. 1= irp d* - Mlb^ M 10 5. Products of inertia can be evaluated in a similar way ; but as they are generally eliminated from the equations of motion by a proper choice of axes, their absolute values in terms of known quantities are seldom required. 6. Although integration gives directly the values of moments and products of inertia, yet the process becomes tedious for many bodies ; and the following propositions will be found use- ful for their determination, when one knows the position of the centre of inertia. RIGID DYNAMICS. Proposition I. — To connect moments and products of inertia of a rigid body about any axes ivitli moments and products of inertia about parallel axes through the centre of inertia. Fig. 5. Let the plane of Fig. 5 represent any plane of the body per- pendicular to the two parallel axes, of which one cuts this plane in the point O, and the other passing through the centre of inertia cuts it in G. Then for any point P in this plane, we have r^=:p'^+r'^ + 2P' GM. Hence for the whole body we must have tmr^=tm{p'^-\--t^'^ + 2p • GM) = ^mp^+Smr'^-{-2p '^mGM = Mp'^ + tmr'\ since tmGM^^o. Or, as it may be written I=r,+Mp^ where / is the moment of inertia about any axis, and /<, is that about a parallel axis through the centre of inertia, and / is the perpendicular distance between ihe axes. If three parallel axes be taken in a body, of which the third passes through the centre of inertia, and a plane be taken cut- iiicts of inertia mi products of crtia. the body per- one cuts this ugh the centre this plane, we M=o. and Iq is that \, and / is the ?hich the third : be taken cut- MOMKNTS OF INERTIA. 7 ting these axes perpendicularly at the points O, O', G, then we can prove for the whole body, as before, that / about axis through O =r'-\-Ma\ /' about axis through 0' = lo + Mb'^, OG=a, 0'G=b. and where If G happens to be in the line 00\ this relation is much sim- plified. Also, if OG is at right angles to 00\ then which relation is sometimes useful in the case of symmetrical bodies. It is evident, moreover, from these relations that, of all straight lines having a given direction in a body, the least moment^'of inertia is about that one which passes through the centre of inertia. Fig. 6, In the case of products of inertia, similar results may be obtamed. Thus, if we require the product of inertia with regard 8 RIGID DYNAMICS. to any two coordinate planes of a body, let parallel planes be taken passing through the centre of inertia. Let the plane of the paper in Fig. 6 be any plane of the body perpendicular to these four planes. Then, if P be any point whose coordinates referred to the two sets are {x\ y) and (-i',/'), we must have for the whole body Xmxy = tm {x' +/) (/' + <]) = ^vix'y +f^inx' + q^mv' 4- ^vipq. . : %vix}' = %>fix'y + M - pq. Proposition II. — /// t/ic case of a lamina, tJie viomcnt of inertia about any axis perpendicular to its plane is cqnal to the sum of the moments about any two perpendicular lines drazun in the plane through the point ivhere the axis meets the lamina. For /= ^m{x'^ +_;/2) = Imx"^ + ^my"^. Proposition III. — To find the moment of inertia of a body about any line, kmnving the moment and products of inertia about any three rectangular axes drawn through some point on this line. In Fig. 7 let the three rectangular axes be OX, OY, OZ, and let P be any point of the body {x, y, z), and ON -diWy line drawn from O, inclined at angles «, /S, 7 to the axes. Then /about ON=tmPN'\ /W being perpendicular to OY, and /\V- = (9P2 _ OX^ = (.1-2 +f + ,c'-^) - {x cos a -\-y cos /3 -f .- cos 7)2 = (x^ +J'2 + ."2) (cos^ a + cos^ ^ + cos^ 7) — (.r cos a +y COS/3 + .C' cos 7)2 = (j'2 + ,:;2)cos2 «+ ••• + ••- — 2J',CC0S /3cos 7 — •••. . •. /= "^M l{y^ + ^^) cos^ a H — -!- J — 2 Sm \ yrj cos j3 cos 7 + ••• + ••• MOMENTS OK INEKTIA. I planes be he plane of Midicular to coordinates ist have for V moment of qual to tJic s drawn in imina. I of a body of inertia le point on OZ, and ine drawn ar to OX, i + .:r cos 7)2 a 1 4-- cos 7)2 3s/3cos7 •• + ••• Fig. 7. = A cos^a + Z? cos^/3 + Ccos2 7 — 2 D cos /3cos7 — 2E cos 7 cos « — 2 /'cos rt cos ^, A, /), C being moments of inertia about the three axes, and D, E, F products of inertia with regard to the coordinate planes. In this expression, it will be seen that if the axes of coordi- nates be so chosen that D, E, I'' vanish, then /= A cos^ a + B cos^ ^+C cos^ 7. Axes for which this holds are called Principal Axes, and A, B, C Principal Moments In many cases such axes can be found by inspection. Thus, if a body be a lamina, one principal axis at any point is the perpendicular at that point. Also, if a body be one of revolution, the axis of revolution must be a principal axis at every point of its length. And it may be stated as a general rule that axes of symmetry are principal axes. 10 RKilD DYNAMICS. 7. In most of the problems dealing with the motion of c'xtentled bodies the axis about vvhieh the moment of inertia is to be found usually passes throujjjh the body ; but it is apparent that the precedinj;- propositions apply equally to all cases where the axes about which moments of inertia are required do not cut the body. Thus in the first proposition the axes parallel to that passing through the centre of inertia need not cut the body ; in the case of a lamina, the moment of inertia about any line perpendicular to the lamina and yet not intersecting it will still be the sum of the moments about any two perpendicular lines drawn at the point where the axes meet the plane of the lamina produced ; and similarly the moment of inertia about any line outside of a body will be known when we know, at any point on this line, the moments and products of inertia with respect to any three rectangular axes drawn through this point. ! : ; I 8. TozvuscncV s Theorem. A closed central curve, of any magnitude and form, being supposed to revolve round an arbitrary axis in its plane not intersecting its circumference ; the moment of inertia with respect to the axis of revolution of the solid generated by its area is given by the formula /=J/(rt2 + 3/,2), where M is the mass of the solid generated, a the distance of the centre of the generating area from the axis of revolution, and Ji the arm length of the moment of inertia of the area with respect to a parallel axis through its centre. For, if dA be an element of generating area, p being the density, and x a variable coordinate. MOMKNTS OF INKRTIA. II motion of [ inertia is s apparent ises where eel do not parallel to 3t cut the about any tinj; it will ■pendicular ane of the ;rtia about lovv, at any nertia with this point. 5rm, being plane not lertia with ited by its istance of revolution, area with Ikit, by the symmetry of the generating area with respect to its centre, i;(.iv//i)=o and i^(.rV//)=o. Illustrative Examples on Miwients of Inertia. I. Find the moment of a rectangular plate about a diagonal, the sides being 2a, 2b. In this, applying Pruj^osition III., we have /=/lcos2 6? + />'sin2^, the centre of the plate being the origin, and A, B principal moments. 3 aVf^ I=M Id'^l)^ 2. A sphere or a circular plate, about a tangent. Apply Proposition I. 3. Find the moments of inertia of a rectangular parallelo- piped and of a cube, about their axes of symmetry ; also about a diagfonal. 4. The moment of inertia of a right circular cone about a s 1 /;2 5^2 I /,2 slant side is M „ „ , a being the height and b the radiu 20 a^ + lP' of the base. 5. If a is the length and b the radius of a right circular cylinder, the moment of inertia about an axis through the cen- M la'^ \ tre of inertia perpendicular to its axis is — ( — \-b^\. 6. The moment of inertia of a pendulum bob, density p, in the form of an cqui-convex lens of thickness 2/ and radius a, ibout its axis is irp I {2ax—x'^)^dx. la ki(;nj DYNAMICS. 7. Find the moment of inertia of an anchor ring about its axis. 8. The moments of inertia of an ellipsoid about its three axes arc j\f — ^^^, M — - — , M - — 5 5 5 To find these, either divide the solid ellipsoid up into elliptic jilates, or deduce from the case of a sphere. 9. A trian^^ular plate of uniform density. (I) To find the moment of inertia about the side HC. t II l¥//////////////////^^^^^ JC a. Fig. 8. c In Fig. 8, divide the triangle into strips of mass pydx, where y=B'C. Then /about BC=^nydx • x'^= p \ ~ — '-ax^dx = M -^ • '■^ '^Jo p 6 (2) About a line through the centre of inertia parallel to BC. (3) About a line through A parallel to BC. MOMENTS OF INKRTIA. •3 about its three axes to elliptic ?C. iv, where 2l to BC. (4) About a median lino. ,/7 In Fig. 9, divide the triangle into strips parallel to BC, as before, and \et }' = B'C'. Then the mass of a strip is pj/t/.v sin JJ, and its moment of mertia about AD is pjv/.r sin D • - sin''^ D. 12 Hence /of triangle about AD=— | y^dx, and j' = — '^. rd'^ sin^ y; I=M- 24 ./ ^2_|_^2 \ ^^ ^^ ^ being the three sides. (5) About a line through A, perpendicular to the plane of the triangle. Use Fig. 9, and the moment of inertia will be found to be m - -2N 4 \ 3. (6) About a line through the centre of inertia, perpendicular to the plane of the triangle. 36 10. Find the moment of inertia of a hemisphere about (i) Its axis. (2) A tangent at its vertex. (3) A tangent to the circumference of its base. (4) A diameter of its base. 14 RIGID DYNAMICS. II. The moment of inertia of an ellipsoidal shell of mass M fi^-c^ For a spherical shell about a m about the major axis is M diameter, I=^M\a^. ^ Deduce, by differentiation, from the ellipsoid and the sphere. 12. For an oblate spheroid (such as the earth), of ex^en- tricity e, composed of similar strata of varying density, he moment of inertia about its polar axis is fTrVi— e^j p.-^dx, where a is the equatorial radius and p the density at a distance X from the centre. This can be integrated when the law of p is known. 13. The moment of inertia of a paraboloid of revolution about its axis of figure is M • —, where r is the radius of the ■I base. 14. The moment of inertia of the parabolic area cut off by any ordinate distant x from the vertex is M^ x"^ about the tan- gent at the vertex, and M — about the axis, where j/ is the ordi- nate corresponding to x. 15. The radius of gyration of a lamina bounded by the lem- niscate r^^a^' cos 26, (i) about its axis is -Vtt — |; (2) about 4 a line in the plane of the lamina through the node and perpen- dicular to the axis is -Vtt + I ; (3) about a tangent at the node 4 IS -Vtt. n ii; 16. To find the radius of gyration of a lamina bounded by a parallelogram about an axis perpendicular to it through its cen- tre of inertia. (Euler.) If 2a, 2 b, be the lengths of two adjacent sides of the parallel- ogram, then, whatever be their inclination, 3 MOMENTS OF INERTIA. '5 17. To find the radius of gyration of a hollow sphere about a diameter. (Euler.) 5 a^ — b^ a and b being the external and internal radii. 18. To find the radius of gyration of a truncated cone about its axis. (Euler.) 3 a^-b^ 10 = 10 a^ — b^ a, b, being the radii of its ends. 19. The moment of inertia of a lamina bounded by a regular polygon of n sides, each of length 2 a, about an axis through its centre perpendicular to its plane is f(. + 3co.^) And from this it can be seen that the moment of inertia about any line in the plane of the lamina through the centre 12 \ ^ ;// 20. A quantity of matter is distributed over the surface of a sphere of radius a, so that the density at any point varies in- versely as the cube of the distance from a point inside distant b from the centre. Find the moment of inertia about that diame- ter which passes through the point inside, and prove that the sum of the principal moments there is equal to 2 M {a^ — b'^). What if the point be outside .? :; CHAPTER II. ELLIPSOIDS OF INERTIA AND PRINCIPAL AXES. 9. Ellipsoids of Inertia, At any point C? in a rigid body let there be taken three rectangular axes OX, OV, OZ, as in Fig. 10. Describe with O as centre the ellipsoid, Ax^ + By^ + C:.^ — 2 Dyz - 2 Ezx — 2 Fxy = c, z Fig, 10. 16 ELLIPSOIDS OF INERTIA AND PRINCIPAL AXES. 17 three vith O where A, B, C, D, E^ F, have the meanings already attached to them, and are positive. Then, if OP be any line drawn from O, and cutting the ellipsoid in the point P, the moment of inertia of the body about OP is A cos" a + B cos^ /9 + ^'cos^ 7 — • • • = /, where «, /3, 7 are the angles which OP makes with the coordi- nate axes. But if X, y, z, are the coordinates of the point /*, and if OP = r, we must have, since the point is on the ellipsoid. And since this relation is true for any position of OP, we see that the moment of inertia about any line drawn from O will be inversely proportional to the square of the corresponding radius vector cut off by the ellipsoid. Any such ellipsoid is called a Momental Ellipsoid. 10. If we refer the ellipsoid to its axes OA, OB, OC, then D, E, F disappear, and the axes of the ellipsoid are therefore what we have defined as Principal Axes. 11. It is evident that any set of principal axes at a point might be found in the foregoing manner, namely, by construct- ing a momental ellipsoid at the point in question and trans- forming to the axes of figure, which would therefore give the directions of the principal axes. And it may be stated also that three principal axes necessarily exist at each point in space for a rigid body, since the above [)rocess can always be performed. 12. From the properties of the momental ellipsoid it follows that at any point there is, in general, a line of greatest moment and also one of least moment ; if the ellipsoid degenerates into a spheroid, the moments of inertia about all diameters perpen- dicular to the axis of the spheroid are the same ; if it becomes a sphere, as in the case of all regular solids at their centres, the moments of inertia about all lines through the centre are vmtm i8 RIGID DYNAMICS. ill 1; equal, a proposition which can be applied with advantage to the cube, proving that the moments of inertia about all lines through the centre are the same. 13. For a lamina, at any point, the section made by the cor- responding momental ellipsoid is called the Momenta/ Ellipse of the point. Illtistrativf Examples. 1. To construct a momental ellipsoid at one of the corners of a cube. Taking the edges as axes, A==B=C, D=E=E, and the equation for the momental ellipsoid becomes A {x^ + j'2 -I" z^) — 2 Dixy -\-yz + ::-x) = c, which on transformation would give a spheroid of the form and it can be seen that one principal axis is the diagonal through the corner in question, and any two lines at right angles to one another and to the diagonal will be the other two principal axes. 2. To find the momental ellipsoid at a point on the edc, j of a right circular cone. Choosing axes OX, OV, OZ, as in Fig. 11, it is evident by inspection that D=^F=o, and the axis OY is one principal axis. Then, if AB = a, OB = b\ BG^la, and A=m(^-+-\ * \20 10^ B = A-\-Mb\ C=M^-^, E = M—, and the equation of the 10 4 momental ellipsoid at O is (3 <^H 2 rt2).i-2 -I- (2 3 /;2 + 2 rt2)j2 ^ 26 ^2-2 _ I o abxz = c. t t r c V a r The momental ellipsoid .,t the point A, or at any point along the axis AB., is a spheroid. of y2\ i ELLIPSOIDS OF INERTIA AND PRINCIPAL AXES. 19 Fig. 11. 3. The momental ellipsoid at a point on the rim of a hemi- sphere is 4. The moi.iental ellipsoid at the centre of an elliptic plate is 5. The momental ellipsoid at the centre of a solid ellipsoid is 14. The Ellipsoid of Gyration. If at a point in a body an ellipsoid be constructed such that the moment of inertia about any perpendicular drawn from the origin on a tangent plane is equal to Mp'^, where M is the mass of the body and p the length of the perpendicular, it is called an ellipsoid of gyration. And, since, referred to its axis, we have by definition A=Ahfi about the axis of x, B = Mb'^ about the axis oi y, and C=Me'^ about the axis of s, its equation must be ^-2 1 ,2 „2 .+ •'- + - = -—. ABC M This ellipsoid may also be used to indicate the directions of the principal axes ; and, from the form of its equation, it is 20 RIGID DYNAMICS. apparent that it is co-axial, but not similarly situated, with a momental ellipsoid. 15. When ellipsoids are constructed at the centres of inertia, it is customary to speak of them as ccjitval ellipsoids. 16. Equimonicntal Systems. Two systems are equimomental when their moments of inertia about all lines in space are equal each to each. And from this definition, taken along with the two fundamental proposition.s already proved, — /= A cos^ a -\- B cos^ ^+C cos^ 7, it follows that systems will be equimomental when they have 1. The same mass and centre of inertia. 2. The same principal axes at the centre of inertia. 3. The same principal moments at the centre of inertia. In some particular cases we may, instead of considering a system or single body, use a simple equimomental system in determining its motion ; but generally the labour of proving that systems are equimomental, or of finding a simple system which will be equimomental with a complicated one, is greater than that of solving the problem directly. The following examples, however, will serve to show how the process is carried out. > Illustrative Examples. M I. Show that three masses, each equal to — .placed at the 3 middle points of the sides of a triangular plate of mass lil, are equimomental with the triangle. If this equimomental system be assumed, all the problems in connection with a triangular plate, such, for example, as finding moments of inerlia about the sides, perpendiculars, and median lines, are very much simplified ; but the difficulty of proving ELLIPSOIDS OF INERTIA AND PRINCIPAL AXES. 2 1 this assumption is greater than that of solving the problenis, as has already been clone by a direct process. 2. In an elliptic plate, find three points on the boundary at which, if three masses each equal to — be placed, they will form 3 a system equimomental with the plate, whose mass is M. 3. Show that three points can always be found in a plane area AT of mass J/, so that three masses, each equal to — , placed at 3 these points will form a system equimomental with the area. The situation of the points is shown in Fig. 12, which repre- sents the momental ellipse at the centre of iner'.ia of the area. A may be anywhere on the boundary of the ellipse ; B and C are so situated that BD = DC ?i\\(\ OD=DE. Fig. 12. 4. Find the momental ellipse at the centre of gravity of a triangular area. 22 RIGID DYNAMICS. 5. Th2 niomental ellipse at an angular point of a triangular area touches the opposite side at its middle point, and bisects the adjacent sides, 17. Pyincipal Axes. To find the principal axes at any point of a rigid body, three rectangular axes might be chosen, and the conditions Xmxy = o, '^mj',a=o, ^;«,av=o, would be sufficient to solve the problem, cither by direct analysis or by the construction and subsequent transformation of the equation of the momental ellipsoid. But this process would often be tedious, and is generally unneces- sary. Usually, by inspection, one at least of the principal axes can be found, as has been already mentioned, and then the other two may be obtained by the following propositions. Given one principal axis at a point, to find the other two. Let O be any point in the body, and let OZ, drawn perpen- dicular to the plane of the paper be one perpendicular axis. Take any two lines, OX, O Y, at right angles to one another as ■ Fig. 13. ELLIPSOIDS OF INERTIA AND PRINCIPAL AXES. 23 axes in this plane, and let OA'', OV be the other two principal axes at O. Then if P be any point (x, y) or {x'y'), and the body extends above and below the plane of the paper, we must have as a condition that OX',OY' shall be principal axes, ^wx'y' = o throughout the body. But x'=x cos e-\-y sin 6, and / = -x sin 6 +y cos 6. Therefore the condition becomes tm\ -x"^ sin 6 cos O+y'^ sin 6 cos 6+xy cos=^^-sin^ 6] =0, which becomes, on reduction, tan 2 ^ = — ---^Z_=^Z_, :Lnix'^-^iny'^ B-A according to our previous notation. If, then, A, B, F be found in respect of any two rectangular axes OX, OY, 6 is known, and therefore the position of OX', OV. 18. The condition that a line shall be a principal axis at some point of its length is, that taking the line as axis of s and the point as origin, the relations 'Ej;n's = o, lmys = o shall be satisfied. It is not true, however, that if a line be a principal axis at one point of its length, it will be a principal axis at any other, or at all points of its length. For example, in Fig. 1 1, the line OX is a principal axis at the point ^ on account of the sym- metry of the cone, but it is not a principal axis at the point O. Similarly, in a hemisphere, any diameter of the base is a prin- cipal axis at the centre of the base, but not at a point on the rim. There is one case, however, in which a line is a principal axis throughout its length, and as this is of some importance, the following statement and simple proof are given. 19. If a line be a principal axis at the centre of inertia, it will be a principal axis at every point of its length. H RIC.IU DYNAMICS. Let a portion of the body be represented in Fip;. 14, O being the centre of inertia, 00' the principal axis at the centre of inertia, OA", OY any rectani^ular axes at O, perpendicular to 00\ and 0X\ OV parallel axes through O'. Then we have, by a previous proposition : Imx'a' 3X.0' = t}nx:::xt 0-\-M{/ix), and tmy's' at O'^^tmy::: at (9 + M{hy). But x=y = o = 'lmxa = 'ljiiy:;, by hypothesis. .•, at (9' Smxy = o = -iny'c\ and therefore 00' is a principal axis at O', and therefore also at any point in its length. Conversely, it may be shown that if a line be a principal axis at aP. points in its length, it must pass through the centre of inertia. 1 1 bcinpj trc of lar to ELLIPSOIDS OF INKKTIA AND PRINCIPAL AXES. 2$ 20. To dctorminc the locus of points at which the momcntal elhpsoid for a given body de;;enerate.s to a .spheroid, and the points, if such exist, at which it becomes a sphere. Let the body be referred to its principal axes at the centre of inertia, and let A, />*, and C be its principal moments, and J/ its mass : — (i) If all three moments be unequal, say A > B > C, there will be no point at which the momental ellipsoid for that body will be a sphere, but at any point P on the ellipse ■»*JL_L '■' -J_ -n A-C^B-C~M' ' ' or on the hyperbola. A-B B~C J/' j = o. ■e also n that ; must it will be a spheroid with axes of revolution touching the conic at P. The momental ellipsoid at all other points will have three unequal axes. (2) If two of the moments be equal, and each less than the third, say A >B=C, there will be two points at which the mo- mental ellipsoid for that body will be a sphere, viz., the points 'A-C on the axis of .r, distant ± ve M from the centre of inertia. At every other point on the axis of x, the momental ellipsoid will be a spheroid with the axis of x as axis of revolution. At all points not on the axis of x the momental ellipsoid will have three unequal axes. (3) If two of the moments be equal, and each greater than the third, sa.y A = B>C, the momental ellipsoid for that body will be a spheroid at every point on the axis of c, or on the circle, .H/=^^, ^=0. At all other points it will have three unequal axes. 26 KIGID DYNAMICS. (4) It /] = /)'= 6^, the: momcntiil ellipsoid will be a sphere at the centre of inertia and a spheroid at every other point. From the above it is seen at once that in the majority of bodies there is no point for which all axes drawn through it are principal axes. Illustrative Examples. I. To find the principal axes of a triangular lamina, at an angular point. One principal axis is the line drawn through the angular point perpendicular to the lamina, and the other two are found in the following way. In Fig. 15, let OA, OB be two rectan- gular axes, OX, OY the principal axes in the plane of the lamina. Then the angle which OX makes with OA will be 2 F given by the formula tan 2 ^= ^ —,» where Fig. 15. ^ = moment of inertia about OA i ' ,.„(« *) =JoU" /'sin'^y^'''^v/j, -Jl and KLLIi'SUlDS OK INERTIA AND PRINCIPAL AXLS. 27 /?= moment of inertia about t?Z? » \^ y^ /3 s I n (0 (.1- +_;/ cos (afdxdy, [ y^ fJsina)(.r+jcosa))>'sin W.tv/;/, p being: the density of the lamina, and the axes of x and y lying along the sides of the triangle. It will be found, on evaluating these integrals, that tan J ^ - ^ ''^'" *" ^'^ '^"^ ^"^ "^ d^ + r?/^ cos tu 4- b^ cos 2 w As a simple case, let w = ^; then the triangle is right angled, and tan 2 ^ = -^— _, as can easily be found independently of the above formula. 2. To find the principal axes at any point of an elliptic lamina. In Fig. 16, let O' be the point (a, /S) at which we require the "! ^rl »n Fig. 16. 28 RIGID DYNAMICS. ; i I principal axes. Then the angle 6 which O'X' makes with the principal axis at O' is given by tan 2 0- 2F B-A where A = I about 0'X' = I about QX^ M^^ ^ = / about C>' 7'=/ about OY+Ma^ :^g + ,4 and F= tmxy = %7nxy + Ma^ = Ma^. 2Ma^ .'. tan 2 6= ^ 8«/3 (rt2-^2)+4(«2_^2)- The third principal axis is, of course, at right angles to the lamina, through the point O'. 3. To find at what point a side of a triangle is a principal axis. Fig. 17 shows the construction and proof. BC is the side in question, and is bisected at B. AD is drawn perpendicular to BC, and DE is bisected at O. Then, taking the equimomental svif-em — at the middle points of the sides, in order that the . . 3 inertia-pri:)duct F may vanish, the principal axis perpendicu- lar to the side BC must bi.sect the join of the mid-points of the sides AB and AC, hence BC, OY are the principal axes in the plane of the lamina at the point O. ELLIPSOIDS OF INERTIA AND PRINCIPAL AXES. 29 4. Find the principal axes at any point of a square or a rec- tangular plate. 5. Find the principal axes at any point within a cube or a rectangular parallelopiped. 6. The principal axes at any point on the edge of a hemi- sphere are, one touching the circumference of its base, and two others, given by the relation tan 2 ^ = |. 7. The principal axes at any point on the edge of a ri — — — r: — ^ dfi dfi)_ dh , dlv dfi df' 7)l[ = N, (B) V dfi -^ dfl)_ where L, M, iV^are the couples produced by the external forces. 25. It may be stated b-re that D'Alembert's principle holds also in the case of a system of bodies moving under their mutual actions and reactions, and applies to the motion of liqp.ids. It is a direct consequence of Newton's Third Law of Motion. 26. Deductions from D'Alembert's Principle. Taking any one of the equations (A), we have But, by definition of the centre of inertia, '^mx=Mx, and 2;/.^ = J/^'-^' dt"^ '" df^ Therefore the above relation becomes M dt^ and similarly for the other two. (i) Hence, the motion of the centre of gravity of a sy stein under the action of any forces is the same as if all the mass tvere col- lected at the centre of inertia and all the forces were applied there parallel to their former direction. And so the problem of finding the motion of the centre of inertia of a system, however complex, is reduced to finding that of a single particle. ill 'Ak 4 v:\\ 'i'f ■if •■( \k 34 RIGID DYNAMICS. Moreover, taking- one of the equations (B), siii^c we may choose the origin of coordinates at any point, let it be :.D chosen that at the time of forming these equations the centre of inertia is coincident with it, but moving with a certain velocity and acceleration. Then, evidently, we must obtain a relation of the same form as the foregoing, just as if we had considered the centre of inertia as a fixed point. In other words, such a relation as the above will hold at each instant of the body's motion, independently of the origin and of the posi- tion of the body. (2) Hence, the motion of a body, under the action of any finite forces, about its centre of inertia, is the came as if the centre of itiertia were fixed and the same forces tvere acting on the body. 27. The two previous deductions are known as the principles of the Conservation of the motions of Translation and Rotation, and show us that we may consider the two motions indepen- dently of one another. 28. Impulsive Equations of Motion. Since an impulse can be measured only by the change of momentum it induces in a body, in applying D'Alembert's Prin- ciple to impulsive forces we must alter the expressions for the effective forces, which will be represented not by the products of masses and accelerations, but by the products of masses and changes of velocity. All the preceding relations will hold equally for impulsive forces if we then write changes of velocity for accelerations. Thus, such a relation as 2w — '- = "^mX, dt^ \ D'ALEMBERT'S PRINCIPLE. 35 for finite force: will become 2;;/ fch\ Kit) ~ dx dt = 1X, for impulsive forces where the velocity of each particle of m-^ss m is changed from -^- abruptly to ( -^ dt ^ \dt by the action of an impulse X. And it may be said, generally, that equations of motion for impulsive forces can be obtained from the corre- sponding equations for finite forces by substituting in the latter changes of velocities for accelerations. 29. In forming any relations for impulses, it must be borne in mind that all finite actions, such as that of gravity, are to be neglected ; after the impulse has acted, the subsequent motion will, of course, be found by applying the equations for the finite forces which usually are called into play after the impulse has operated. ''1 I' , Illustrative Examples. I. A rough uniform board, of length 2 a and mass rn, rests on a smooth horizontal plane. A man of mass M walks from one end to the other. Determine the motion. This example furnishes an excellent illustration of the truth of D'Alembert's principle, which asserts that the motion of the centre of inertia of the system will be the same as if we applied there all the forces external to the system, each acting in its proper direction. All the forces at the centre of inertia are then downwards, and as the centre of inertia cannot move downwards, it must therefore be at rest ; and as the man walks along the whole board, he will therefore advance relatively to the fixed horizontal plane through a distance board will recede through a distance ^^ 2 ma , and the ',\ * ill 36 RIGID DYNAMICS. Analytically, we have for the motion in a horizontal direction, since there are no horizontal forces external to the system, the equation 2;/^--— =0. -f- and (it = o or constant. If the man and board start from rest, as we have supposed, then ^-_ dt' = o. ,". .*•= constant. which means that the position of the cen're of inertia remains unaltered throughout the motion of the two parts of the system. 2. Two persons, A and B, are situated on a smooth horizon- tal plane at a distance a from each other. If A throws a ball to B, which reaches B after a time t, show that A will begin to slide along the plane with a velocity — -, where M is his own mass and in that of the ball. 3. A person is placed on a perfectly smooth surface, may he get off } How 4. Explain how a person sitting on a chair is able to move the chair along the ground by a series of jerks without touching the ground with his feet. 5. How is a person able to increase his amplitude in swing- ing without touching the ground with his feet .-' ;ction, n, the )osed, lains :em, izon- ball n to own D'ALK.MI'.KKT'S I'RINCII'LE. 37 6. Explain dynamically the method of high jumping with a pole; and show that a man should be able to jump as far on a horizontal plane without a pole as with one. 7. Two coins, a large and a small one, arc spun together on an ordinary table about an axis nearly vertical. Which will come to rest first, and why.'' 8. A circular board is placed on a smooth horizontal plane, and a dog runs with uniform speed around on the board close to its edge. Find the motion of the centn^ f the board. 30. TJic Principle of Energy. Before entering upon the discussion of the motion of a rigid body, what is known as the principle of energy will be explained, as it is exceedingly useful, and often gives a partial solution of a problem without any reference to the equations of motion, and in many cases furnishes solutions which are both simple and elegant when compared with those obtained by the use of Cartesian coordinates. If a single particle of mass ;;/ be moving along the axis of x, under the action of a force F in the sa. e direction, we have, as the equation of motion, iPx ^ I 'I 'Mi '■'I 1 'A low lOve Ung And multiplying both sides by ~ and integrating, we get dx where V is the initial value of v or dt ng- The expression on the left-hand side of the equation is the change in kinetic energy, which is equal to the zvork done by the force from o to x. 38 kldlD DYNAMICS. !l What is true of :i sinj^le force acting in a definite direction and of a single jxirticl^' of mass m is also true of a number of forces acting on a rigid body or on a system. Then the analytical expression for the work done by a system of forces becomes ^;;/J \Xiix + ) 'dy + Zih), which must be equal to In the general case, where bodies move with both translation and rotation, the total kinetic energy can easily be shown to be that due to translation of the whole mass collected at the centre of inertia, together with that due to rotation about the centre of inertia considered as a fixed point. For if Xf y, a be the coordinates of any particle of mass m and velocity v at time t, and x, y, 1: be the codrdinates of the centre of inertia, |, ?;, ^ the coordinates of the particle referred to the centre of inertia, then the total kinetic energy is equal to ^^vriP' =\^in ,A2 1(f)- ^^-m \ = J5;. dt] \dtJ '\ (f)i--' -'"Id + IJHfJI' since by definition of the centre of inertia the other terms disappear. This proves the proposition. 31. According to the kind of motion and the choice of coor- dinates and origin, this expression for energy will assume various forms which will be given under the discussions of the special cases throughout the treatise. Twice the energy is termed the vis viva. D'ALEMMERT'S PRINCIFLE. 39 32. To find the work clone by an impulse; let Q be the measure of an impulse which, acting on a particle of mass ;// moving with velocity F, changes its velocity suddenly to v ; then the kinetic energy is changed from ,] m V^ to \ mvK Work done by the impulse = \Q'{v+V), since the impulse is measured by the change of momentum and Q is therefore equal to mv — m V. A similar relation will evidently apply to a rigid body where V and V are the velocities of the point of application of the impulse resolved in the direction of the action of the impulse. Illustmth'c Examples on Energy. r. A rod OA, of length 2.7, fixed ;'t (9, drops from a horizon- tal position under the action of gravity : find its angular velocity when it is in the vertical position OB. (See Fig. 18.) o a. ,111 1 1 '■ii + ■.oi .1^ B Fig. 18. 40 lUC.lD DYNAMICS. Here, the work done by gravity in moving the rod from the position OA to OB is Jlf^'-a, M being the mass of the rod. The rod starts from rest in the position OA, so that when in the position OB the change in kinetic energy is measured simply by the energy in the position OB. This kinetic energy is equal to the expression .] ^)m>^, 2> being the velocity of any particle m ; and the linear velocity of any particle in OB is tor, where ft) is the angular velocity and r the distance of the particle from O. Hence and Stnr^, the moment of inertia of the rod about O, is Mz — ; 4a^ .-. ft,2=3_C 2a which gives the angular velocity of OB. This example may serve also to show the independence of the motions of translation and rotation ; for, taking the expres- sion just found, this may be put in the form Mga = } M{a ; and the two energies t.iken together would be the equivalent of the work tlone, or of Jlj.nt. 2. A uniform stick of length 2 a hangs freely by one end, the other end being close to the ground. An angular velocity in a vertical plane is then communicated to the stick, anil when it has risen through an angle of 90', the end by which it was hanging is loosed. What must the initial angular velocity be so that on falling to the ground it may pitch in an upright position ? Figure 19 shows three positions of the stick. It starts with an angular velocity &>, communicated to it in some way, and reaches its second position with an angular velocity to', such that 1 (O" 2 a (^^) a relation which may be obtained at once from the principle of energy. Then, the stick being freed, the centre of inertia has a ; from ion of posed lue to «/ cv I I A I I I 4- ■ • t' -Si Fig. 19. ii :'!l i 42 RIGID DYNAMICS. motion of translation upwards represented by aco', and at the same time the stick keeps on rotating about the centre of inertia. Owing to the action of gravity, the motion of translation ceases, alters in cUrection, and finally the stick drops to the ground in an upright position. The time it takes the centre of inertia to move from its second position to its final position when the stick pitches upright is found from the well-known formula for space described under the action of gravity, which, in this case, becomes a= —aoi'- 1+\ (^fi. {b) The condition for pitching upright is evidently to be found from the condition that the rod after leaving position (2) must 7r rotate through (2//+ i)— before touching the ground, and there- fore w / = (2«+l) rr if) (a), {b), and {c) give the result „2=^ (-7^) where TT /=(2«+i)-. 2 3. A uniform heavy board hangs in a horizontal position suspended by two equal parallel strings fastened to the ends. If given a twist about a vortical axis, pro"e that it will rise a'-o) through a distance — — , where 2 « is the length of the board, and o) the vertical twist. 6^'- 4. A cannon rests on a rough horizontal plane, and is fired with such a charge that the relative velocity of the ball and cannon at the moment when the ball leaves the cannon is V. If iM be the mass of the cannon, in that of the ball, and fi the coefficient of friction, show that the cannon will recoil a dis- tance f-^ \M-{-in 2^lg on the plane. I; D'ALEMBERT'S PRINCIPLE. 43 (^) , and fired and is V. the I dis- 5. A fine string is wound around a heavy grooved circular plate, and the free end being fixed, the plate is allowed to fall freely. Find the space described in any time. 6. A coin is spun about an axis nearly vertical upon an ordi- nary table. Form the equation of energy at any time as the coin descends to its position of rest. 7. A narrow, smooth, semicircular tube is fixed in a vertical plane, the vertex being at the highest point ; and a heavy flexi- ble string, passing through it, hangs at rest. If the string be cut at one of the ends of the tube, to find the velocity which the longer portion will have attained when it is just leaving the tube. If a be the radius of the tube, / the length of the longer por- tion, then, on equating the kinetic energy at the time the string is leaving the tube to the work done by gravity up to that time, it will be found that the required velocity is given by the relation '=^rt|2 7r-^(7r2-4)|. 8. Explain why the grooving in a rifle barrel diminishes the force of recoil. 9. A rough wooden top in the form of a cone can '•otate about its axis, which is fixed and horizontal. A fine string is fastened at the apex, and wound around it until the top is com- pletely covered. A small weight attached to the free end is allowed to fall freely under the action of gravity, unwinding the string from the top which rotates about its axis. Find the angular velocity of the top when the string is completely unwound; also, the equation of the path of the descending weight. 10. Two equal perfectly rough spheres are placed in unstable equilibrium, one on top of the other ; the lower sphere resting on a perfectly smooth horizontal surface. If the slightest ' 1 I ;:i It 44 RIGID DYNAMICS. disturbance be given to the system, show that the spheres will continue to touch each other at the same point, and form the equation of energy at any time. Figure 20 shows the solution of this problem. D'Alembert's principle asserts that the centre of inertia must descend in a straight line, since the only external force is gravity. ii ^'''" ~ ~''»v^ ^ ■V • \ / N / \ / \ 1 \ 1 \ 1 \ \ \ 1 1 1 \ ^^"^"^ /^^X. \ ^r 1 ^V \ / 1 ^V \ . \ \ , y \ N 1 / V •V I X I ■^ -'"^'^ 1 ■^ <-' "^ ""^^ ^^ 1 Nv \^^^^ 1 / / 3 y<^\ 1 / ^ V \ / 1 jb: M^ Fig. 24. If //be the horizontal intensity of the earth's magnetism, m the magnetic strength of either pole, and 2 / the length of the needle, then it is evident, from Fig. 24, that when the needle makes an angle with the magnetic meridian, we have MJc^ df -Hvi 2 /sin 6?= -HM' sin 6, where Mk"^ is the moment of inertia about the axis of rotation, and Al' is the magnetic moment. Hence, for small oscillations, iPe HM' dfi I e=o, and the time of a small oscillation is Tryj I HM' 5. A circular wire carrying a current and freely suspended, as in Ampere's experiment, places itself at right angles to the magnetic meridian. If slightly disturbed, find the time of an oscillation. V. a solenoid be used, find also the time of a small oscillation. 6. Find the equation of motion of a metronome and the time of a small oscillation. MOTION ABOUT A FIXED AXIS. FINITK FORCES. 53 7. Find the least axis of oscillation for a sphere and an ellipsoid. 8. A right circular cone makes small oscillations about a diameter of its base. Find the time of one of these oscillations, a being the altitude, and b the radius of the base. Find, also, the least axis of oscillation when a — 2 b. 9. A helix of wire, with the ends bent inwards and ending; on the axis, is fastened at the upper end, and on being pulled slightly by the lower end vertically downwards, and then freed, oscillates under gravity. Find the time of an oscillation. 10. A uniform beam rests with one end on a smooth hori- zontal table, and. has the other end attached to a fixed point by means of a string: of length /. Show that the time of a small 7 g oscillation in . vertical plane will be 2 7r\/- II. A sphere rests on a rough horizontal plane with half its weight supported by an extensible string attached to the high- est point, whose extended length is equal to the diameter of the sphere. Show that the time of a small oscillation of the sphere parallel to a vertical plane is 2 7r'\/-^, a being the radius of the sphere. ^^ 12. A uniform beam of length 2^ is suspended by two equal parallel strings, each of length b, fastened at the ends, and attached to fixed points in the same horizontal line. Show that if given a slight twist about a vertical central axis it will ^3^ make small oscillations in time 2 tt^ 39. Determination of g by the Pcndnlmn. If a pendulum of any form be allowed to make small oscilla- tions under the action of gravity, wo have the time of a com- iff ' ii •' ■ i 54 RIGID DYNAMICS. plctc oscillation given by the relation / = 2 7r'Y . where / is the A'' /i^4-P length of the equivalent simple pendulum and equal to — y — If, now, / be observed by means of a clock, and // and /- be found, we have the value of ^ given. This method is one of the most accurate known for finding the intensity of the earth's attraction at different points on its surface. Various forms have been given to these pendulums, from time to time, in order to ensure accuracy of measurement ; and the most important of those which have been used for the scientific determination of gravity are described below. (a) Berth's Pendulum. Borda (1792) constructed his pendulum so as to realize as nearly as possible the simple pendulum. It was made of a sphere of known radius, equal to a. To render it very heavy it was composed of platinum and was suspended by a very fine wire about twelve feet in length. The knife edge which carried the wire and sphere was so arranged by means of a movable screw as to oscillate in the same time as the complete pendulum. The time was determined by the method of coincidences^ and g was found from the relation t=2lT where / is the length from the knife edge to the centre of the sphere, a the radius of the sphere, and a half the angle through which the pendulum swings at each oscillation to or fro. (b) Katers Pendulum. In 18 1 8, Captain Kater determined the value of gravity at London by applying to the pendulum the principle discovered by Huyghens, that the centres of suspension and oscillation are reversible. He made a pendulum of a bar of brass about an MOTION AIJOUT A I'lXHD AXIS. FIMTK KOUCKS. 55 inch and a half wide and an LM;;hth of an inch in thiciZ 34 38 40 41 43 44 45 48 50 51 51 51 52 54 55 55 57 59 59 60 I'S. 24 N. 32 N. 27 S. 55 S- 29 N. 38 N. 46 N. 4 N. 56 S. 56 N. iZ N. 55 S- 2 S. 54 S. 28 N. 44 N. 41 N. 7 N. 50 N. 24 N. 50 N. 37 N. 28 N. 28 N. 31 N. 30 N. 46 S. 51 s. 58 N. 3 N. 46 N. 56 N. 45 N. /. 99.0966 99.1134 99.1019 99.0948 99.1217 99.1104 99.1091 99.1227 99.1168 99.1581 99.1497 99.1712 99.1712 99.2500 99.2641 99-3097 99.3191 99.3190 99.3402 99-3470 99-3623 99.3858 99.4042 99.4169 99-4143 99,4140 99-4235 99.4501 99-4565 99-4550 99.4621 99.4854 99.4876 99-4959 Okskrver. Freycinet Sabine Hall Foster Sabine Basevi and Heaviside Basevi and Heaviside Sabine Basevi and Heaviside Liitke Foster Biot Duperrey Biot Biot Kater Foster Foster Lutkc Sawitsch MOTION ABOUT A FIXED AXIS. FINITE FORCES. 57 :s of ding the ition may be used, where / is the length of the seconds }iendulum in centimetres. See also Geodesy, by Colonel A. R. Clarke, Chap. XIV. The places are arranged geographically in order of their lat;- t tides, and show thereby the gradual increase in the length of the seconds pendulum as we go from the equator to the pole. Those places, in the preceding table, for which the lengths of the seconds pendulum have been calculated from a number of observations made by different observers, are indicated by a dash. iide iide iide 41. During the past few years several observers hav? made observations on the value of g at different points in North America. Professor Mendenhall, of the U. S. Coast Survey, during the summer of 1891, visited a number of places on the Pacific coast between San Francisco and the coast of Alaska, and in his report of the expedition gives a table of the values determined, with the places and corresponding latitudes. He made use of a half-seconds pendulum enclosed in an air-tight chamber which could be exhausted with an air pump. A spec- ial method was used for noting the coincidences (see U. S. Coast and Geodetic Sun>ey. Report for 1891, Part 2). Defforges, one of the greatest living authorities on methods of gravity determination, crossed from Washington to San Francisco during the summer of 1893 and made a number of observations which are given in the following table. The value of g alone is given. Washington , 980.169 Montreal 980.747 Chicago 980.375 Denver 980.983 Salt Lake City 980.050 Mt. Hamilton 979.916 San Francisco 980.037 These are all reduced to sea level. *1 58 RIGID DYNAMICS. ll t'l :- 42. Experimental Determination of a Moment of Inertia. In many cases of small oscillations under gravity, where it is difficult to calculate the moment of inertia of a body from its elements, the time of oscillation is observed ; and, the moment of inertia being increased by the addition of a mass of definite figure, the time of oscillation is again noted. The required moment of inertia may then be calculated. This method is particularly useful in the case of magnetic oscillations about a vertical axis. lli, J' I?' 11' s 'I 43. Pressure on the Fixed Axis, rical. Forces and Body Symmet- If a body be moving about an axis, and it is symmetrical with respect to a plane passing through the centre of inertia and perpendicular to the axis, and at the same time the forces acting on the body are also symmetrical with respect to this plane, then we may suppose that the pressures on the axis are reduci- ble to a single one which will lie in the plane of symmetry and will cut the axis of rotation. To determine, in such case, the direction and magnitude of the resultant pressure, we proceed in the following way. Let the body, Fig. 25, surround the point O and let it be symmetrical with respect to the plane of the paper which con- tains C, the centre of inertia : the axis of rotation being perpen- dicular to the plane of the paper, and passing through O. Let the forces acting on the body also be symmetrical with reference to this plane. And let the body, moving about the axis through O, be situated at any time t as represented, Q being the angle which the line OC fixed in the body and moving with it makes with the line OA fixed in space. Then the resultant pressure on the axis will be in the plane of the paper, and its direction will pass through O. Let its components measured along two rectangular axes OX, 0Y\\\ the body, be /'and Q. Let CO = Ji. Then, X, V, being the accelerations on unit mass in the MOTION ABOUT A FIXED AXIS. FINITE FORCES. 59 I -! directions OX, OY, we have, by D'Alembert's principle, the relations Fig. 25. Fig. 26. Now, if Q) be the angular velocity, any particle such as ;;/ will be acted on by the forces wwV, mwr, as is indicated in the figure; and these forces resolved along OX, OY, as shown in Fig. 26, would give in -7- = — mw^x — niwy, 'ib' 2 , • tn —f- — — viwy + inoix. dr The values of ~, ^---^ may also be obtained by direct differ- entiation from .r=rcos^, j = rsin^. Thus, ^'V • .d9 dy ^de ~-^- — r cos 6 . =,i-ft). - .- = — ^' Sin 6 — z= -yco, dt dt -^ ' dt dt \ I ■ ii H l! M \ 60 RIGID DYNAMICS. Hence, our relations for determining the pressi'"es become P + '^mX-\- l.m{oP'x + wj') = o, Q + 2;« F+ 2;;/ (tu'-^j — iox) = O. . '. /-' = — 2;;/ A'— 2 w (tu^.r + &>J')> Q= —"^jii Y— 2 w (w^j — w.i'). But, by definition of the centre of inertia, 2wa)^.r=ft)22 'nx=MJioiP', '^vmy = io^niy = 0, ^mvP'y = (jiP'^my = o, '^inwx — io^inx= Mhto. .: P=-^mX-M]m\ Q=-'LmY+Mhio, which equations determine the pressures P, Q, and therefore the direction and magnitude of the resultant pressure when we know q>, which is found from the relation already given, . d^0 N ft) = — ~ = where A^ is the moment of the external forces about the rota- tion axis, and '2nir^ is the moment of inertia about the same axis. This, on integration, gives Af P -f .P' = - 2;;/ («' - ft))j/ = - (w' - a))iW>, 2 F+ S + g' = 2;/^ («' - ft)),r= (o)' - (o)M~x, '1Z + R + R'=0, L + C-^ + C<2^= — S w.c(&)' — (o)x= — (o)' — ft)) ^nixs, M+ C^ + C2' = 2;«^(ft)' -(w)j= (ft)' -ft))Swj.?, 7V= (ft)' — ft)) • "^Mf^. 48. If the body starts from rest, then co = o, and the sudden angular velocity generated by an impulse which tends to turn a body about a fixed rotation axis is obtained from the relation '1' n 'f •'] ft) )lacc, is where N is the moment of the impulse about the axis, and ^m)'^ is the moment of inertia. As before, the problem is sim- plified by choosing the origin at a point where the rotation axis is a principal axis. I ! 'mi i 72 RIGID DYNAMICS. 49. Centre of Percussion. In the general equations just found, let us suppose that the impulsive actions ?re those caused by a blow Q represented by components X, F, Z\ and that the blow is struck at some point on the surface of a body, capable of motion about a fixed axis, which either passes through it or to which it is rigidly con- nected. What is the condition that there shall be no impulsive pressure on the axis ? Or, in other words, is it possible to strike the body at a certain point in such a way as to produce Fig. 32. no strain upon the axis about which it is free to rotate } Let the body (Fig. 32) surround 0\ let ZZ^ be the axis of rota- tion, and let the plane of zx, which is the plane of the paper, contain G, the centre of inertia of the body. Suppose that the blow Q is applied at the point whose coordinates are ^, ?;, ^ (the coordinate r\ not being shown, being drawn upwards perpen- dicular to the plane of the paper). If there be no resulting b P' C( Let rota- aper, It the ^(the rpen- ilting MOTION ABOUT A FIXED AXIS. IMPULSIVE FORCES. 73 pressure when the body is struck, the general relations become : X=o, Y={(o'-co)Mv, Z=o, L = qZ— ^ V= — {(!)' — oi)^mx3y M= ^X- ^Z= - {co'-o})'2mj'::, N=^V-r]X= {(o' -(o)^m>'^={oi' -(o)Mk\ where k is the radius of gyration about the axi.s. From these it will be seen that, since X=o, Z=o, we have also 2wjxr=o. And also, ^ Y= («o' — Q))^mxc, Y=:{(0'-0))MX. -, _ 2 ;;/xc _ 2 jhx::: Mx "Lmx And f is given by the last relation, t^ jco'- o>)MP ^ {(o'- co)MP ^ P V {a)'-(o)Mx X The above conditions holding, and there being no pressure on the axis, the line of the blow is called a Line of Percussion, and any point in this line is termed a Centre of Percussion. 50. By an inspection of the foregoing relations, we have, I. A'=o, Z—0\ and therefore one condition, that there may be no strain upon the axis, is that the line of the blow must be perpendicular to the plane containing the rotation axis and the centre of inertia. ;;i:i. n I 74 RIGID DYNAMICS. 2. 2;;y^ = o, and S;;/;r^ = ^' 2;«;r. Now, since O may be chosen anywhere on the axis, let it be so chosen that ^=o. Then for that origin so chosen ^niyz would be zero, and ^mxs also zero. Therefore, an essential condition, to be first satisfied for a line of percussion, is that the axis of rotation must be a prin- cipal axis at some point of its length. 3. f =--> which shows that when a centre of percussion does exist, its distance from the axis is the same as that of the centre of oscillation. If ^=0 and 3=0, then the line of percussion passes through the centre of oscillation, which may be stated in the following way : // tJic fixed axis be parallel to a principal axis at the centre of inertia, the line of action of the bloiv will pass through the centre of oscillation. Illustrative Examples. I. A uniform rod, fixed at one end and capable of motion in a vertical plane, is hanging freely under .i. action of gravity, and being struck perpendicular to it^ length, rises into the position of unstable equilibrium. Find the magnitude of the blow that there may be no strain nt the fixed point. In order that there may be no strain on the axis, it must be struck at the centre of percussion, which point will be at a distance 4.- from the fixed end, if the length of the rod be 2 a. 3 Then, if w be the angular velocity produced by the impulse, we ha^c from the equation of moments. 3 \,vir (O. B=Maw. MOTION ABOUT A FIXED AXIS. IMPULSIVE FORCES. 75 Also, — = — --'^ sin 6 is the equation of motion of the rod as it rises upwards, being acted upon by gravity, and starting with an angular velocity «. «/(o 2il*^^ 0) 2_ 3^, a .'. B = LI aw = J/V 3 ga. From this it may be seen that generally whe:^ a body is struck at the centre of percussion, the value of the impulse is measured by the product of the mass and the velocity of the centre of inertia. 2. A cu'cular plate free to move about a horizontal tangent is stuck at its centre of percussion and rises into a horizontal position. Find the blow. As before, and B^Maoi, a being the radius, d(iy AiT • a — ^-^ sm 6 gives o). dt 5^ ^ 5 3. A sector of a circle, whose radius is a and ingle «, is capable of turning about an axis in its plane which is perpen- dicular to one of its bounding radii. Find the coordinates of the centre of percussion. f'^'g- 33 shows the position of the centre of percussion C, whose coordinates are ^mx 2.1HX- 2,7/ /X > »>-Vll u v.\^'^,",. |! il 76 RIGID DYNAMICS. On transforming to polar coordinates it will be found that ^= \a sin «, f=^^b ( — 1- cos « ). Vsin / Fig. 33. 4. To find the centre of percussion of a triangular plate capable of rotation about a side. Fig. 34. MOTION ABOUT A FIXED AXIS. IMPULSIVE FORCES. 77 Fig. 34 shows the position of the centre of percussion. AB is the rotation axis, PD perpendicular to AB, E the middle point of AB, F the middle point of DE. Then AB is a prin- cipal axis at the point F, and G being the centre of inertia of the plate, and PD—p, C is the centre of oscillation, C is the centre of percussion, and h p 2 3 When the triangle is isosceles, C and C coincide. 5. ABCD is a quadrilateral (Fig. 35), AB being parallel to CD. Show that, if AB^—iCD^, the point /* is a centre of per- cussion for the rotation axis AB. (Wolstenholme.) ill Fig. 35. 6. A uniform beam capable of motion about one end is in equilibrium. Find at what point a blow must be applied per- pendicular to the beam in order that the impulsive action on the fixed end may be one-third of the blow. 7« RIGID DYNAMICS. 51. Initial Motions. Changes of Constraint. If a body, moving about a fixed axis with known angular velocity, is suddenly freed from its constraint and a new axis fixed in it, or if a body at rest is disturbed so that there is a sudden impulsive change of pressure, we can determine the new angular velocities and changes of pressure by reference to the impulsive equations of motion already found. Sometimes, however, solutions which are more instructive may be obtained by cc idering elementary principles ; and the following exam- ples are given to illustrate the methods to be employed in various cases. II !!> Illustrative Examples. I. A uniform board is placed on two props; if one be sud- denly removed, find the sudden change in pressure at the other. Fig. 36 illustrates the problem. The board is of length 2 a, and rests on the props A and B, which are fixed in position in ['I a, ^ Cty M^ M B Fig. 36. JR' A My W i I the first figure, so that R = \Mg. If B be now removed, the board begins to turn about the upper end of A under the action of gravity, and to each element of the board an acceleration wr is given suddenly ; so that if we communicated to each element 7n an acceleration ar in the opposite direction (upwards), we MOTION ABOUT A FIXED AXIS. IMPULSIVE FORCES. 79 would have, by the p,pplication of D'Alembert's principle, A", 2L(;;/aj;'), and Mg in equilibrium with one another, as indicated in the second figure. Also, taking moments about O, just when the prop is removed, we have '^{mcor) • r=Mg- a. (0 = 3f 4 a .: R' = Mg - Maco = \I\fg. 2. The extremities of a heavy rod are attached by cords of equal length to a horizontal beam, the cords making an angle of 30° with the beam. If one of the cords be cut, show that the initial tension of the other is two-sevenths of the weight of the rod. 3. A uniform rod is suspended in a horizontal position by means of two strings which are attached to the ends of the rod. If one of these strings be suddenly cut, find the sudden change in tension of the other string. 4. Two strings of equal length have each an extremity tied to a weight C, and their other extremities tied to two points A, B in the same horizontal line. If one be cut, the tension of ACB the other is instantaneously altered in the ratio i :2 cos^ . 5. A particle is suspended by three equal strings of length a from three points forming an equilateral triangle of side 2 ^ in a horizontal plane. If one string be cut, the tension of each of the others is instantaneously changed in the ratio ^- —^ — ^ "" 2{d'-0'') 80 RIGID DYNAMICS. r ! 6. A rod of length 2 a falls from a vertical position, being capable of motion about one end in a vertical plane, and when in a horizontal position, strikes a fixed obstacle at a given dis- tance from the end. Find the magnitude of the impulse, and the pressure on the fixed end. Fig. 37. Let the rod (Fig. 2i7) <^^^op from the vertical position and strike an obstacle when in the position OB with a blow Q. Let R be the impulse on the fixed end O, and then we have, taking moments about O, Q- 3 ■ — 7/ — ' and since the rod falls from the vertical position, its angular velocity when in the horizontal position is found in the usual way to be given by 2a ■• ^= d-^ 3 ' The impulsive pressure on the fixed end is obtained from the relation Q = R + ^ iinro)) =R-^ Maw. MUTIUN AliOUT A FIXED AXIS. IMl'ULSIVE FORCES. 81 ^ 2 = < 3 ^, or as Q. 4a Hence, if the obstacle is beyond the centre of percussion, the impulsive strain at O is downwards. If at the centre of per- cussion there is no impulsive action on the a.\is, and when d * ■<^ >> ^ 9) I W/jt o^ :) ' 82 RIGID DYfJAMICS. 7. A rod is moving with uniform angular velocity aboat one end fixed ; suddenly this end is freed and the other end fixed. Find the new angular velocity. O O' T .2L. O /' O' Fig. 38. 1 . Fig. 38 indicates the solution. In the first figure each particle has a linear velocity wr in the direction indicated, on account of the angular velocity w. In the second figure both ends are free, and the velocities remain as before. In the third figure O' is instantaneously fixed, which does not affect the velocities of tne other elements of the rod, by the definition of an impulse. And hence w', the new angular velocity about 0\ will be as shown in the figure in direction, and its magnitude will be found by using the formula for moment of momentum. Thus And if .v + r=a, and p is the density, . •. 60 I p(a —x)x(Lv=M • — G)', */o 3 . '. to' = ), (O. 8. A rod of length a is moving about one end fixed with uni- form angular velocity, when sud(lenly this end is freed, and a point distant / from it is fixed. What in general will be the direction and magnitude of the new angular velocity .-* This is an extension of the preceding i^roblcm, and the method of solution will be similar. Let O (Fig. 39) be the first point fixed, and the angular velocity be co, as indicated. Then this point being freed, let the second point O' be fixed. MOTION ABOUT A FIXED AXIS. IMPULSIVE FORCES 8^ The new angular velocity will be obtained by equating the moments of momentum before and after the fixing of the point 0'. Thus p \ (ox{l-x) . dx-p f a)x{/+x) ■ (ir=AU'^ (about O') x &>'. For, the linear velocity of an element at P is {/-x)(o before O' is fixed, and its moment of momentum about O' will there- A A jr 71 I « O' JP o \ Fig. 39. fore be vi{l—x)(xi'X\ while the moment of momentum of an element at Q will be vi{l-\-x)w • x in an opposite direction to the former with reference to the point 0\ If p be the density and a the length of the rod, we then get the above relation which determines the sign and value of 6)'. It will be found on integrating the above expressions that w' will have the same sign as co, the opposite sign, or will be zero, according as which shows that if a rod be moving about an axis, and this axis be freed and a new axis fixed through the centre of percus- sion, it will be reduced to rest. I' I' HI 9. An elliptic lamina is rotating with uniform angular velocity about one latus rectum, when suddenly the axis is freed and the other latus rectum fixed ; find the new angular velocity. w = ^—,w. 1+4 e'' 84 RIGID DYNAMICS. l 3 'I; .1 10. A circular plate rotates about an axis through its centre perpendicular to its plane with uniform angular velocity. If this axis be freed, and a point in the circumference of the plate be fixed, find the new angular velocity. Fig. 40 gives the solution. For an element at P the linear ^•elocity is /l=&)„, OB = (o,,. Fig. 42. Then, owing to &),., the point P would be displaced down- wards in an infinitely small time dt, a distance coa-PA/.dt or (o^y sin AOB • dt. Due to Wj its displacement would be upwards (above the plane of the paper) and equal to o) PAWt or Q)iX sin AOB ■ dt. Therefore the total displacement of P is sin A OB{y(i)^—X(o)di, m and this is zero when RICH) DYNAMICS. .r y -t V or = - -• (O,, 0) OA OB ii * ';.! -li which is the equation of the straii;ht Hnc OC. And thus for all points along OC there is no displacement ; that is, the body is turning about OC, due to rotations about OA and OB. That the line OC represents the magnitude of the resultant angular velocity may be shown by considering the displace- ment of the point A. Let w, be the resultant angular velocity about OC. The displacement of A, due to w^, is zero. The displacement of A, due to wj, is OA sin AOP • Jt, and therefore OA^\v\AOC • w,dt = OA sin A OB • oj^df ,.p ii'mAOB .-.^ sm AOC Proposition 3, — ^/ ^ body fixed a! a point have anovular velocities o)^, oiy, oi, coinmnnieated to it about three rectangular axes passing tlirough the fixed point, the resultant angular veloc- ity is given by Also, if a body have an angular velocity w about an instanta- neous axis it may be said to have three angular velocities w,, w^, (1)^ about three rectangular axes ; and if u, /3, 7 be the angles which the instantaneous axis makes with the coordinate axes. then and (Or (O,. w cos « cos j3 cos 7 = &), X ft). y fi>. give the equations of the instantaneous axis when w,, eo^, tu, are known. ^i^ MOTION ABOUT A FIXKD POINT. 9« 55. That a poi»it may have at the same instant three angular velocities can be seen by means of the apparatus shown in Fig. 43- Fig. 4 3. are To an upright stand is attached by means of pivots a system ol two rings and a sphere. The outer ring can rotate about an axis passing through the points ^l, /> ; the second ring may be made to rotate about CD ; and the inner sphere about JiF. Now, the axis AB is initially in a horizontal position, and coincident with the axis of x drawn from O, the centre of the sphere ; and if CD be made coincident with the axis of y by placing the plane of the outer ring in the plane of xj', then it is evident that by turning the inner ring the axis £F may be made initially coincident with the axis of 3. This having been done, rotations may be given first to the sphere, then to the inner ring, and lastly to the outer ring; and thus any point on the sphere will have simultaneously the ij2 KIt.lD IJVNAMICS. f ' ■' » three anj^iilar velocities ^iven to the system, and the sphere will rotate about a result. int axis in space, which would be lixed were there no friction at the pivots ami no resistance of the air. The arrangement also siiovvs how a heavy boily may be fixed at its centre of gravity and at the same time be given rotations about axes fixed in sjjace. 56. Li mar Velocity ixnd Auij^nlar Velocity. In the case of a body moving with one point fixed we may replace the angular velocity co about the instantaneous axis by ft),, w^, (o, about three rectangular axes drawn through the fixed point. The next thing to be done is to connect the expressions for the effective forces with these component angular velocities and the coonlinates of any element of the body, and in order to do this we must obtain an expression for the linear velocities '-^, '-^, ' f of any element at the point dt lit dt ^ ' (.r, y, d) in terms of ,v, y, a, and gj,, w^, oj, ; on differentiating these expressions, we shall then obtain the linear accelerations. We may proceed either geometrically or by direct analysis. y ■< i I. By Gco))ictrical Displacctticnt. Fig. 44 shows how the linear displacements arise from the rotations about the coordinate axes. In the first figure the body is supposed to be fixed at O, and 01 is the instantaneous axis about which the boily is moving with angular velocity w. The body may be supposed to have three rotations &)^, w^, (o, about the three coordinate axes instead of ay about the instantaneous axis. Then, con- sidering positive rotations as those in the direction of the motion of the hands of a watch, and taking the displacements of the point P{x, y, ::) due to a rotation w„ we have, in the second figure, P moving along a small arc PQ in time dt, due MOTION AIIOUT A Kl,\i:i) I'UINr. 03 to ft),, This small (lisplaccnu-nt /'(J is f(|uivaleiU to two J'K, RQ in the directions indicated. Ilciuc \vc have, and Fig. 44. And, by considering the other planes, we should get the dis- placements of P due to ft), and to <>',, thus : Along Ox Oy Oz Displacements due to o), —yw^dt xw^dt Displacements due to tu, —zoa^dt y/odt Displacements due to w^ zw^dt -xw^dt These are written down symmetrically ; and from them we see that the linear displacement along Ox, which we call dx, is I < !| i^ H J, I 94 RIGID DYNAMICS. equal to {sot^—yoi^.dt, and, therefore, in the limit the linear velocity dt = z(o^-y(0,. dy dt^'^'^'~"^'* and d^ dt ^yco,-XQ)^. 2. By Direct Analysis. Let the body (Fig. 45) be fixed at the point O, and let 01 be the instantaneous axis as before, and the angular velocity ro be fi. \A .'I i i\ ) I Fig. 45. equivalent to w,, m^, «„ as shown. Then an element at P is tending to move at any instant in a circle about (9/, and its be linear et 01 be city to be / MOTION ABOUT A FIXED POINT. lis 95 absolute velocity is wp — -~, where/' is the perpendicular from P on the instantaneous axis. And, if «, /S, 7 be the angles which 01 makes with the coordinate axes, then f^={pzQ^^—y cos 7)2 H h ••• ; also ^, -^, — are the direction cosines of the tangent at P, ds ds ds -, •^, " are the direction cosines of OP, r r r cos «, cos /8, cos 7 are the direction cosines of 01. And, since OP is perpendicular to the tangent at P, and 01 also perpendicular to this tangent, we have dx X . dy y , dz z ^ ds r ds r ds r dx dv dz ax , nv a I "^ ^ — cos « + -^ COS l3-\ cos 7 = o. ds ds ds r dx ds dy ds dz ds z COS ^—y cos 7 x cos ^^ — z cos « y cos « —x cos /3 / ds And, therefore, since — = «A we have, multiplying each dt quantity by --> dx = {z cos yS —J cos 7) ft) = rft>„ — ji'ft)^, : at /* is ', and its dt dz _ 'dt~ as found betore. anal, anal. --ao)^ — ^ft),, -.yw^-xojy, 96 RIGID DYNAMICS. 11 fi-' ■i; !^1: 57. The former of the two investigations in the preceiling article may be presented in purely analytical form thus : (i) From the point P (Fig. 45) let fall perpendiculars on the coordinate axes OX, OV, OZ, and let 9, , yfr be the angles which these perpendiculars make with the coordinate planes XV, YZ, ZX. The angular velocity of P about the axis OX will be — -, and the resolved parts of this parallel to the coor- dt dinate axes Now .„ , (bx\de fdv\dd , fdz\de ,. , and y = ^{>^—x^) cos, 6, z-=^{i'^—x^) sin^, bx dd =0. and And by definition, 80' 83 89 ^/{r^—x^) sin 6= —z, = V (r^ — x^) cos 6 —y. dd dt '8x\de =&), (8x\de and (: '^_z\d±^ 89) dt r'^~z^) cos i/r= y(/^-j'2) sin ^, y= V('''^--0 cos 9= ^(;'2_y^) sin 1^, z= V(r2_y2) cos <^.= y(r2 -.1-2) sin 9 : I MOTIOxN ABOUT A FIXED POINT. (dx\dd (Oy\!e__ fd.:r\Jdt ' \d(\>)dt ' '' \d(l>jdt 97 # ._.. 3£\^_ / dx \dyfr _ ~~ ~ ' .dfjdt ~ ( ds \d'\}r \d^J dt -y(o. dfj dt The total velocity parallel to OX is the al<;cbraic sum of the partial velocities, that is dx ^ (bx\ie (dx Y4> f dx\d± dt \ddjdt \o'4>rdt V)fjdt' dx dd> d-^ Similarly, and dy d-^ dO dt dt dt ds dd d(f> dt dt (it (2) The second investigation in Art. 56 may also be pre- sented in a purely analytical form thus : ,^^2+y+.2^;.2^ (I) .i-cos«4-j/ cos/3 + ,:r cos 7 = »'Cose, (c = angle /(9/^), /)2= {p cos /3-j/ cos 7)2+ (.r cos 7 -.7 cos a)'^ + {y cos a-,r cos 7)"'^. Also and and «o. "^JL-^^^^n, cos a cos y8 cos 7 . •. xci)^ + J'ft>» + -<«, = ^'« cos e, (2) (fT-^+<-">^+*-^--^-"'''^-'^' H l>' .1 :U If;.' (; 98 RIGID DYNAMICS. The body being rigid and O a fixed point in it, r and e are constants ; also w, o)„ w^, a>, are independent of x, jy, a, the coordinates of P, therefore from (i) and (2) we obtain, by differentiation. dx , dv , dz dt dt dt dx , dy , d:; dx; dt dy dt da dt :;a)^ —ya^ .rco, — aa^ yw^ — xco^ = ± ^ by (3). (4) The ambiguity of sign in (4) arises from the fact that in equations (2) and (3) there is i othing to determine whether the rotations to,, (o^, cd,, are in the directions x to y, y to ;y, z to x, or in the opposite directions x to ^, j: to y, y to x. If they are in the former directions, the value + 1 must be taken, if they are in the latter directions, the value — i must be taken. 58. General Equations of Motion. Let the body be in motion, with one point O (Fig. 46) fixed, and let three rectangular axes be drawn from O, OX, OY, OZ^ to which we may refer the position of the body at any time dur- ing its motion. And let it be acted upon by external forces, producing on each element of the body m, accelerations X, V, Z in the directions of the three fixed axes. Then, if P be the pressure on the fixed point, and X, /li, v, the angles which the direction of the pressure makes with the fixed axes, we have, by D'Alembert's principle, the relations 2;// — '^ = Sw A' + P cos \, dr 2;// -j-^=1;n Y -\- P cos fi, 2;// — ^=1mZ +Pcosu. dt^ (I) (2) (3) MOTION ABOUT A FIXED POINT. 99 And, also, (4) S//^ d^y\ Im d\v d'h :;;/ .x dfi d^V ^'d^-y d^x\ dt'^J (4) (5) (6) Fig. 46. where L, M, A^are the couples due to the external forces. Now, since the body, when we form the above equations, is moving about an instantaneous axis 01 with some angular velocity d(o, o , / , , x 'dfi^^dt~^'di~ '""''*' <»A-^«x +J'&>, + ^&),), and, similarly, we should get, by symmetry, d"^)! da), d(o^ ^ ~dt'^ "■^' dt ~ ~dt~ ^^' '^^^^''^'^ ^y^^ + ■^^'"')- Therefore relation (6) becomes Ar V ( dh d\v\ = 2, /« {x"^ -^y^) ' — - — 2 7nxs — - — 2 myc — »f dt dt dt it'. MOTION ABOUT A FIXED POINT. lOI IS of cir- rc ft) (2), t in and by analo;j;y J/ and L can be written down. Since L, M, .V are given, these results would give the values of &>„ eB„, to, on integration, after calculation of the moments and products of inertia required. But this calculation, as can be seen, would be tedious, and we can avoid most of it by choosing axes which although still fixed in space are in coincidence with the princi- pal axes of the body when we form the equations of motion. This device enables us at once to disregard the products of inertia, and makes a great simplification in the problem. It is due to Elder, and the equations thus obtained are known as Elder s equations of motion. 1 on 60. Elder s Equations of Motion, Instead of choosing any three rectangular axes fixed in space at the instant under consideration, let axes be so chosen that they coincide with the principal axes of the moving body; and let ft)j, ft)2, 0)3 be the angular velocities about these principal axes, which will then be the rame as ^(o^ = L, B'^^^-{C-A)a>^,. dt dt ^ ' The relations between ^J, ^, -y\ the angular accel- dt dt dt erations around axes fixed in the body, and — ^ ^^ — ', dt dt dt the angular accelerations around axes fixed in space, may be determined for any given position of the moving body, as follows : Let /j, Wp «j ; /g, Wg, «2 5 4> '^^a* ''3> ^^ the direction cosines |5i 1 1 Ml 104 RK'.ID DYNAiMICS. of axes fixed in the body referred to coordinate axes fixed in space. Then will (U2 = /aft), 4- f^i^fOy + n^di. ,^ + ''3<«3);77 + (''^1®1 + W2«2 + ^"Z^^ ~Jl 1 / . 1 \^^h ( ,dL dm-, , duA f ,dL wr dnt] dn. \ ^^''^dt) + / , dL dm^ d)i\ v^-dj-^'"^ dt'-^'^'-dih = ;d in (I) (2) (3) le of ^ '•)«2 (Oo MOTION AIIOUT A FIXED POINT. 105 as appears at once on diffcrentiatinf; the equations in groiij) (3). (4) r/o). (/(o, , (io), ,._, ..„, I". From the second and third equations in group (i) \vc may in liicc manner obtain and lit ^ dt ^(it ^ lit '' dt ' Hence the acceleration around any axis may be projecteil on coordinate axes just as angular velocities and as segments of the rotation axis may be projected, and all theorems on the projection of segments of a line may be interpreted as theorems on the projection of angular accelerations about the line. If the axis of to^ coincide at any moment with the axis of tw^, then will /i= i, ?;/i = o, «i = o, a>i = a),, and by (4) above dw^_d(ii^ dt dt 62. Angular Coordinates of the Body. The equations of motion known as Euler's enable us to find ft)j, ft)2, Wg, the angular velocities of the body with reference to the principal axis drawn through the fixed point about which the body is moving. As these principal axes, however, are in the body, and move with it, we must have some means of determining the position of the body with reference to axes fixed in space, because the values of the angular velocities found by solving Euler's equations tell us nothing whatever as yet of the situation of the body with regard to any known directions in space. In order, then, to fix the position of the body at any time and give us a definite idea of its situation with reference to some initial position, three angles 6, , -yjr ':! u. ./ lit io6 KUWU DYNAMICS. cc^sirvf ^sinS 0)2 COfCOSO Fig. 48. MOTION AUOin" A IIXEIJ I'OINT. 107 cos^ arc chosen, known as the angular coordinates ; they define the situation of the i)rincii)al axes, and therefore of the body itself, beinj; measured from some initial fixed axes of reference which, at the i)ej;innin<; of the motion, coincide with the principal axes of the body. Relations can be easily found between 0, 0, y^r, and , yjr \ they also indicate how the relations existinj^ between these displacements and the an<;ular velocities about the principal axes are to be found. Let a spherical surface of radius unity be constructed at the fixed point C) (I'i^C- 4'^)» about which we suppose a body to be moving. Initially, let the body, which we may represent by its principal axes O/l, OB, OC, be in such a position that OA, (U\ t^C coincide with OA', OV, 0/ respectively. Then, by suppos- ing the body to turn through the angles i/r, 0, ^ in order, so that the point A travels in the directions indicated by the arrows, it is evident that af/y position of the body will be fully known in respect of the fixed axes 0.\\ O )\ OX, when we know three such angles as 6, <^, i/r. At any instant the body has angular velocities Wj, &)„, 0)3 indi- cated by arrows ; and in order to connect these with the angular coordinates, consider the motion of a particular point such as C. The velocity of the point C at the instant in question, may be considered as the resultant of the angular velocities co^, Wo, wg, or as due to changes in 6, (fy, yjr, i.e., to velocities '-—, ^— , ^ J-; at at at and by expressing in the two systems of change the velocity of C resolved in three determinate directions, and equating the lesults, we shall arrive at the relations between (Up Wg, Wg, and dp d(f) (i± lit' dt' dt' \ io8 RIGID DYNAMICS. til ?!;'■ The auxiliary figure shows the motion of the point C due to the two systems. The line ZCZ^ is the tangent to the line of the great circle, and the point C will evidently have angular velocities w^ co.^ in the directions indicated by the arrows ; it has also a motion — along the tangent to the great ci xle at C dt and a motion —f- sin d perpendicular to this former. This at velocity ^ sin 6 arises from the fact that C, owinij: to the -v|r lit 'or motion, has a velocity along a tangent to a small circle with CC as radius, and its velocity perpendicular to ZCZ^ must be CC • ^-^=OC^\x\. 6 • — ^ = --J^ sin 6, since we have agreed to call (U dt dt *= the radius OC unity. Hence, we have from the auxiliary figure, remembering that the radius is unity, the relations, dQ velocity of C along ZC=—- = (i>^ sin A + Wocos 6, dt (I) velocity of C perpendicular to ZC='^^ sin 6 dt ^d± ~ dt - — Wj cos (f> + o)2 sin , i/r. The complications in the former figure are onitted. m (I MOTION ABOUT A FIXED POINT. 109 This (I) Fig. 49. 63, Pressure on the Fixed Point. The pressures on the fixed point, measured along three fixed rectangular axes, will be given by the equations, (3) ^w^^SwA'+Z'cosX, dt^ 11 I 2;«^ = 2wF+Pcos/ti, dt^ dt^ d^V where 2;;/—- is now to be expressed in terms of the coordi- dfi nates of the centre of inertia, the mass of the body, and the no RIGID DYNAMICS. I, m angular velocities. Thus, if we evaluate as formerly — ^ in terms of 0),, a>y, ft),, we get 2;// ci^x dt^ — Sw -I , dt dt G)2.r-f GJx(^«x+/a>y+"a)J \ ; and if x, y, z be the coordinates of the centre of inertia, we have, on reduction, to determine the three pressures. Mass •] ~5-^— j/^^ — &)2i' + to,(.rG)^+3'a>j, + 5wJ [ =/'cosA, + 2w/A', l dt dt ) and two similar relations for P cos /a, P cos v. These equations are with reference to axes fixed in space ; but if we refer them to the principal axes moving with the body, we may use Euler's equations, and substitute for — >', — ', — - dt dt dt their values in terms of A, B, C, L, M, N, w^, co^, co^. The equations when finally reduced in this way become Mass . |coi(^+6^-^)(^+^)-K2+ft)32)I-} =/'cos\4-2;«JSl'— Mass • ]77^"~7;j[' with the two analogous expressions for P cos fi, P cos v. In these expressions x, j, 2 are the coordinates of the centre of inertia, L, M, iVthe couples due to the external forces, A^ B, C the principal moments at the fixed point. And it is evident that if x—y='z = o, the pressure on the fixed point will be the resultant of the external forces ^mX, 2/;/ F, 2///Z ; as, for example, in the case of a heavy body fixed at its centre of gravity, where the pressure must be simply the weight of the body. iil MOTION ABOUT A FIXED POINT. Ill Illustrative Examples. 1. If cu,, coj,, w, be the angular velocities about the coordinate axes by which the motion of a body about the origin may be exhibited, find the locus of the points the magnitude of whose velocity is aw^. 2. The locus of points in a body (which is moving with one point fixed) that have at any proposed instant velocities of the same magnitude, is a circular cylinder. 3. A body fixed at one point moves so that its angular velocities about its principal axes are a sin ;//, a cos ;//, in which t represents the time, and n and a are constants. Show that the instantaneous axis describes a circular cone in the body with uniform velocity. 4. A uniform rod, length 2 a, turns freely about its upper end, which is fixed, and revolves so as to be constantly inclined at an angle « to the vertical. Find the direction and magni- tude of the pressure on the fixed end. 5. Any heavy body, for which the momental ellipsoid at the centre of inertia is a sphere, will, if fixed at its centre of inertia, continue to revolve about any axis around which it was origi nally put in motion. 6. A right circular cone, whose altitude is equal to the diam- eter of its base, turns about its centre of inertia, which is fixed, and is originally put in motion about an axis inclined at an angle a to its axis of figure. Show that the vertex of the cone will describe a circle whose radius is ^ a sin «, a being the altitude. This is evident, since the momental ellipsoid at the centre of inertia of the cone is a sphere ; therefore the cone will revolve about the original axis permanently (Ex. 5 above), and its axis ,:'i,| 112 RIGID DYNAMICS. will describe another cone, and its apex will trace out a circle of radius | « sin «. 7. A circular plate revolves about its centre of gravity fixed. If an angular velocity qj were originally impressed upon it about an axis making an angle a with its plane, show that a normal to the plane of the plate will make a revolution in space in time 27r .^w^ = Mgh sin 6 sin (^, dt B'^'^-{C-A)oi^(o^ = Mghs,mecos, dt C'^-(A-B)co,co^ = 0, (I) (2) (3) and we have also the relations ill de dt = (o^ sin (f> + (02 cos <{), ■ — ^ sin^=ft)2 sin>i-Tf + c^a ^f ) = ^^S^'' ^^^ ^(*^i sin 4- <»2 ^^^ ^)» which, by aid of (4), becomes This, on integration, gives A \ 2 w^rtfcoj -{-A \ 2 (o^^dw^ = 2 Mgh I sin ^^^. .-. Ai(o^-\-w^) = 2Mgh{co^0Q-cos6). But, taking (4) and (5), and squaring and adding, we get . •. a(^J + A sin2 df^y = 2 3fgh{cos Oq-cos 6), (a) which, as will be seen hereafter, is one form of the equation of energy. 6^. In order to obtain another relation between -, , —f' ^ dt dt we may proceed as follows. Multiply (i) by cos^ and (2) by sine/) and subtract, and we get , dw„ . do), C— A dd . sin ^{^^=2Mgh{zo^ ^^-cos 0), — , ^^, and then the position and the motion of the (// dt top at evf^ry instant are knov A sin2 6/'^ = 0/(cos <9o-cos 0). lit ^ Ihese give —, — f- d/ dt top at ev^ry instant are known. As we have seen, — -^ depends for its sign on ;/ and the posi dt tlon of the centre of gravity; it also changes with Q\ and, or eliminating -^, we get dt A sin O-j- = Vcos Q^ — cos 6 y/2Mgh'Am\^6 , de . and -— wi .rj - C'^;i^{cos 6q — cos 6) ; JQ and — will also change in value, and will have minimum values dt (o) when = 0^^ and 6 = dp $1 being a root of the quadratic 2 A Mgh sin^ — C'^ifi (cos 9^ — cos Q) = o. The top will then, as it is first placed on the plane, tend dQ to drop down, and — will go on increasing until, having dt passed some maximum value, it reaches its minimum value MOTION AHOirr A FIXKO POINT 119 J when B = Oy Meanwhile ' j has also been going through peri- odic changes, being a maximum when 6=$^. The top then oscillates between the positions $^ and 6^, and at the same time is carried about a vertical axis with a pre- cessional motion (not constant) '-^^ ' lit To an observer placed above the top and watching the pro- jection of its centre of gravity on the horizontal plane, that point would describe the curve indicated in Fig. 52, lying between two circles whose radii are // sin ^^ and // sin ^j. Fig. 52. The curve described will not necessarily be closed ; that will depend on « being an integral part of 2 tt. It is evident also, from the fact that maximum and minimum values exist at I20 KKJII) DYNAMICS. the cusps and the outer points, that the curve described touches one circle and cuts the other at ri;;ht angles. The maximum value of j^ may be found by putting , =o in the equations on page 1 18, which will give II.. d s i n*"^ e f "'^ Y = 2 Mgh (cos e^ -cos 6), A sin^ B J = Cn{cos ^,, — cos 0). i< I (it Cn Cn ^F being the weight of the top. dO dyjr When 6 = 0,., it can be seen that both — and — "^ vanish identically. 68. Top spinning ivit/i Great Velocity on a Rough Horizontal Plane. In most cases the top is spun with a very great velocity, and then placed on the plane. By taking the value of - already found, A sin e~ = Vcol^l'T^^co^ V3 A/gliA s'^u'^0- CV(cos ^^-cos 6), it will be seen that if ;/ become very great, cos ^q — cos ^ must become very small in order that the expression under the radi- cal may remain positive, hence the axis of the top, instead of per- forming large oscillations, will depart but little from 0^, its initial position, and -^ will approach a constant value, and the motion at will therefore become steady. The time of a small oscillation may be found in the following way : til ' = MOTION AlUH'T A FIXKI) POINT. Let = 0^ + //^ ,f being small. 121 COS 0^ — COS im0 = u approximately, and the foregoing relation for '^^ becomes A<^ __-^ /t"S 0^ — C( )S dt 'i^ MgliA sin 0~ ChP''^^^ ^o-cos sin where Hut sin , ^({0 — ^ • "^ Jr^'^ '^Ig'tA sin 0^11 - ChihiK .A d0_ . ^_MghA9:m0, d0 _ dn dt ~ dt die A = at. ^'^ V2 an — ir H = a vers — /", A and <— f') Q=0Q-\-a\\ — cos^^/ This is a periodic function which repeats values of 6 every time / is increased by 2 7r A and therefore the time of a complete small oscillation is 277 A Cn 122 Also, RIGID DYNAMICS. dyjr _ Cn co s 0^^ — cos 6 _ Cn dt A •An^e Cn dt A sin 6. Chi' ii yi sin ^n . . .ai I— cos-f'/ A sni ^„ V A dyjr^ Cn _ MghA?:\x\0,, f Cn . •cos-— t A fcj I \\ i ) ,1 I = I JVfgh Cn I —cos Cn A , Mirk ^ ]\T8:h . . Cn ^ and consist of two terms, one increasing uniformly with the time, the other very small, and a periodic function of the time. If n be extremely large, we have, approximately, , Mgh . and the precession is then nearly constant and equal to Wh Cn W being the weight of the top. 69. If, then, a top be spun with very great velocity and placed on a rough horizontal plane, inclined at an angle to the vertical, it will make small oscillations in time — - — , and at the Cn same time will revolve about a vertical axis with an angular velocity very nearly equal to Wh Cn In the ordinary case, the oscillations will be so rapid at first as to be barely visible to the eye ; is the speed diminishes, owing to resistance of air and friction at the apex, they become more noticeable ; until finally, MOTION ABOUT A FIXED POINT. 123 when the top is ^^ dying'' n becomes comparable with the other quantities, the oscillations become wider, and the formulas of Art. 6"] apply. 70. Top spinning on a Smooth Hori-zonial Plane. Let a top {Fig. 53) be spun and placed in any manner on a smooth horizontal plane, and let its position after any time Fig. 53. t has elapsed be that shown in the figure. It is acted upon only by the reaction R of the plane and its weight Mg acting at G, the centre of gravity ; and if ^, 77, ^ be the coordinates of G, the equations of motion of translation are Im'^^M'^- dt"^ dfi O, dt^ dt^ m li I' ■< B 124 RIGID DYNAMICS. d^ From these it is seen that — ^ = constant = initial value ; dt -^ = constant = initial value ; and if therefore any horizontal dt motion be imparted initially to the centre of gravity, it will preserve that velocity at every instant thereafter. And, since ^=//cos^, CG being equal to h, and d being the inclination of the axis of the top to the vertical, the third relation becomes ^./V^cos^)^ dt^ g' ... /e=,/|^.+f^(i|?i^|. If! ,i The equations of motion of the top about the centre of gravity considered as a fixed point are (i) ^^ + (C-^)&)2«3 = ^'^sin<9sin^ -J/^2 sin ^cos ofjj^.de. . : A (coi^ + 0)32) = 2 Mg/i{cos 6^ - cos 6) - M/fi sin^ {^^. . , A W + /^ sin2 e('^ + Mh^ sin2 9 ("^j \dtj \dt J \di Jt) — 2 Mgh (cos Q^ — cos 0) , and the other relation will be as before : ^ sin2 ^ -"^ = 0/(cos ^^ - cos 6?). dt These two relations give the solution of the problem. 126 RIGID DYNAMICS. M And it is evident that, independent of its motion of translation in a horizontal direction, the centre of gravity can only move up and down with an oscillatory motion while the apex describes on the plane the fluted curve already obtained in the case of a rough plane (Fig. 52), the values of Oq and 6^ being as before j0 those which make — a minimum. ^/ If 0)3 = n be very great, the discussion is the same as before, and it can easily be seen that the apex of <"he top will describe a simple circle (approximately) on the plane, and the motion will be steady, the time of a small oscillation and the period of precession being obtained as formerly. \y hi 1 •!' I y I ■ ! I li: 71. All the previous results obtained theoretically in the case of motion of a top on a smooth or rough plane can be verified experimentally by having a number of tops made similar to that shown in Fig. 54. Fig. 54. A circular plate of brass, a quarter of an inch in thickness, and from three to five inches in diameter, has a steel axis through the centre. The centre of gravity of the top may be from one to two inches from the apex on which it spins, and the point may have varying degrees of sharpness. Everything should be symmetrical and made true, so that MOTION ABOUT A FIXED POINT. 127 The top is most readily set spinning by using a two-pronged handle with openings through which the axis may pass : a cord put through a hole in the axis and wound about it, is pulled rapidly, and the top drops with a high speed from the handle. A little practice enables one to spin the top and let it drop on a smooth or rough surface at any required inclination. The following problem may also be examined by using several of these tops of various sizes, and with points of varying degrees of sharpness : A common top, zvJicn spun and placed oji a rougli horizontal plane, at an angle to the vertical, gradually assumes an nprigJit position. Explain this. This is the case of the ordinary peg top of the schoolboy, which is usually made of a cone of wood through which passes a steel axis ending in a sharp point ; when spun upon a rather rough surface, it gradually becomes upright and 'sleeps.' It will be found, after a little experimenting, that this appar- ently paradoxical rising of the top to a vertical position against the force of gravity depends on two things : 1. The degree of sharpness of the apex on ivhich the top spins. 2. The position of the centre of gravity. If the point be very sharp so that the top in spinning is not able to form a small conical bed for itself and thereby be acted on by a couple arising from friction at a considerable distance from the point, it cannot possibly become erect. When, however, the point is rather blunt, and the centre of gravity not too high, the top will slowly rise up under the action of the friction (which tends to diminish the angle of inclina- tion), and 'sleep.' The equations of motion are similar to those obtained in Art. 64, with the additional relations introduced by friction. The solution of the equations shows that the top rises to the vertical, on the supposition that the point of the top is a portion ■" ■■ ■ 1 :'■ "M ; s t, . !|-i>' 128 RIGID DYNAMICS. .il r i of a spherical surface and that friction is thus enabled to act in the proper manner. A complete analytical solution of the problem is given in Tellett's T/u'ory of Friction, Chrp. VITI., where the top is sup- posed to be a symmetrical pear-shaped cone with a spherical surface as the apex upon which it spins. Fig. 5£ •HI 72. The Gyroscope nwving in a Horizontal Plane about a Fixed Point. If a gyroscope bfe put in rapid motion and placed so that the prolongation of the axis of rotation can rest on a fixed point of support, and if, at the same time, an initial angular velocity MOTION ABOUT A FIXED POINT. 129 ;t in a in sup- rical about the point of support be given bodily to the gyroscope (in the proper direction) in a horizontal plane, it will revolve about a vertical axis, and the apparently paradoxical motion is pre- sented of a body whose centre of gravity moves in a horizontal plane although its point of support is at quite a distance from the vertical through the centre of gravity. In Fig. 55 the gyroscope is supposed to be set rotating and started in a horizontal plane with its centre of gravity at the point G, the weight acting vertically downwards in the direc- tion indicated by the arrow. It is supported only at the point O, and, if rotating rapidly enough, will keep on moving uniformly in this horizontal plane in a direction hereafter determined. Its position at any time is given by the position of its />riu- cipal axes at O : these are OA, OB, OC. It is evident that ^ = — and that C moves along XNN\ 2 NON' being the line of nodes, and the angle BON = . At each instant the gyroscope tends bodily to turn about NON' under the action of gravity, and the value of this turning couple is mgh, tn being the mass of the gyroscope and OG = h. Resolving this couple mgli into two, we get ; \, i ! mgh cos ^ about OBy and 7ngh sin ^ about OA. Then Euler's equations become : vit a t the nt of ocity J A — - -f {C— /4) W2«3 = nigh sin ^, A 2 dt (C— ^)a),a)3 = tngh cos 0, ^ dt °' from which it is seen that a»3 = constant = «. ■ ' U' :ttJ V ' l' ''i fl' - i' 130 Also, we have RIGID DYNAMICS. - =0 = ,^ cos(j>, dt ; =&)., sin

dt = ;i, since 6— -, and therefore cos^=o. h it- ■• i m ■f 1 4 ■h i I ! I From the preceding relations we have : a)j sin (fy + o)^ cos ^ = o, ft). sin <^ — ft)j cos<^ = rA/r .-. squaring and adding But since 2^ 2 ^^W ft>f + ftJ3^= - r ^ -7- + {C— A)(o^(o^ — vtgh sin <^, A ,^ — {C—A)(i),COo = JUg'/lCOS(l). dt Therefore, multiplying the former by (o^, and the latter by q)^, and adding, we get Aco^-^ +Aco^-~ = 7ng/i{co^ sin (fi + w^ cos(f>)=0. /^ (&)j2 + 0)2^) = constant. .•. G)j2 + &)./ = its initial value, = «^ say. Vi- : f I Then d-\lr , d(b —y- = a, and — - = «. dt dt MOTION ABOUT A FIXED POINT. 131 3y «2' since both may be taken zero when / is zero. . ■. (o^— —a cos «/, (Of^ = a sin nt, And, substituting these values in the relation for the first couple, we get A{an sin «/) + {C— A)n[^n sin «/) = m^/i sin «/. . '. Cnu sin ;// = w/^// sin « A and d^ _jngh _ Wh ~ dt ~ Cn Cn ' TF being the weight of the top, and ;/ being the initial velocity of rotation. Hence the axis OC moves around in a horizontal plane with uniform velocity ---^ and the direction of revolution is indicated by the sign of 71 or Wg ; that is, /o an observer looking dozvn in the direction ZO, the gyroscope ivill revolve bodily in the same direction as the gyroscope rotates about its axis luhen viewed by an observer at C. It is important to observe that the necessary condition for the motion of the gyroscope bodily about OZ is that it receives an initial angular velocity, so that O) i+(>>'i-\--f-] =some finite quantity. If this initial velocity be not given to it, it will act in the same way as a top, tending to drop down and oscillate as it moves around the vertical. 132 RIGID DYNAMICS. ii ^ I U'l i< I' ' Usually w is very great, so that « is small, and the preces- sional motion is slow. I'^or a complete discussion of the experiments which can be performed with the Gyroscope see Chap. X. 73- ^^ fi'^^^ ^/'^' Pt'cssurc on the Fixed Point in the Case of the Gyroscope. As an illustration of the use of the equations of Art. 63, we may find the pressure on the point about which the gyroscope revolves. In this case we shall have, calling the mass of the gyroscope 5 to avoid confusion, = P cos X + S/z/A'- s{^z-^^^, S \ CO, . C+A-b(^ +-^^) - (0,3'-= + a,,V I = Pcos^ + ^mV-s(^.v-^cy . = P cos v + 'LmZ—S(—y -'^xj, which become, since A = B, and (x, y, z) are (o, o, h), s\rsin<^ + 5|^"-^./.2|. S\—{(i)^-\-(t)^)h\=PC0'iiV, the last of which can be obtained from elementary consider- ations. \- ■ }: MOTION AHOUT A FIXKI) I'OINT. 133 D N (///;'■ COS (i ,,, , , /^hn) ' // ' A ) /' cos /A = /;/ - -^ ,— ^ • /r -.V sin + w./ , ' A A ) . P cos 1/ = ;;/ 1 — ((Uj- + w.^)/t \ , the mass being denoted by ni. These relations taken in conjunction with (Uj sin (f) + 0)2 cos = 0, tocsin — ^1 cos 0= ," = — J - CO,) = M, CicoJ - ft,,) - E{C0J - CO,) - D{C0J - COy) = jV, CO,, ft)y, 0), being the digular velocities about axes fixed in space at time /, and these being suddenly changed by the impulsive actions to coj, coj , coj. 75. Taking the foregoing expressions for the impulsive' couples, we can simplify them by choosing principal axes, which make D, E, F vanish ; if, at the same time, the body starts from rest, oi„ co^, ft), are zero, and the equations become Aco',=:L, Bco\ = M, Cco\==N. The equations of the instantaneous axis are X _ y _ c or or «x «'r CO,, X A y M B z C Ax .By_ Cz L M N !>-.' Ik: . V ' 136 RIGID DYNAMICS. The plane of the impulsive couple is and therefore the instantaneous axis (that is, the line about which the body will begin to rotate under the action of the impulse) is the line conjugate to the plane with regard to the ellipsoid The equations of the instantaneous axis are L M N' and the equations of the axis of the impulsive couple are X _ y _z Hence it will be seen, by comparing these two sets of rela- tions, that if a body fixed at a point be struck, it will not begin to rotate about the axis of the impulsive couple induced by the blow, unless A—B=C, or unless tlie plane of the impulsive couple be a principal plane or parallel to a principal plane. For the two sets cannot reduce to a single set unless A = B=C\ or unless two of the quantities, x, y, z, vanish, (which means that the axis of the couple is one of the principal axes). It v/ill be seen from the preceding investigation that, if a rigid body be free to turn about a fixed point, the problem of determining the change produced in the motion of the body by the action of a given impulse, is equivalent to determining the change in its motion when the body is acted on by a given impulsive couple. This equivalen:e also appears from the fol- lowing considerations. The impulse may be resolved into an I \ i MOTION ABOUT A FIXED POINT. 137 * equal and parallel impulse acting at the fixed point and an impulsive couple. The impulse acting at the fixed point will have no influence on the motion of the body, and therefore only the couple need be considered. Resolving the latter with respect to the coordinate axes we obtain the equations on page 135. Illustrative Examples. I. A cube is fixed at its centre of inertia, and struck along an edge. In this simple case it is evident, without forming the equa- tions of motion, that, since the momental ellipsoid is a sphere, A = B=C, and the cube begins to rotate about the axis of the impulsive couple. Thus, in Fig. 56, the cube is fixed at O, its centre of inertia, and on being struck by a blow Q, begins to rotate about the axis of the impulsive couple A OB. \ N ^ q/^ n \ \ Fig. 56. 2. A homogeneous solid right circular cylinder is rotating with given angular velocity about its centre of inertia, which is fixed ; the cylinder receives a blow of given intensity in a direc- ill m 138 RIGID DYNAMICS. li:,' ' ' 111 tion perpendicular to the plane in which its axis moves. Deter- mine the subsequent motion. 3. A lamina in the form of a semi-ellipse bounded by the axis minor is movable about the centre as a fixed point, and falls from the position in which its plane is horizontal ; deter- mine the impulse which must be applied at the centre of inertia, when the lamina is vertical, in order to reduce it to rest. If this impulse be applied perpendicularly to the lamina, at the extremity of an ordinate, through the centre of inertia, instead of being applied at the centre of inertia itself, show that the lamina will begin to revolve about the major axis. 4. A triangular plate (right angled) fixed at its centre of inertia and struck at the right angle perpendicularly to the plate. u i Fig. 57. In Fig. 57 let G be the centre of inertia of the trangle, and C the point where the blow is struck at right angles to the plane MOTION ABOUT A P^IXED POINT. 139 of the paper. Then if we construct the momental ellipse at G, it touches the three sides at their middle points. The impulsive couple in this case contains the line CG in its plane ; but since AB is a tangent to the ellipse, A'GB' is the diametral line con- jugate to CG. The triangle therefore commences to rotate about A'GB', which is drawn parallel to the hypothenuse. 5. A solid ellipsoid fixed at its centre is struck normally at a point/, g, r. If /, m, n, be the direction cosines of the line of the blow whose magnitude is Q, and if the equation of the ellipsoid be /V'M Mill rvj a^ Ir c^ then the equations of the instantaneous axis will be Ax^By_^Cz L M N' or ^^{l^ + c\r Q{qn — rvi) ' qn — nn rl —pn p in — ql ' and since the blow is normal to the ellipsoid at /, q, r, I VI n du d?i dii dx dy dz or / _ ;« _ n p q r a" ^2 Therefore the equations of the instantaneous axis will be ^ U^^c^ ^q_ c^ + d^ ^^r_ a^^b'^ ^2 • ^2_,.2'^ ^2 • ^_^iy ^2 • a^-lP- ii i:i t: ! 'I t CHAPTER VIII. MOTION ABOUT A FIXED POINT. NO FORCES ACTING. 76. Heavy Body fixed at its Centre of Gravity. The simplest case of motion under no forces which ordinarily presents itself is that of a body acted on by gravity and fixed in such a manner that it can only rotate about its centre of gravity considered as a fixed point. Here we have at And, multiplying these three equations by eDj, ©g, 6)3, respec- tively, and adding, we get dt dt dt .'. Aw^+B(o^-\-C(o^= a constant (I) Similarly, multiplying the three equations by A^i, Bm,^., Cco^, respectively, adding, and integrating, we get A^a)^-hB^o)^^-\-C^(o^^= a constant 140 (2) '3' (2) MOTION ABOUT A FIXED POINT. 141 (i) States that the kinetic energy is constant, as might be expected, since no forces act ; this can be seen by taking I" Smir ■-^MO'-m^m\ = |-Swf(ra.2— J<»3)^+ ••• + ••• }2 since the products of inertia vanish. (2) is another way of expressing the constancy of the moment of momentum. For (moment of momentum)^ where 0)1 '2' 77. Now, since h^, h^, h^ are constant at all times, the plane h^x-^h^y-\-h^z-o, or A(ii^x-\-B(d^y^Cai^z=o is an Invariable Plane fixed in space ; the line X _ y A( Bo)o Ci Wc iO)i x^t«2 '-"'3 is perpendicular to this plane, and is an Invariable Axis. The instantaneous axis is given by X 6)1 0)2 0)3 0) i till I .:, '* ! ( I ill I' ' ' Ir't'' HIT 142 RIGID DYNAMICS. 78. If we now construct the momental ellipsoid at the fixed point O, as in Fig. 58, O '', OB, OC being the principal axes, Fig. 58. and POP' the instantaneous axis at any time /, the equation of the ellipsoid will be and those of the instantaneous axis ft)j Wg a>3 ft) Now, X, y, 3 being any point on this line, let it represent the point P ; then at P we have MOTION ABOUT A FIXED POINT. 143 i- — Ji — _£. — '' Wj ft)^ (Ug ft) and .'. G) = _ • r, ,r = G) ^■/&' J = «..^, Ml r=«Wo k Therefore the angular velocity at any instant is proportional to the radius vector of the ellipsoid. Moreover, taking the tangent plane at P to the ellipsoid, its equation is ax dy dz where which becomes ^=«i|, y^^i\' ^^^^j' ^-"4)l+-+-=°' or or Ao).^ ■ ^ + B(o^ . rj + Cco^ • ^=kc. And, if we construct the plane Aa^x + Bw^y + C(o^s = o and represent It by XYX'Y', this is the invariable plane ; and we see that the tangent plane to the momental ellipsoid at the 1^1 il ItiH ;'';' If! 144 RIGID DYNAMICS. point where the instantaneous axis cuts the ellipsoid is always parallel to this invariable plane. Hence, the motion of the body fixed at O, and under the action of no forces, is completely represented dy tJiv rolling of the momcntal ellipsoid on a plane fixed in space and parallel to the invariable plane, and at a distance from it equal to 00\ sir ^ , If..' J'-l: II'. [I »■ ',1 I' I ' 79. The ellipsoid in rolling on the fixed plane traces out a curve on that plane, and also one on its own surface. The curve traced out on the surface of the ellipsoid is called the Polhode, and its equation is found by taking the condition that the perpendicular from the centre of the ellipsoid on a tangent plane at x, y, a is constant, and combining it with the equation of the ellipsoid itself. The equation of the Polhode is, therefore, Ax'^+By'^+Cz'^^c^ A\v'^ + E^y'^^-(?d^=c'\ The curve traced out on the plane is called the Herpolhode^ and its equation is found from the relation a pi = p2 = Opi -00'^ = t^ -p\ and will vary with r, and therefore with ta and with p. It is apparent that any one of the central ellipsoids might be chosen instead of the momental ellipsoid, and the motion of the body exhibited in a similar manner by the changes in motion of the ellipsoid chosen. Innumerable problems may be constructed from the preceding representation ; but they are all dependent on properties of the ellipsoid, and are not problems in Dynamics. CHAPTER IX. MOTION OF A FREE BODY. 80. We have already seen, in discussing DAlcmbcrfs Prin- ciple, that the general equations of motion of any body are M^-^=- X 2;;^( Z and -;« w- S;«{.( X dfl d\x -dt-n^ y _d^y dt"^ ^m[x{ Y dfi d'^y — X Z dfi -Ax dty\ dh =0, =0, d\x\ ) If M be the whole mass, x, y, ^ the coordinates of the centre of inertia at time t, and x', y\ 2' the place of m relatively to a system of axes originating at the centre of inertia and parallel to the original set of axes, then the equations of motion become d^ H5 if \ \ «ll i : !-! I 146 and RIGID DYNAMICS. 2;;/ 2;//!,y[.r ..{.(K-'|^)-y(.v--)}=o, which latter can be transformed in the ordinary way so as to determine the angular velocities. These equations theoretically give a complete solution of the problem. But the most important case of free motion of a body, and the only one which admits of simple solution, is that in which o Fig. 59. ■t/de particles of the body 7nove iji parallel planes. Here it is evi- dent that we need only consider the motion of one particular plane of particles, and that containing the centre of inertia is as to f the , and /hich .^ > evi- :ular ia is MOTION OF A FKEK I50DY. H7 chosen, and the position of the body at any time determined in the following way. Let the plane in which the centre of inertia moves be repre- sented by the plane of the paper, the same section of the body being represented at any two times as in Fig. 59. Let the body be referred to fixed axes OX, OV, and let AC/> be any line in the body passing through the centre of inertia C, and in its initial position let this line be parallel to OV, as shown. Then, after any time f, the body has reached its second position, and it is evident from elementary geometry that the body can get from its first position to the second by translation of the centre of inertia C, and by rotation about C through an anjrle 0, equal to that which ACB in Its second position makes with the axis O V, or with a parallel line fixed in space. For translation of the centre of inertia, we have, by D'Alem- bert's principle, dt'' dt'' 2.1)1 — -^ = Zm V = AT — 4- dt^ d*^ And for rotation about the centre of inertia considered as a fixed point, we get i dt" ^ dt^S dt^ \ X Therefore, at any time, the motion of the body will be fulIyjQ known when we know 1. The initial conditions, so that 6 is known. 2. The coordinates of the centre of inertia with reference toi* "I ar some axes fixed in space ; this gives — '--, —^. dt^ dfi 3. Mk"^ about the axis of rotation through the centre of inertia. 4. Geometrical relations between x, y, i iK: I f' ' I r'' II ' •';!!! 148 RIGID DYNAMICS. In cases of constraint where bodies roll or slide on others, geometrical relations are easily found, and the unknown reac- tions eliminated by taking moments. Illustrative Examples. I. A heavy sphere rolling down a perfectly rough inclined plane. In this problem gravity, by the aid of friction and the reac- tion of the plane, produces both the translation of the centre of inertia and the rotation. Let OXy O V (Fig. 60) be the axes of the coordinates fixed in space, the sphere starting to roll from O. Then at any time /, Fig. 'jO. the, position of the sphere, is given by.r, j, the coordinates of C, the centre of inertia, and the angle through which the sphere has rolled ; that is, through which it has rotated about C, considered as a fixed point. The initial conditions, combined with the geometrical condi- tions for perfect rolling, give x=ad, y=a. (i) MOTION OF A FKEi: BODY. For the translation of the centre of inertia, \vc have rif-X at' M'-^ J + J4'-cosrt-A' = o. if' The rotation about C is given by 149 (2) (3) (4) These four relations give a complete solution of the problem, for we have and, therefore, from (2) and (4), M{k^^a^)'^^^=Arga^mii., (5) from which it is seen that dfi = f^sin a. and also, :^=i\^sin «• /2; R = Mg cos a, F=| i^i^sin a. These results give the space passed over in time t, and show that five-sevenths of gravity is used in translation, while two- sevenths is used in turning the sphere about the centre of inertia. The relation (5) may also be obtained at once by forming the equation of energy. For the sphere has fallen through a dis- tance X sin u., and therefore the work done by gravity is ['' \ Ik ill ' l!i ll i I It If : > 150 or RIGID DYNAMICS. M^v sin a, MgaO sin «, which must be equal to the kinetic energy at time t, and there- fore 1 MirJ^ + /.■2a)2) = J/^rt^ sin «. .-. |i^/(.?H/f'2)C^Y = i]/^«(9sina, which gives, on differentiation, M(a^^k')^^=Mga sin a. as before. 2. If a heavy circalar cylinder rolls down a perfectly rough inclined plane, one-third of gravity is used in turning and two- thirds in translation. 3. A very thin spherical shell surrounds a sphere, both being perfectly smooth and consequently no friction between them, and the system rolls down a rough inclined plane. In this case, if we neglect the mass of the outer shell, the inner sphere acts just as if it slid down the plane, because, since there is no friction between it and the shell, as the shell rolls it slips around, and therefore tli° equation of motion is ar M being the mass of the sphere, which is s,o large that the mass of the outer shell is negligible in comparison. If, however, the shell and sphere were united, the system would roll down, and then the equation of motion would be M dt'^ : J/|-^sin a. And the times occupied in rolling a given distance in the two cases would be to one another as Vs : V7. lere- MOTION OF A FREE BODY. I5t In the case of a cylinder surrounded by a cylindrical shell, gravity would be diminished to two-tliirds of its value, and the times occupied in rolling a given distance would be as V2 : V3 under similar circumstances. ugh two- 4- To determine whether a sphere is hollow or solid by roll- ing it down a rough plane. This could be done by observing the space passed over in a given time, and by calculating the moments of inertia and forming the equations of motion (i) on the supposition of a solid body ; (2) on the supposition of a shell of radii a, b. 5. A homogeneous heavy sphere rolls down within a rough spherical bowl ; it i^, required to determine the motion. )oth ^een the use, hell the em wo .A£ Fig. 61. I . 1 iii! lr.ll l\' I ii iV' S': r^ I ^ « 152 RIGID DYNAMICS. Let the radius of the spherical bowl (Fig. 61) be d, and of the sphere, a ; and let the sphere start with AP coincident with BQ. Then, at time t, the circumstances are as shown in the figure. Let G)= angular velocity about P, OCP = , OM=x, DPA-.= d, PM^y, BCO = a, then will x={b—a)s\n^, and y=^b — {b—a)cos-\- F sin (f) — m£; (I) (2) F being the friction, and R the reaction at the point D, acting in the directions indicated by the arrows. dt dt dt dt Moreover, MPA being the exterior angle at P, and a dt (3) Along with the foregoing relation we have, also, taking moments about P, (4) MK'^—^F-a. dt It is then easy to find R and F by taking the values of x and p, and differentiating twice and substituting in (i) and (2). of the \xBQ. ire. (2) acting (3) taking (4) and y, MOTION OF A FREE BODY. It will be found on reduction, that 153 w/r, and R= ^(i; cos <^- 10 cos a), (^-«)(^J = -W(cos becomes small, this gives the time of a small oscillation of a sphere . ithin a spherical bowl. For S^ g{b-a), a being the radius of the ball, and g of the roller. m |i i: Pi: (i rii; 'f ;■■,! I- I' 154 RIGID DYNAMICS. 7 A uniform straight rod slips clown in a vertical plane between two smooth planes, one horizontal, the other vertical ; find the motion. Let OX, OY he the horizontal and vertical planes, and let the rod starting from its upper position when /=o assume the position AB at time /, as in Fig. 62. Fig. 62. Then we have two reactions at the points A and B, and the weight Mg acting at the centre of gravity, C. s'o that if x,y be the coordinates of C, and 6 the angle of inclination of the rod AB to the horizontal, v/e get rfi MOTION OF A FREE BODY. 155 ane cal ; let the \^ o d the gle of and x=acos,d, y = a sin 6, J where 2 rt- is the length of the rod. Also, taking moments about the centre of gravity, we would have and we may suppose that 6 is initially equal to «. These four relations give a complete solution of the problem. It will be found that the rod leaves the vertical plane when sin ^ = I sin cc, and then the motion becomes changed, the rod moving with a constant horizontal velocity along the horizontal plane equal to \\— — -, until it finally drops and lies in the 3 plane. The problem may also be solved by aid of the principle of energy. 8. A circular disc capable of motion about a vertical axis through its centre perpendicular to its plane is set in motion with angular velocity 12. A rough uniform sphere is gently placed on any point of the disc, not the centre ; prove that the sphere will describe a circle on the disc, and ♦•hat the disc will revolve with angular velocity ^— — ^- : • H, where MB is ^ 7M/c^ + 2inr^ the moment of inertia of the disc about its centre, m is the m.ass of the sphere, and r is the radius of the circle traced out. 9. A homogeneous sphere is placed at rest on a rough inclined plane, the coefficient of friction being jjl ; determine whether the sphere will slide or roll. 10. A homogeneous sphere is placed on a rough table, the coefficient of friction being fi, and a particle one-tenth of the mass of the sphere is attached to the extremity of a horizontal 156 RIGID DYNAMICS. Ill in' ■i f J I Mr m J ■ V 1 M ■ ■ 1 : ' , diameter. Show that the sphere will begin to roll or slide II iin... ...:ii 1- ic II according as /4>or< What will happen if u — 10V37 81. Impulsive Actions. Motion of a Billiard Ball. 10V37 The complex motions of a homogeneous sphere moving on a rough horizontal plane are well illustrated in the game of billiards, where an ivory sphere is struck by a cue and made to perform evolutions that seem to the unscientific little short of marvellous. In the general case the course which the billiard ball takes depends on the initial circumstances, that is to say, on the way in which it is struck by the cue ; and the motion is made up of both sliding and rolling, so that the centre of the ball moves in a portion of a parabola until the sliding motion ceases, when it rolls on in a straight line. If struck so that the cue is in the same vertical plane with the centre of the sphere, then the motion is purely rectilinear; which is also the case if the cue is held in a horizontal position. It may also happen that if the ball be struck by the cue at a certain oblique inclination to the table, its path, after sliding ceases, will be opposite to the horizontal direction of the stroke, and it will roll backwards. For a complete solution of the problem, then, we should know the direction, intensity, and point of application of the blow struck by the cue, so that the velocity of translation of the centre of gravity is known, and the initial angular velocity. 82. In ordinary blows, the initial value of the rolling friction will be very small compared with the sliding friction, so that at the beginning the former may be neglected, and the equations of motion for sliding found in the following way. Let the plane in which the centre of the ball moves be the plane oi xy, so that {x, y, —a) are the coordinates of the point of contact at time t. Let F be the value of the sliding friction, and /Q the angle it makes with the axis of x. MOTION OF A FREE BODY. 157 Then evidently the pressure on the table is equal to the weight of the ball, so that R = Mg and F=ixR. The equations of motion of the centre of gravity are 7l/^'= -Fcos/3, M'^,--^ -Fsin(3, dt" M^^=o = R-Mg. For rotation about the centre of gravity we have ^^=_rt/rsin/3, dt dt dt aF cos /3, ' Si ^3 = ^3' where u^, Vq are the axial components of the initial velocities of the centre of gravity, and fl^, fig' ^3 ^^^ ^^^ initial angular velocities about axes through the centre of gravity. The above give a complete solution of the motion during sliding which, however, in the case of an ordinary billiard ball, lasts but for a small fraction of a second. 83. At the instant the ball is struck by the cue the hnpidsive equations will evidently be formed as follows. 158 RIGID DYNAMICS. I' 'Jf , I if ( f l.^:•^T! r Let Q be the value of the blow struck by the cue, and « the angle the cue makes with the table ; also, let F be the impulsive value of friction at the instant of striking, and ^ the angle which it makes with the axis of x. Then, the axes being chosen as in the preceding problem, and the line and angular velocities being denoted as formerly by Uq, Vq, Hj ^^, 1> we have ' ,4/,^ z Q cos a — Fcos /3, \Mvq= -Fsin/3, AD.-^= —Qh sin«— rt/^sinyS, Ail^= Qk -\-aF cos ^, , AD,^= — Q/i cos a, where /i is the horizontal distance from the centre of the ball to the vertical plane containing the line of blow, and k is the per- pendicular on the line of blow from the point where /i meets the vertical plane containing that line. And the impulse on the table must be equal to Q sin a. See, Theorie mathhnatique des effcts du jeu de billard, par G. Coriolis, Paris, 1835. 84. Impulsive Actions. Free Body. Illustrative Examples. 1. A uniform rod is lying on a smooth horizontal table and is struck at one end in a direction perpendicular to its length. Determine the motion. What if it be struck at the centre, or at the centre of per- cussion for a rotation-axis through one end of the rod } 2. Two uniform rods of equal length are freely hinged together and placed m a straight line on a smooth horizontal plane. The system is then struck at one end in a direction perpendicular to its length. Examine the motion initially and subsequently,. MOTION OF A FREE BODY. 159 l> Here, the circumstances are a little more complicated than in the preceding problem, so that it is well to form the equa- tions of motion of the two rods separately. Let in be the mass of each rod, 2 a the length, C, C the centres of gravity, v, v' the velocities of translation of C, C\ and o), w' the angular velocities. Then if (9, O' be the instantaneous centres so that CO=a. and C'O' =x', we get cox = 7', (o'x' =v', and (a —x)co = {a -{-x')q)'. And if Q be the blow, and R the react'. ; a the free hinge, the equations of motion of the two rods ar . mv=Q-\-R, ma(o ^ o mv' = R, mao^ = K per- from which it will be found that 6) = 2 ft)', and the initial velocity of the end struck is four times that of the other end. 3. Three uniform and equal rods AB, BC, CD arc arranged as three sides of a square having free hinges at B and C; the end A is struck in the plane of the rods and at right angles to AB by a blow Q. Determine the motion, and show that the initial velocity of A is nineteen times that of D. • ,-S.v i6o RHAD DYNAMICS. 41 ii' This is solved in the same way as the former problem by considering each portion separately. Thus, if /v be the reaction of B, we ha'^e mv =Q + R, maw and 3 tax—v. Q-R, Also, if R' be the reaction at Q R — R' = m{a —x)w = m{a 4-.t-')itely arranged in respect of polarity, to render the instrument astatic. An insulated stud projects from the middle of the lower end of the frame to receive an index that extends nearly to the periphery of the circular base piece and moves over a graduated semicircular scale. An iron point project:, from the insulated stud into a mercury cup in the centre of the base piece, and is in electrical communication with the platinum pointed screws of the current breakers. The current-breaking springs are con- nected with the terminals of the magnet wires, and the magnets are in electrical communication with the wheel-supporting fraine. One of the binding posts is connected by a wire with the irer- cury in the cup, and the other is connected with the stand- ard. A drop of mercury is placed in the cup that contains the agate step to form an electrical connection between the iron cup and the pointed screw. The current breaker is contrived to make and break the current at the proper instant, so that the full e' Jct of the mag- nets is realized, and when the binding posts n connected with four or six Bunsen cells the wheel rotates at iiigh velocity. The wheel will maintain its plane of rotat n, and when it is brought into the plane of the meridian, the index will appear to move slowly over the scale in a direction t ntrary to the earth's rotation, but in reality the earth and the scale with it move from west to east, while the index remains nearly stationary. 90. FesscTs Gyroscope. Another most useful and instructive for of gyroscope is that known as Fessel's, which is represented ni Fig. 71. " (2 is a heavy fixed stand, the vertical shaft of which is a i 'S f ■■ i 1 f 'i 1 ! I : ■ 1 1 f ilii iii 'i' 170 RIGID DYNAMICS. cylinder bored smoothly, in which works a vertical rod CC, as far as possible without friction, carrying at its upper end a small frame BB'. In BB' a horizontal axis works, at right angles to which is a small cylinder /?, with a tightening screw 7/, through which passes a long rod G(7', to one end of which is affixed a large ring AA', and along wliich slides a small cylinder carrying a weight IV, which is capable of being fixed at any point of the rod ; and so that it may act as a counterpoise to the ring, or to the ring and any weight attached to it. An axis AA' works on pivots in the ring, in the same straight line with GG' ; to AA' a disc, or sphere, or cone, or any other body, can be attached, and thus can rotate about AA' as its axis ; to the body thus attached to A A' a rapid rotation can be given, either by means of a string wound round A A' or by a machine contrived for the purpose when A A' and its attached body are applied to it. It is evident that the counterpoise JV can be so adjusted that the centre of gravity of the rod, the ring, the attached body, and the counterpoise, should be in the axis BB' ; or at any point on either side of it ; that is, /i may be positive, or be equal to o, or may be negative. Also by fixing BB' in the arm of CC which carries it, the inclination of the rod GG' to the vertical may be made constant ; that is, may be equal to 6^ throughout the motion. When the counterpoise is so adjusted that the centre of gravity of the rod GG' and its appendages is in CC', then // = o, or, what is equivalent, 7/1/1^ = o." (Price, Calculus ; vol. iv.) It is evident that with such an instrument, with its various re, as I small gles to trough fixed a irrying of the g, or to orks on to AA' ttachcd, dy thus ' means for the ) it. It that the )dy, and ooint on to o, or r' which may be lOut the centre of m h = o, »1. iv.) i various THE GYROSCOPE. 171 adjustments, all the motions about a fixed point can be fully dis- played and examined ; and the results already obtained in the case of the top (Art. (yG) and the gyroscope (Art. 72) thereby shown. 91. Another form of gyroscope worthy of notice is that first constructed by Professor Gustav Magnus of l?crlin, and de- scribed by him in Poggendorff's Annalcn dcr PJiysik uud Clicmic, vol. xci., pp. 295-299. The instrument consists of two rings and discs such as AA\ Fig. 71, connected by a rod sup- ported in much the same way as the rod GQ in Fessel's gyro- scope. There is a binding-screw at B, to arrest, when so desired, motion about the horizontal axis BIV , and also a short rod projecting horizontally from the upper part of the vertical axis CC by which the motion about that axis may be accel- erated, retarded, or completely arrested at will. By means of two cords wound round their axes and simultaneously pulled off, the discs can be put in rapid rotation with nearly equal veloci- ties either in the same or in opposite directions. The follow- ing phenomena are exhibited by this apparatus : If tlie connecting rod be supported midway between the discs, and if the discs be made to rotate rapidly with equal velocities in the same direction, and no weight be suspended at W (Fig. 71), the connecting rod will remain at rest. If a weight be suspended at W, the rod and discs will slowly rotate about the vertical axis CO . If the motion round the vertical axis be accelerated, the loaded end of GG will rise, if the horizontal rotation be retarded, the loaded end will sink. If the binding-screw be tightened so as to arrest this rising or sinking, the rotation about the vertical axis will also cease, to commence again as soon as the binding-screw is loosened. If the discs rotate with equal velocities in opposite directions, the loaded end of GG' will sink. If the connecting-rod be sup- ported at a point nearer to one disc than to the other, and the discs be made to rotate with equal velocities in opposite direc- tions, the instrument will still be found extremely sensitive. 'f- — ii ■ I 4 1 '!) 1 NOTE ON THE PENDULUM AND THE TOP. ^'n> I. In Art. 35, pp. 47 to 49, we have found the equation (//^ + /f'^) ( - ) =2 ^// (cos 6 — cos «), or, as it may be written (sec page 50), H~r) = 2^ (cos c/— cos a) (0 for the oscillations of a rigid body about a fixed horizontal axis, and have applied it to the case of a pendulum making extremely sma!' oscillations. We shall here consider the general case, when the arc of the oscillations is not necessarily small. Let and Differentiating, cos ^ — cos «=(i — cos«) cos^ (/>. .•. I — cos ^ = (1 — cos«) sin^ ^ cos 6 = cos^0 + cos a sin^ ^. (ii) sm (/— - =2 sm 4> cos (6(1 —cos ct)~. :. (i+cos^) de\^ (it = 4(cos^ — ccjsa), \-T.] SubstitAiting in (i), l\-j-\ =^(i-sin2|asin2<^). 172 NOTE ON THE PENDULUM AND THE TOP. 173 (i) (ii) Let and «2=sin*^ a, (iii) and (ii) becomes (iv) \dt) J"* d^ an elliptic integral of the first kind. .'. ^ = am(i//), cos ^ = cn2 (i;/) + cos a ^v?{yi). Equation (ii) may be written in the form sin ^^ = sin?, f; consequently (iv) may be written in the form sin .] = sin \ u sn vt. This equation determines the position of the pendulum at any given instant, and, by inversion, the dmes at which the pendulum is in a given position. If The the period of the pendulum, i.e. the length of time required for the pendulum to make a double swing through the arc 2 a, Integrating and writing ^ for 1^ and sin | « for k, r=27r^(^) 1 1 +(|)2(sin|)2 + (^ll^y (sin \ af {y} ii M ' I I : I I ll'i '74 RIGID DYNAMICS. 2. In Art. 67, p. 118, wo have found the ccpiation ( A sin ^y Y= (cos (9„-cos e)\2AMg/i sin2 9 - C-//2(cos ^„-cos e)\, (n) for the nutation oscillations of a top spinning about a fixed point, and in Art. 68 we have determined the approximate period of small oscillations. The period of oscillations of any magnitude anu the value of d at any given instant may be determined as follows : Let A = Jr(//' + P) = jr/i/, (see page 50) and 2 A Mgh si n^ ^ - C V (^os ^0 - cos (9) = 2 A Mgh (cos e - cos ^i)(cosh 7 - cos (9), which requires that cos ^i + cosh 7 = 2 AMdi and cos ^,. cosh 7=^'^^'"^^^" -I. ^ ' 2 AMo-h Substituting in {a\ that equation becomes sin^^,~j =2^i^(cos^o-cos^)(cos^-cos^i)(cosh7-cos^). (i) Let cos Bq — cos 6= (cos ^^ — cos ^j^ ) cos^t. .-. cos^ — cos^i = (cos^o — cos^i)sin2T, and cos^ = cos^iCosV + cos^ySin2T. (2) Differentiating, — sin ^ — - = 2 sin T cos t (cos 6^ — cos 6,)—- .: ('sin6'^Y=4(cos^,-cos^)(cos^-cos6'jC^ Substituting in ( i ), Hy.) = '] c?" fcosh 7-cos ^1— (cos ^^--cos 6^) sin^rf a <> (n) (2) ^• Let and NOTE ON THL PENDULUM AND THE TOP. ,75 1/1 zi \ I cos 0,) — cos 6^1 . ., 1 = \ ^'•(cosh 7-C0S ^1)1- , J siii-'t ^ i cosh 7 -cos 6^1 I =^(co.h» . 7-co.s»l 0,)} , - -^^^^^^^^ X ,„,v{. . AC" = cosh^^7-C()s^^<9i' /i/2=^'-(cosh2.]7-cos2.](9j). lit) ' »/ii V ( I — K^ sinV V ( I - «"'* sinV) an elliptic integral of the first kind. .'. T=am(i^/), and (2) becomes cos Q = cos 0^^ en- (i^/) + cos d^ sn^ (^ /), (4) thus determining the inclination of the axis of the top to the vertical at any given instant. The period of a complete oscillation will be y._4 ri itr vjo •%/( I — «''^ sin' V( 'r) \\^(cosh-A 7 — eos-^ ^i)y ( - \2'4J I-3-5 ^'i^J'^^^-i (5) Comparing equations (4) and (5) with (iv) and (v), it wi:l be seen that the top's oscillations in nutation are of exactly the same character as the oscillations of an ordinary pendulum. Note, however, that in the discussions of the oseillations of the pendulum, 6 is measured from an initial axis directed straight downwards, while in the discussion of the motion of the top, IMAGE EVALUATION TEST TARGET (MT-S) // /. &J [A 1.0 I.I 11.25 1^ IIM 2: lis ilM 1.8 U 111.6 V] *^./ / Photographic Sciences Corporation L17 \ :\ ^v \ '<«*>. 6^ ^ #V ^ 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 872-4503 '^'- ? «-^<^ <; i- u.. ^ ci^ -ill f 'hi 'M li i 176 RIGID DYNAMICS. is measured from the vertical axis as initial, so that 0, «, an J in the former discussion should be replaced by tt — O, tt — u, and TT — T, to bring it into strict conformity of notation with the discussion of the movements of the top. On makin<; these chanfijes, it will be found that the pendulum oscillating about a fixed horizontal axis is merely the special case of the top in which ^o = 7r — «, d^ = Tr, n = o, and, therefore, 7 = and ^ is constant. Equation (4) enables us to find the value of at any given in.stant /, but to completely determine the position of the top, it is necessary to be able also to find yjr. To do this requires the integration of the equation (d) of Art. 65, p. 1 16, which may be reduced to the integration of two elliptic integrals of the third kind, as follows : A Gin2^ "^f- = Cu (cos e^^ - cos 0). (^) dt'' A \i-\- COS d I— cos (9 cos2 1 <9,, A \i +COS 6^ cos'-^T + cos ^f) sinV sina.V^', I —cos 6^ cos'-T — cos 6q sin'-T C0S2 1^,^ Cu A \\ -|-coso:*:c (^) 1. Find the principal axes of a quadrant of an ellipse at the centre. 2. If a rigid body be referred to three rectanj^ular axes such that A=B and ^{mxy) = o, show that the mean principal moment of inertia = A. 3. Determine the position of a point O in a trianjjjular lamina, such that the moments of inertia of AOll BOC, CO A, about an axis through O, perpendicular to the plane of the lamina, may all be equal. 4. Find the moment of inertia of "^he solid formed by the revolution of the curve r = a{\ +cos^) about the initial line, about a line through the pole perpendicular to the initial line. 5. A uniform wire is bent into the form of a catenary. Find its moments of inertia about its axis, and its directrix. 6. Find the moment of inertia of a paraboloid of revolution about a tangent line at the vertex; the density in any plane perpendicular to the axis varying as the inverse fifth power of its distance from the vertex. 7. Find the moment of inertia of a semi-ellipse cut off by the axis minor about the line joining the focus with the extremity of the axis minor. 8. If the moments of inertia of a rigid body about three axes, passing through a point and mutually at right angles, be equal to one another, show that these axes are on the surface of an N I'J 1/8 RIGID DYNAMICS. . I elliptic cone whose axis is that of least or greatest moment according as the mean moment of inertia is greater or less than the arithmetic mean between the other two. 9. Show that the moment of inertia of l regular octahedron about one of its edges is | cfiM, where a is the length of an edge and M is the mass of the octahedron. 10. Prove that the moment of inertia of a solid regular tetra- hedron about any axis through its centre of inertia is M — , a being the length of an edge. 11. If /3, 7 be the perpendiculars from B and C on a principal axis at the angular point A of the triangle ABC, show that (,,2 _ ^2 _ ^2)(^2 _ ^2) = ^2^2 _,. ^2^2 ^_ 2 (/,2 _ ^2)^^, 12. Show that if a plane figure have the moments of inertia round two lines in it, not perpendicular to one another, equal, a principal axis with respect to the point of intersection bisects the angle between them. 13. Determine the points of an oblate spheroid with respect to which the three principal moments are equal to one another. 14. Show that the conditions which must be satisfied by a given straight line in order that it may, at some point of its length, be a principal axis of a given rigid body, is always satis- fied if the rigid body be a lamina and the straight line be in its plane, unless the straight line pass through the centre of inertia. 15. If a straight line be a principal axis of a rigid body at every point in its length, it must pass through the centre of inertia of body. 16. Assuming that the radius of gyration of a regular poly- gon of ;/ sides about any axis through its centre of inertia and in its own plane, is ^-AMl 2-hCOS 11 ifM; -fcos— yf I- ^°^?)}' where c is the length of any side ; find the radius of gyration MISCELLANEOUS EXAMI'LES. 179 less .. '^ of a circular disc about a line through any point in its circum- ference and perpendicular to its plane, and show thai it is e(|ual to the radius of a circular ring about a tangent. 17. The locus of a point such that the sum of the moments of inertia about the principal axes through the point is constant, is a sphere whose centre is the centre of inertia of the body. 18. li A, B, C be the moments of inertia of a 'oody about principal axes, A cos^ a^ B cos^ ^ •\- C cos^ 7 will be the moment of inertia about any other a.\is passing through the origin and having cos a cos /3 cos 7 for its direction- cosines. 19. If the centre of inertia of a rigid body be the origin, and the principal axes at that point the axes of coordinates, then at an umbilicus of the ellipsoid 1-2 1/2 r.2 + -.'^^^ + = I, A^\ B+\ C+\ two of the principal moments of inertia will be equal. 20. If the density at any point of a right circular cone be proportional to the distance from the exterior surface, show that the radius of gyration about the axis of figure is j-, where a is the radius of the base. 21. Find the moment of inertia of the solid (,l-2 +y ^ .2 _ ,^^^.)2 = ^2( 1-2 ^j,2 + .2) about the axis of .v. Find also the moment of inertia of the surface of this solid. 22. The locus of points at which one of the principal axes passes through a fixed point in one of the principal planes through the centre of inertia, is a circle. 23. If a and /' be the sides of a homogeneous parallelogram, 6 and (/> the inclinations of its principal axes in its own plane, r n I ' ■ I iMJ So RIGID DYNAMICS. through its centre of ineitia, to these sides respectively, show ^^^^ a^ sin 2 ^ = />'^ sin 2 4>- 24. Find the moments of inertia of a uniform circular lamina about its principal axes through a ^ivcii point in its plane. 25. Show that two of the principal moments of inertia with respect to a point in a ri^id body cannot be equa". unless two are equal with respect to the centre of inertia and the point be situated on the axis of unequal moment. 26. Prove that in any rigid body the locus of the point through which one of the principal axes is in a given direction is a rectangular hyperbola whose plane passes through the centre of inertia, and one of whose asymptotes is in the given direction ; unless the given direction be that of one of the prin- cipal axes through the centre of inertia. 27. A series of parabolas are described in one plane having a common vertex A and a common axis, and from a point P in one of them an ordinate PJV is drawn to the axis. Show that if the moment of inertia of the curvilinear area APN about an axis through A perpendicular to the plane of the parabolas be proportional to the area APN, the locus of P is an ellipse. 28. Show that if the momcntal ellipsoid at a point not in one of the principal planes through the centre of inertia be a spheroid, it will at the centre of inertia be a sphere. 29. Find the moment of inertia of a segment of a circle about its chord. 30. Find the moment of inertia of an equilateral triangular lamina about an axis through the centre of inertia and perpen- dicular to the lamina if the density of the lamina at any point varies directly as the distance of the point from the centre of inertia. 31. If A, B, C be the moments of inertia about principal axes through the centre of inertia and «, ^, 7 be the moments of inertia about principal axes through a point P, show that MISCKLLANKOUS EXAMPLES. i8l (I) If {a-^l3-r^)=A + B -C, the locus of /' will be one of the principal planes throu<;h the centre of inertia. (II) If rt + /t^ + 7 be con.stant, the locus of P will be a sphere with centre at centre of inertia. (III) If ( v«4- v^+ V7)(V^+ V7- V«) X ( V7 + V« - Vjg)( \Ja + V^ - V7) be constant, the locus of P will be an ellipsoid similar and simi- larly situated and concentric with the central ellipsoid at the centre of inertia. (IV) If ,8 — y, AC are freely movable in a vertical plane about A, /> and C are connected by an elastic string whose natural length is ccpial to AB. The beams are held in a vertical j)osition and suffered to descend. Determine the motion, the coefflcient of elasticity of the string being equal to four times the weight of either beam. 58. A circular wire is revolving uniformly about its centre fixed. If it be cracked at any point, show that the tendency to break at an angular distance a from the crack is proportional to sin^ a 59. A disc which has a particle of equal mass attached to its circumference, rolls on a rough inclined plane. Determine the motion and the friction in any position of the disc, supposing it ill' 186 KKill) IjVNAMICS. 11. to start lioin ilu- |)()siti<)n in which the particle is in contact with the |)l:inc. 60. A spherical shell whose centn- is fixed contains a roii^^h ball which is held at one extremity of a horizontal diametc-r ot the shell and then allowed to descend. IT the radius of tl' • shell be three times that of the ball, and when the ball is next in instantaneous rest the same point of each is a^ain in contact, the anj;ular velocity of the line joinin^^ their centres is 3\/ ]''' [> beinj; the inclination of this line to the horizon, and (r being till' radius of the ball. 61. A circular rinjjf is suspended with its plane horizontal, by three ecpial vertical inextensible strings attachi;d at ecpial dis- tances to its circumference. If the rinj; be twisted till the strin^^s just meet in a point, and be then left to itself, find its ani^ular velocity when the strin<;s are vertical aj^ain. 62. Two rods /i/>, /)C connected by a hinj;e at B are in motion on a. smooth horizontal plane, the end A beinj; fixed. If initially AB has no angular velocity, that of BC being oy, show that when BC has no angular velocity, that of AB will be — and the ancrlc between the rods will be cos' 3^ ' 2n and 2 6 being the lengths of the rods which are supposed equal in mass. 63. A uniform heavy beam of length 2r is supported in a horizontal position by means of two strings without weight, each of length d, which are fastened to its ends, the other ends of the strings being fixed ; in equilibrium each of the strings makes an angle a with the horizontal. Find the time of a small oscillation when the system is slightly displaced in the vertical plane in which it is situated, the strings not being slackened. 64. A lamina bounded by a cycloid and its base has its centre of inertia at the middle point of its axis. It is placed with its base vertical on a perfectly rough horizontal plane, and allowed contact a rouj^h nictii' <>t IS of the IS lu-xt ill itact, the _i,'sin ft .1 ^ = (i^cos2ft, and laid upon a smooth horizontal table; a tly walks alon<^^ the top of the wire, starting; from one vertex. Show that if the masses of the wire and fly be in the ratio d^://', where /(• is the radius of j^yration of the lemniscate about a vertical axis throuL^h the node, then when the flv has arrived at the node the wire has turned throuj;!! an aii<;le ' 43 66. A uniform circular wire of radius 6y. A uniform strin<^ is stretched alonjjj a smooth inclined l)lane which rests on a smooth horizontal table, l^noui^h of the strinf; han^js over the top of the plane to keep the whole system at rest. If the strint; be j^entlv ])ulled over the plane, and the whole system be then left to itself, investigate the ensuing motion, supposinjij the length of the string to be equal to the height of the plane. 68. Two particles of equal mass are attached to the extremi- ties of a rigid rod without inertia, movable in all directions about its middle point. The rod being set in motion from a given position with given velocity, find equations to determine its sub- uent motion. scq md 69. A rod of length 2 a movable about its lower end is inclinec at an angle a to the vertical, and is given a rotation w about the 1 I il' i i •"■i 1 !t i' ■■^^<•■ !^l ;i 8 ! if i88 RIGID DYNAMICS. vertical. If be its inclination to the vertical when its angular velocity about a horizontal axis is a maximum, show that 3 j^ sin^ tan 6 + 4 im^ sin* a = o. 70. The time of descent, down a rough inclined plane, of a spherical shell which contains a smooth solid sphere of the same material as itself is ^j, the time of descent down the same plane of a solid sphere of the same material and radius as the shell is /g. Determine the thickness of the shell. 71. A heavy chain, flexible, inextensible, homogeneous, and smooth, hangs over a small pulley at the common vertex of two smooth inclined planes. Apply d'Alembert's principle to deter- mine the motion of the chain. /2. A perfectly rough right prism, whose section is a square, is placed with its axis horizonial upon a board of equal mass lying on a smooth horizontal table. A vertical plane containing the centres of inertia of the two is perpendicular to the axis of the prism ; a horizontal blow in this plane communicates motion to the system. Show that the prism will topple over if the momentum of the blow be greater than that acquired by the I 3 TT system falling through a height — tan ~a, where « is a side of 12 o the square section of the prism. 73. Determine the small or.ciliaiions in space of a uniform heavy rod of length 2 a, suspended from a fixed point by an inextensible string of length / fastened to one extremity. Prove that if X be one of the horizontal coordinates of that extremity of the rod to which the string is fastened X = A sin {n^t + a) + B sin {n^t + ^), where ;/j, «2 are the two positive roots of the equation aln^ - (4^ + 3 l)g>fi + T^if = o and A, B, a, ^ are arbitrary constants. hL MISCELLANEOUS EXAMPLES. 189 ts angular lat lane, of a ' the same ime plane he shell is leous, and tex of two le to deter- s a square, ^qual mass containing the axis of ates motion over if the Ted by the is a side of a uniform )oint by an lity. Prove .t extremity 74. The bore of a gun-barrel is formed by the motion of an ellipse whose centre is in the axis of the barrel and plane per- pendicular to that axis, the centre moving along the axis and the ellipse revolving in its own plane with an angular velocity always bearing the same ratio to the linear velocity of its centre. A spheroidal ball fitting the barrel is fired from the gun. If V be the velocity with which the ball would have emerged from the barrel had there been no twist, prove that the velocity of rotation with which it actually emerges in the case supposed is 2 77;/ 7' »n the number of revolutions of the ellipse corresponding to the whole length /of the barrel being ;/, and i' being the radius of gyration of the ball about the axis coinciding with the axis of the barrel, and the gun being supposed to be immovable 75 A plane lamina moving either about a fixed axis or in- stantaneously about a principal axis, impinges on a free inelastic particle in the line through the centre of inertia of the lamina perpendicular to the axis of rotation at the moment of impact. If the velocity of the particle after impact be the maximum velocity, prove that the angular velocity of the lamina will be diminished in the ratio of i : 2. y6. Two equal uniform rods are placed in the form of the letter X on a smooth horizontal plane, the upper and the lower extremities being connected by equal strings. Show that which- ever string be cut the tension of the other will be the same function of the rods, and initially is |^''sin«, where a is the inclination of the rods. yy. An equilateral triangle is suspended from a point by three strings, each equal to one of the sides, attached to its angular points. If one of the strings be cut, show that the tensions of the other two are diminished in the ratio of 36 . 43. ; I ¥ 190 RIGID DYNAMICS. Iln: V. 78. Apply t//r principle of energy to determine the time of a small oscillation of a uniform rod placed in a smooth, f^ed, hemi- spherical bowl, the motion taking place in a vertical plane. 79. A frame formed of four equal uniform rods loosely jointed together at the angular points, so as to form a rhombus, is laid on a smooth horizontal plane and a blow is applied to one of the rods in a direction at right angles to it. Prove that the frame will begin to move as a rigid body provided the middle point of the rod which receives the blow be equidistant from the line of action of the blow and the perpendicular dropped upon the rod from the centre of inertia of the frame. Prove also that in this case the initial angular velocity of the rod which receives the blow is one-eighth of what it would have been had it been unconnected with the remaining rods. 80. Three equal uniform rods AB, EC, CD, freely jointed at B and C, are lying in one straight line on a smooth horizontal table, and an impulse is applied at the midpoint of BC, perpen- dicular to that rod. Find the stresses on the hiT:ges at B and C in any subsequent positions of the rods, and show that when AB, CD are perpendicular to BC, their midpoints are moving in directions which make an angle cos~^(^) with BC. 81. A parallelogram is formed of four rigid uniform rods freely jointed at their extremities. If the parallelogram be laid on a smooth horizontal table and a blow be applied to any one of the rods at right angles to it, and in a direction passing through the intersection of the lines drawn through its extremi- ties parallel to the diagonals, determine the init'al motioi. of the parallelogram. 82. A circular disc is capable of motion about a horizontal tangent which rotates with uniform angular velocity w about a fixed vertical axis through the point of contact. Prove that if the disc be inclined at a constant angle a to the horizontal, w'bma = -1-2. 4.^ 5«* MISCELLANEOUS EXAMPLES. 191 c of a hcmi- ointcd is laid of the frame )oint of line of the rod 83. A uniform rod is rotating with angular velocity a/ [^-M about its centre of inertia, which is at rest at the instant when the rod, being vertical, comes in contact with an inelastic plane inclined to the horizontal at an angle sin~'\/i. The motion being in a vertical plane normal to the inclined plane, prove that the angular velocity of the rod when it leaves the inclined plane is ^{4. V: r 84. If a rigid body which is initially at rest, and which has a point in it fixed, is struck by a given impulsive couple, show that the vis viva generated is greater than that which would have been generated by the same couple if the body had been con- strained to turn about an axis through the fixed point and not coincident with the axis of spontaneous rotation. 85. A and B are two fixed points in the same horizontal line; CD, a heavy uniform rod equal in length to AB, is suspended by four inextensible strings AC, AD, BC, BD, where /^Cis equal in length to BD, and AD to BC. If two of the strings AC, BD be cut, determine the tension oi the other two immediately after cutting, and find the angular velocity of the rod when it reaches its lowest position. 86. A beam AB is fixed at A. At B is fastened an elastic string whose natural length is equal to AP\ the other end of the string is fastened to a jjoint C vertically above A, AC being equal to AB. The beam is held vertically upwards and then displaced. If it come to rest when hanging vertically down- wards, find the greatest pressure on the axis during the motion. 87. Two equal rods AB, BC are connected by a hinge at />. A is fixed and C is in contact with a smooth horizontal plane, the system being capable of motion in a vertical plane. If motion commence when the rods arc inclined at an angle « to the horizon, show that there will be no pressure at the hinge when their inclination Q is given by the equation 3 (sin^ ^ + sin ^)= 2 sin «. If-':! Ifl .11' i 1 '' i liiS' It; ■ l''f >•, 192 RIGID DYNAMICS. 88. A rod AB is movable freely in a vertical plane about A ; to B is fastened an elastic string, the other end being attached to a point C in the vertical plane at such a distance from A that when the rod is held horizontal the tension on the string vanishes. If the rod be now allowed to fall, find the modulus of elasticity of the string that the rod may just reach a vertical position. 89. A prolate spheroid is fixed at one of its poles, and is allowed to fall from its position of unstable equilibrium under the action of gravity only. Find the pressure at the fixed point in any subsequent position. 90. Every particle of two equal uniform rods, each of length 2cT, attracts every other particle according to the law of gravita- tion ; the rods are initially at right angles and are free to move in a plane about their midpoints, which are also their centres of inertia and are coincident. If angular velocities w, to' be com- municated to the rods respectively, show that at the time / the angle 6 between them is given by the equation y =(.-«')H^iog (3-2V2)^i cos--|-sm -+ 1 2 2 e , . d cos--}-sm — I 2 2 91. Two equal spheres of radius a and mass M slxq attached to the extremities of a rigid rod of the same material, whose length is 4^- and section -^^ of a principal section of the sphere. If the rod can move freely about its midpoint and one sphere be struck by a blow P normal to it and the rod, the time which must elapse before the other sphere takes the place of this one is 44 iraM yP 92. A thin uniform rod, one end of w hich is attached to a smooth hinge, ir allowed to fall from a horizontal position. Prove that the stress on the hinge in any given direction is a maximum when the rod is equally inclined to this direction und to the vertical, and the stress perpendicular to this is then MISCELLANEOUS EXAMPLES. 193 A; ;hed that ;hcs. Jg^/Fcosrt, where JFis the weight of the rod and « is the incli- nation of the given direction to the horizontal. 93. A man standing in a swing is set in motion. Supposing that the initial inclination of the swing to the vertical is given, and that the man always crouches when in the highest position, and stands up when in the lowest, find how much the arc of vibration will be increased after ;/ complete oscillations. 94. Three rods are hinged together so as to form an isosceles triangle ABC, A being the vertex. The whole is rotating with angular velocity to round an axis through the middle point of the base and perpendicular to the plane of the rods, when it is suddenly brought to rest. Show that the impulsive action at A bisects the angle BAC, and find its magnitude. 95. A triangular lamina is suspended at rest horizontally by vertical strings attached to its angular points A, B, C. If the strings at B and C be simultaneously cut, show that there will be no instantaneous change of tension in the string at A, if AD be perpendicular to either AB or AC. AD = CD cos ADC, D being the midpoint of BC. 96. A hollow spherical shell is filled with homogeneous fluid which gradually solidifies without alteration of density, the solidification proceeding uniformly from the outer surface, so that the mass of the solidified portion is proportional to the time. If the shell initially rotate about a given axis with a given angular velocity a) projecting over the edge of the table. Prove that the rod will begin to slide over the edge when it has turned through an angle tan"^ ixa^ a^ + 9{c-af 99. If gravity be the only force acting on a body capable of freely turning about a fixed axis and the body be started from its position of stable equilibrium with such a velocity that it may just reach its position of unstable equilibrium, find the time of describing any angle. 100. If an isosceles triangle move, under the action of gravity only, about its base as a fixed axis starting from a horizontal position, show that the greatest pressure on the axis is seven- thirds the weight. loi. If the centre of oscillation of a triangle, suspended from an angular point and oscillating with its plane vertical, lie on the side opposite the point of suspension, show that the angle at the point must be a right angle. 102. A horizontal circular tube of small section and given mass is freely movable about a vertical axis through its centre. A heavy particle within the tube is projected along it with a given velocity. Given the coefficient of friction between the tube and the particle, determine the terminal velocity of both, and Lhe time which must elapse before that motion is attained. 103. Part of a heavy chain is coiled round a cylinder freely movable about its axis of figure which is horizontal, and the remainder hangs vertically. Determine the motion, supposing the system to start from rest and neglecting the thickness of the chain. 104. Two weights arc connected by a fine chain which passes over a wheel free to rotate about its centre in a vertical plane. miscp:llani:ous examples. 195 axis Given the coefficient of friction between the string and the wheel, find the condition which determines whether the string will slide over the \\heel or will not slide. 105. Two straight equal and uniform rods are connected at their ends by fine strings of equal length a so as to form a par- allelogram. One rod is supported at its centre by a fixed axis about which it can turn freely, this axis being perpendicular to the plane of motion, which is vertical. Show that the middle point of the lower rod will oscillate in the same way as a simple pendulum of length a, and that the angular motion of the rods is independent of this oscillation. 106. A loaded cannon is suspended from a fixed horizontal axis, and rests with its axis horizontal and perpendicular to the fixed axis, the supporting ropes being equally inclined to the vertical. If v be the initial velocity of the ball whose mass is — of the weight of the cannon, and // the distance between the axis of the cannon and the fixed axis of support, show that when the cannon is fired off the tension of each rope is imme- diately changed in the ratio v^ -f ii^gh : 7i{n -f \)g/i. 107. Two equal triangles ABC, A'B'C, right-angled at C and C, rotate about their equal sides CA and A'C as fixed axes in the same horizontal straight line. The distance CC is less than the sum of the sides CA, A'C. The triangles, being at first placed horizontally, impinge on one another when vertical. Determine the initial subsequent motion and discuss the case in which A A' is less than one-fifth CC. :o8. Find the envelope of all the axes of suspension that lie in a principal plane through the centre of inertia of a rigid body, and such that the length of the simple pendulum may be always twice the radius of gyration of the body about one of the axes lying in the plane. 109. A flat board bounded by two equal parabolas with their axis and foci coincident, and their concavities turned towards 196 RIGID DYNAMICS. ■tn III ;„ti' ii li ' i;i ■' ■■'i m each other, is capable of niovinj; about the tangent at one of the vertices. Find the centre of percussion. 1 10. A uniform beam capable of motion about its middle point is in equilibrium in a horizontal position ; a perfectly elastic ball, whose mass is one-fourth that of the beam, is dropped upon one extremity and is afterwards struck by the other extremity of the beam. Prove that the height from which the ball was dropped was |g(2« + i)7r x length of beam. 111. Two equal circular discs are attached, each by a point in its circumference, to a horizontal axis, one of them in the plane of the axis and the other perpendicular to it, and each is struck by a horizontal blow which, without creating any shock on the axis, makes the disc revolve through 90°. Show that the two blows are as V6 : V5. 112. A rigid body capable of rotation about a fixed axis is struck by a blow so that the axis sustains no impulse. Prove that the axis must be a principal axis of the body at the point where it is met by the perpendicular let fall on it from the point of application of the blow. 113. A uniform rod AB of mass A/ is freely movable about its extremity A, which is fixed; at C, a point such that AC is horizontal and equal to AB, a smooth peg is fixed over which passes an inelastic string fastened to the rod at B, and to a body also of mass M which is supported in a position also below C. If the rod be allowed to fall from coincidence with AC, and the string be of such a length as not to become tight until the rod is vertical, the angular velocity of the rod will be suddenly diminished by three-fifths. 114. A piece of wire is bent into the form of an isosceles triangle and revolves about an axis through its vertex perpen- dicular to its plane. Find the centre of oscillation and show that it will lie in the base when the triangle is equilateral. MISCELLANEOUS EXAMPLES. 197 115. A circular lamina performs small oscillations (i) about a tangent line at a given point of its circumference, (2) about a line through the same point perpendicular to its plane. Compare the times of oscillation. 116. A uniform beam is drawn over the edge of a rough hori- zontal table so that only one-third of its length is in contact with the table ; and it is then abandoned to the action of gravity. Show that it will begin to slide Ovcr the edge of the table when it has turned through an angle equal to tan"' -, ft being the coefficient of friction between the beam and the table. 117. A uniform beam AB, capable of motion about A, is in equilibrium. Find the point at which a blow must be applied in order that the impulse at A may be one-eighth of the blow. 118. A rectangle is struck by an impulse perpendicular to its plane. Determine the axis about which it will begin to revolve, and the position of this axis with reference to an ellipse inscribed in the rectangle. 119. A rectangle rotates about one side as a fixed axis. F'ind the pressure on the axis (i) when horizontal, (2) when inclined to the horizontal. 120. About what fixed axis will a given ellipsoid oscillate in the shortest possible time ? 121. A uniform semicircular lamina rotates about a fixed hori- zontal axis through its centre in its plane. Determine the stresses on this axis. 122. If 7"] and T^ are the times of a small oscillation of a rigid body, acted on only by gravity, about parallel axes which are distant a-^ and a^ respectively from the centre of inertia, and T^be the time of a small oscillation for a simple pendulum of length rt-j -f a^y then will {a^ — ct^T"^ = a^ T^ — a^ T^. 123. A uniform beam of mass ;;/, capable of motion about its middle point, has attached to its extremities by strings, each of 198 RIGID DYNAMICS. 'i 'V i ! I' IJ ,1 (if P1 5Mi IcMf^th /, two particles, each of mass/, which haiif; freely. When the beam is in ec|uilibriiim, inclined at an an^le « to the vertical, one of the strings is cut; jjrove that the initial tension of the other string is --.-iv-, and that the radius of curvature of w -\- T,/> sur « . . „ . 9//siiv'« the initial ])ath of the i)article is . ' ■ /;/ cos « 124. A uniform inelastic beam capable of revolving about its centre of inertia, in a vertical plane, is inclined at an angle a to the horizontal, and a heavy particle is let fall ujjon it from a. point in the horizontal plane through the upper extremity of the beam. Find the position of this point in order that the angular velocity generated may be a maximum. 125. A uniform elliptic board swings about a horizontal axis at right angles to the plane of the board and passing through one focus. Prove that if the cxcentricity of the ellipse be V|, the centre of oscillation will be the other focus. 126. A circular ring hangs in a vertical plane on two pegs. If one peg be removed, prove that, P^ P.,, being the instanta- neous pressures on the other peg calculated on the supposition that the ring is (i) smooth, (2) rough, Pj^ ; p^^ : : i : i -|-| tan^w, where « is the angle which the line drawn from the centre of the ring to the centre of the peg makes with the vertical. 127. A uniform beam can rotate about a horizontal axis so placed that a ball of weight equal to that of the beam, resting on one end of the beam, keeps it horizontal. A blow, perpendicular to the length of the beam, is struck at the other end. Investigate the action between the ball and the beam, and the stress on the axis. 128. There are two equal rods connected by a smooth joint; the other extremity of one of the rods can move about a fixed point, and that of the second along a smooth horizontal axis passing through the fixed point, and about which the system is MISCLLLANEOUS KXAMl'LES. 199 v^. revolving under the action of gravity. Find a differential e(|ua- tion to determine the inclination of the rods to the axis at any time. 129. An elliptic lamina whose e.xcentricity is J, Vio is sup- ported with its plane vertical and transverse axis horizontal by two smooth, weightless pins passing through its foci. If one of the pins be suddenly released, show that the pressure on the other pin will be initially unaltered. 130. A plane lamina in the form of a circular sector whose angle is 2 «, is .-n.spended from a horizontal axis through its centre, perpendicular to its plane. Find the time of a small oscillation, and show that if 3« =4sin« the time of oscillation will be the same about a horizontal axis through the extremity of the radius passing through the centre of inertia of the lamina. 131. A hollow cylinder open at both ends, of which the height is to the radius as 3 to V2, has a diameter of one of its ends fixed. Show that the centres of percussion lie on a straight line the distance of which from the fixed axis is eight-ninths of the height of the cylinder. 132. A lamina ABCD is movable aoout AB as a fixed axis. Show that if CD be parallel to AB and AR^=t, CD'^, the centre of percussion will be at the intersection of AC wnd BD. 133. In the case of a rigid body freely rotating about a fixed axis, show that in order that a centre of percussion may exist the axis must be a principal axis with respect to some point in its length. 134. A uniform rod movable about one end, moves in such a manner as to make always nearly the same angle a with the vertical. Show that the time of its small oscillations is Vr 2(1 cos a ) Is.di+Scos'-^'Oi' a being the length of the rod. 1 1 i'l 200 KKJID DYNAMICS. ( || >i .'I I 1 1 . tail m r' HI I 4 135. One end of :i heavy uniforn! rod slides freely on a fine smooth wire in the lorni ol an ellipse of excentrieity - •', and 2 axis minor equal to the lenj^th of the rod ; the other end of the rod slides on a smooth wire coinciding with the axis minor ot the ellipse. The system is set rotatinj^ about the latter wire, which is fixed in a vertical position. Prove that if be the inclination of the rod to the vertical at the time /, u the initial value of 0, and w the initial an^adar velocity of the system about the vertical axis, cos ^ = cos « cos (w/ sin«), 136. A lamina in the form of an ecpiilateral trianj^le rests with its base on a horizontal plane, and is capable of moving; iti a vertical plane about a hin^^e at one extremity of its base. Prove that it will turn completely over if it be struck at its vertex a blow greater than 2mi-^( ' , j in a direction perpen- dicular to that side 'vhich docs not pass through the hinge, w being the mass, tan ^{\ tan B), where />' is the semi-vertical angle of the cone described by the rod. 144. A rigid body, fixed at one point only, is in motion under the action of finite forces. If, throughout the motion, the angular acceleration of the body about the instantaneous axis bear to the moment of inertia about this axis and to the forces acting on the body the same relation as if the axes were fixed, prove that if the three principal moments of inertia at the fixed point be not all equal the locus of the axis relatively to the body is a cone of the second order. 145. A triangular lamina AJ^C has the angular point C fixed, and is capable of free motion about it. A blow is struck at B, I If i I' I M I RIGID DYNAMICS. perpendicular to the plane of the lamina. Show that the instan- taneous axis passes through one of the points of trisection of the side A J). 146. Two equal uniform rods are capable of motion about a common extremity which is fixed, their upper ends bei \g joined by an elastic string. They are set in vibration about a vertical axis bisecting the angle between them. If in the position of steady motion the natural length (2/) of the string be doubled, the modulus of elasticity being equal to the weight of either rod, then the angular velocity about the vertical will be nI{^€^1' where /i is the height of the string above the fixed extremity. 147. A rigid body, of which two of the principal moments at the centre of inertia are equal, rotates about a third principal axis, but this axis is constrained to describe uniformly a fixed right circular cone of which the centre of inertia is the vertex. Prove that the resultant angular velocity of the body is con- stant, that the requisite constraining couple is of constant mag- nitude, and that the plane or the couple turns uniformly in the body about the axis of unequal moment. 148. An ellipsoid is rotating with its centre fixed about one of its principal axes (that of x) and receives a normal blow at a point {/i, ky I). If the initial axis of rotation after the blow lie in the principal plane oi ya, its equation is ^2(^2 -f r2)(rt2 _ b'^)iy + ^2(^2 + ^2)(^2 _ ^2y. _ q 149. A sphere whose centre is fixed has an elastic string attached to one point, the other end of the string being fastened to a fixed point. To the sphere is given an angular velocity about an axis. Give the equations for determining it: motion, the string being supposed stretched and no part of it in contact with the surface of the sphere. If the natural length of the A 111 !! i ;' he instan- ion of the 1 about a . ig joined a vertical losition of i doubled, of either ic :remity. loments at I principal nly a fixed the vertex. »dy is con- istant mag- mly in the about one il blow at a he blow lie astic string ing fastened dar velocity it: motion, it in contact ingth of the I I MISCELLANEOUS EXAMPLES. 203 String be equal to a, the radius o" the sphere, and it be fi.xed at a point (9 at a distance = <'?(V2 — i) from its centre, and if the sphere be turned so that the point on it to which the string is fastened may be at the opposite extremity of the diametor through O, prove that the time of a complete revolution a ^ V5 f'g- 7r + 2V2 where ;, ^ moclulus of elasticity ^ UaM\ weight of sphere \ vs M / where fi — modulus of elasticity. 3 7r + 2V2 150. If the angular velocities of a rigid body, at any time /, about the axes a; j', c, are proportional respectively to cot(w — ?/)/, cot(;/ — /)/, cot(/— w)/, determine the locus of the instantaneous axis. 151. A uniform rod of length 2 a can turn freely about one extremity. In its initial position it makes an angle of 90° with the vertical and is projected horizontally with an angular ve- locity ft). Show that the least angle it makes with the vertical is given by the equation 4 aco'^ cos 6 = T,g sin'^ 6. 152. A rigid body rotates about a fixed point under the action of no forces. Investigate the following equations, the invariable line being taken as the axis of ^ : -7- = — 6" sin ^ sin (i cos be constant, yjr will increase uniformly, and tan 6 — c" tan ^q, where ^=^„, &>„ and those about a system fixed in the body be = o, h, k being the radii of gyration about the principal axes. Show uiSSiS 212 KIGIU DVNAiMlCS. 'I r,i':.! that if nh lie on a certain straight lino, there will be no impulse on the fixed point. 190. A uniform rod of lenjrth 2n and mass ;;/, capable of free rotation about one end, is held in a horizontal position, and on it is placed a smooth particle of mass / at a distance c from the point, r being < ■; the rod is then let go. Find the initial pres- 3 sure of the particle on the rod, and sho-., that the radius of curvature of the particle's path is 191. A lamina in the form of a symmetrical portion of the curve r=a{mr'^ — 6'^) is placed on a smooth plane with its a.xis vertical, then infinitesimally displaced and allowed to fall in its own plane. If the lamina be loaded so that its centre of inertia is at the pole and its radius of gyration = 2 a, find the time in which its axis will fall from one given angular position to another. 192. An elliptical lamina stands on a perfectly rough inclined plane. Find the condition that its equilibrium may be stable, and determine the time of a small oscillation. 193. A perfectly rough plane, inclined at a fixed angle to the vertical, rotates about the vertical with uniform angular velocity. Show that the path of a sphere placed upon the plane is given by two linear differential equations of the form. dh' dfi . dx , „ d\x .,dv , „, ^ the origin being the point where the vertical line, about which the plane revolves, meets the plane ; the axis of y being the straight line in the plane which is always horizontal. 194. The equal uniform beams AB, EC, CD, DE, are con- nected by smooth hinges and placed at rest on a smooth hori- zontal plane, each beam at right angles to the two adjacent, so as to form a figure resembling a set of steps. An impulse MISCKLI.ANKOUS KXAMIM.F.S. 213 I pulse ing the is given at the end //, along .//)'; determine the impulsive action on any hinge. 195. A rectangle is formed of four uniform rods of lengths 2(1 and 2/; respectively, which are connected by smooth hinges at their ends. The rectangle is revolving about its centre on a smooth horizontal plane with an angular velocity o), when a point, in one of the sides of length 2 a, suddenly becomes fixed. Show that the angular velocity of the sides of length 2 /> inimedi- ately becomes jfo. Find, also, the change in the angular 6(1 + 4 d velocity of the other sides and the impulse at the point which becomes fixed. 196. A uniform revolving rod, the centre of inertia of which is initially at rest, moves in a plane under the action of a con- stant force in the direction of its length. Prove that the square of the radius of curvature of the path of the rod's centre of inertia varies as the versed sine of the angle through which the rod has revolved at the end of any time from the beginning of the motion. 197. Six equal uniform rods arc freely joined together and are at rest in the form of a regular hexagon on a smooth hori- zontal plane. One of the rods receives an impulse at its mid- point, perpendicularly to its length, and in the plane of the hexagon. Prove that the initial velocity of the rod struck is ten times that of the rod opposite to it. 198. A uniform rod of length 2 a lies on a rough horizontal plane, and a force is applied to it in that plane and perpendicu- larly to its length at a distance 7' from its midpoint, the force being the smallest that will move the rod. Show that the rod begins to turn about a point distant -\/\(i^-\-/>'^)—p from the midpoint. 199. AB is a rod whose end A is fixed and which has an equal rod 7)C attached at B. Initially the rods AB, BC are in the same straight line, AB being at rest and BC on a smooth 'V'l I ! I i !i ; I, i ;i < ;i I ii 214 RI(;iI) DYNAMICS. horizontal plane having an anj^ular velocity fo. Show that the greatest an^^le between the rods at any suhse{|uent time is cos"' j''^ and that when they are a^iiin in a straight line, Ihcir angular velocities are r^ and — -g- respectively. 200. A rectangular board moving uniformly without rotation in a direction parallel to one side, on a smooth horizontal plane, comes in contact with a smooth fixed obstacle. Determine at what point the impact should take place in order that the angular velocity generated may be a minimum. 201. b'our ecpial uniform rods, freely jointed at their extremi- ties, are lying in the form of a scpiare on a smooth horizontal table, when a blow is applied at one of the angles in a direction bisecting the angle. Find the initial state of motion of each rod, and prove that during the subsequent motion the angular veloc- ity will be uniform. 202. A sjihere is moving at a given moment on an imperfectly rough horizontal table with a velocity ?-, and at the same time has an angular vtilocity w round a horizontal diameter, the angle between the direction of v and the axis of w being «. Prove that the centre of the sphere will describe a parabola if rt'/'W -f (a^ — P)vQ) sin <■ = ai'^. 203. Two rods, OA and OB, are fixed in the same vertical plane, with the point O upwards, the rods being at the same angle « to the vertical. The ends of a rod AB of length 2 a slide on them. Show that if the centre of inertia of AB be its middle point, and the radius of gyration about it be /', the time of a complete small oscillation is Mrt^tan^< -f X-^) \ i (7 if cot a i 204. One end of a heavy rod rests on a horizontal plane and against the foot of a vertical wall ; the other end rests against a parallel vertical wall, all the surfaces being smooth. Show that, if the rod slijD down, the angle <}>, through which it will turn tl h e( a: NflSCKI.I.ANI'OrS KXAMI'I.KS. '5 lat the itnc is I, their [)tiiti()n pUmc, line ut lat the xtremi- rizonlal ircction ich rod, r vcloc- crfcctly nc time ic an<;lc Prove vertical "le same igth 2 a B be its the time iane and igainst a low that, will turn round the common normal to the vertical walls, will he j;iven by thi' eciualion '—r-rl i + "^cos'-A)-!- , .. ,., sin d> = C, where 2ii is tlu leiif^lh of the rod and 2 b the distance betwecti the walls 205. Two e(|ual uniform rods, loosely jointed together, are at rest in one line on a smooth horizontal table, when one of them receives a horizontal blow at a {^iven jjoint. Determine the ini- tial circumstance of the motion, and prove that, when next the rods are in u straight line, they will have interchanged angular velocities. 206. One end of a uniform rod of wei;;ht w can slide by a smooth ring on a vertical rod, the other entl sliding on a smooth horizontal plane. The rod descends from a position inclined at an angle fi to the horizon. Show that the rod will not leave the horizontal i)lane during the descent, but that its maximum pressure against it is J7ccos''*/i and that its ultimate pressure is \ w. 207. A lamina capable of free rotation about a given point in its own plane, which point is fixed in space, moves under the action of given forces. If the initial axis of rotation of the lamina coincide very nearly with the axis of greatest moment of inertia in the plane of the lamina, the angular velocities about the other principal axes will be in a constant ratio during the motion. 208. A sphere of radius a is partly rolling and partly sliding on a rough horizontal plane. Show that the angle the direction of friction makes with the axis ot .v is tan~' , // and ■:> V — (la).^ being the initial velocities, w^, w,^ the initial angular velocities. 209. A perfectly rough circular cylinder is fixed with its axis horizontal. A sphere is placed on it in a jiosition of unstable equilibrium, and projected with a given velocity parallel to the axis of the cylinder. If the sphere be slightly disturbed in a I' BHB ill ! I ^' .:1 l;;M l*>i t l»^[ 111 '' 2l6 RIGID DYNAMICS. horizontal direction perpendicular to the direction of the axis of the cylinder, determine at what point it will leave the cylinder. 2 i D. A parabolic lamina, cut off by a chord perpendicular to its axis, is kept at rest in a horizontal position by three vertical strings fastened to the vertex and two extremities of the chord ; if the string which is fastened to the vertex be cut, the tension of the others is suddenly decreased one-half. 211. Three equal, perfectly rough, inelastic spheres are in contact on a horizontal plane; a fourth equal sphere, which is rotating about its vertical diameter, drops from a given height and impinges on them simultaneously. Investigate the subse- quent motion. 212. A rod of length (7, moving w'th a velocity v perpendicu- lar to its length on a smooth horizontal plane, impinges on an inelastic obstacle at a distance c from its centre. Show that its angular velocity when the end quits the obstacle is 3^/c a' 213. A solid regular tetrahedron is placed with one edge on a smooth horizontal table and is allowed to fall from its position of unstable equilibrium. Find the angular velocity of the tetra- hedron just before a face of it reaches the table, and the magni- tude of the resultant impulsive blow. 214. A uniform sphere of radius a, when placed upon two parallel, imperfectly rough, horizontal bars, has its centre at a height ^ above the horizontal plane which contains the bars. It is started with a velocity 7> parallel to the bars, and an angular velocity to about a horizontal axis perpendicular to the bars in such a direction as to be diminished by friction. In the case in which 2a^D. > sdv, the sphere will begin to roll after a time yw^(2 rtM- 5 ^^y where fi is the coefficient of friction. What will at that instant be the velocity and position of the sphere ? 3 axis of ylindcr. licukir to 2 vertical ic chord ; 2 tension 2S are in which is m height ;he subse- ;rpendicu- ges on an )\v that its /c 2 ■ le edge on ts position [ the tetra- the magni- iipon two :entre at a le bars. It an angular :he bars in the case in a time that instant MISCELLANEOUS EXAiMl'LES. 217 215. A heavy uniform rod slips down with its extremities in contact with a smooth horizontal floor and a smooth vertical wall, not being initially in a plane perpendicular to both wall and floor. Prove that if 6 be the inclination to the horizon and (j) the angle which the projection of the rod on the floor makes with the normal to the wall, (/1-2 + ..2)sin(^ ^^'^"^^ ^,""^ ^^ = /C-2cos0 ^'(^°^ ^ ^^" ^\ and (/i'2 + rt2)cos^sin at right angles to the string. Prove that the greatest angle which the string makes with the rod is 2 sm" 12^ and that the angular velocity at the instant is 7' a-\-b 245. A rough sphere is projected on a rough horizontal plane and moves under an acceleration tending to a point in the plane and varying as the distance from that point. Show that the centre of the sphere will describe an ellipse, and find its com- ponent angular velocities in terms of the time. 246. Three equal uniform rods, AB, BC, CD, freely jointed together at B and C, are lying in a straight line on a sraooth 'j; circle il plane nt of it he hori- id mass be in )port be he com- um at a ;entrc of the new le point, )ola attached rod and :al plane, angles to ig makes tal plane he plane that the its com- Y jointed L smooth MISCELLANEOUS EXAMPLES. horizontal plane and a given impulse is applied at the midpoint of BC at right angles to BC. Determim; the velocity of BC when each of the other rods makes an angle with it, and prove that the directions of the stresses at B and C make with BC angles equal to tan"^(| tan 0). 247. Three equal uniform straight lines, AB, BC, CD, freely jointed together at B and C, are placed in a straight line on a smooth horizontal plane and one of the outside rods receives a given impulse in a direction perpendicular to its length at its midpoint. Compare the subsequent stresses on the hinges with the impulse given to the rod. 248. A homogeneous right circular cylinder of radius a, rotating with angular velocity w about its axis, is placed with its axis horizontal on a rough inclined plane so that its rotation tends to move it up the plane. If u be the inclination of the plane to the horizontal and tan « the coefficient of friction, show that the axis of the cylinder will remain stationary during a period T = aw 2 s: sm a and that its angular velocity at any time / during this period is equal to w — ."''' a 249. A hoop is hung upon a horizontal cylinder of given radius. Determine the time of a small oscillation I. When the cylinder is rough. II. When the cylinder is smooth. 250. Prove the following equations for determining the mo- tion of a rigid bod" whose principal moments of inertia "t the Cjntre of inertia are equal : X dn L dw. — = -j- vB^ + TC'^o, etc., -^ = —1 - ©./g -f 0)3^2, etc. ; Cj- at A at It, V, 10 being the velocities of the centre of inertia parallel to the three axes moving in space, w^, co.,, tUg the angular velocities about these axes, 6^ 6,^, 6.. the angular velocities of these axes about fixed axes instantaneously coincident with them, X, Y, Z 224 RIGID DYNAMICS. ' . »r 1i ^1 ! ' the resolved forces, L, M, N their moments about the axes, G the mass of the body, and A its moment of inertia about any axis through the centre of inertia. 251. A uniform rod of mass ;// and length 2 a has attached to it a particle of mass / by means of a string of length /; the rod and string aie placed in one straight line on a smooth horizontal plane, and the pn.rticlc is projected with a velocity V at right angles to the string. Prove, then, when the rod and string, make angles B, cf) with their initial positions, } ^/e P + al? cos(<^ - 6) \ y +\d^ + ad cos(<^ 0) (it about a hori- zontal axis, and the direction of the former make an angle a with the axis of the latter, show that the angle through which the direction of motion of the centre has turned, when perfect rolling begins, is _i 2 aw cos a tan' $v — 2 am sin a 253. If a homogeneous sphere roll on a perfectly rough plane under the action of any forces whatever, of which the resultant passes through the centre of the sphere, the motion of the centre of inertia will be the same as if the plane were smooth and all the forces were reduced in a certain constant ratio ; and the plane is the only surface which possesses this property. 254. A smooth ring of mass ;;/ slides on a uniform rod of mass M. Determine the velocity of the ring at any point of the rod which it reaches, no impressed forces being supposed to act. MISCELLANEOUS EXAMl'LES. 225 L' axes, )iit Liny ttachcd nj^th /; smooth velocity the rod r -t- d)v, ^rizontal e initial ; a hori- angle a h which 1 perfect ^h plane resultant le centre and all and the 1 rod of nt of the ;d to act. If when the ring is distant c from the centre of the rod, the angle at which its path is inclined to the instantaneous position of the rod be greater than cot"M 2 -f ■{-' v/c- I ,.,i » show that it will never reach the centre of the rod, /c'^ being the radius of gyration of the rod about its centre. 255. A uniform rod of weight W^ and length 2^ is supported in a horizontal position by two rtne vertical threads, each of length c, and each is attached at a distance c from the centre of the rod. The rod is slightly dis' ' d by the action of a horizontal couple whose moment is ./K, and which does not move the centre of the rod out of a vertical line. Show that the time of a small oscillation of the rod will be 256. A circular lamina, rotating about an axis through the centre perpendicular to its plane, is placed in an inclined posi- tion on a smooth horizontal plane. Give a general explanation of the motion deduced from dynamical principles, and show that under certain circumstances the la* ^ina will never fall to the ground, but that its centre will perform vertical oscillations, the time of an oscillation bein'r TT / 1 4- 4 cos^ a\h V 2 U)' a sm a a being the inclination of the lamina to the horizon at first, a its radius, and to its angular velocity. 257. A beam rests with one end on a smooth horizontal plane, and has the other suspended from a point above the plane by a weightless, inextensible string ; the beam is slightly displaced in the plane of beam and string. Find the time of a small oscil- lation. Q 1 * 1 III tr' 226 RICAD DYNAMICS. I u .11 .11 .iiif [/' ifll 1 258. Find the condition that :i free rigid body in motion may be reduced to rest by a sin<;le blow. 259. A perfectly roii<;h horizontal plane is made to rotate with constant an<.;ular velocity about a vertical axis which meets the jjlane in 0. A sphere is projected on the plane at a point P no that the centre of the sphere has initially the same velocity in direction and magnitude as if the sphere had been placed freely on the plane at a point Q. Show that the sphere's centre will describe a circle of radius OQ, and whose centre K is such that O/'^ is parallel and equal to OP. 260. If a free rigid body be struck with a given impulse, and any point of the body be initially at rest after the blow, show that a line of points will also be at rest, and determine the con- dition that this may be the case in a body previously at rest. 261. A free rigid body of mass /// is at rest, its moments of inertia about the principal axes through its centre of inertia being A, B, C. Supposing the body to be struck with an impulse R through its centre of inertia, and with an impulsive couple G, prove that it will revolve for an instant about an axis whose velocity is in the direction of its length and equal to LX ^ MY XZ Am Bin Cm A^ B' cy X, V, Z being the components of R, and Z, M, N the com- ponents of G, in the principal planes. 262. A sphere with a sphere within it, the diameter of the latter being equal to the radius of the former, is placed on a perfectly rough inclined plane, with the centre of inertia at its shortest distance from the plane, and is then left to itself. Find the angular velocity of the body when it has rolled round just once, and determine the pressure then upon the plane. It IIP MISCKLLANKOUS EXAMI'LKS. 227 )n may rotate li meets (a point velocity placed f> centre is such Ise, and vv, show ;hc con- rest. lents of • inertia impulse 3uple G, s whose he com- • of the ed on a ia at its f. Find md just 263. Two equal rods of the same material are connected by a free joint and placed in one straij^ht line on a smooth horizontal plane; one of them is struck perpendicularly to its len<;th at its extremity remote from the other rod. Show that the linear velocity communicated to its centre of inertia is one-fourth greater than that which would have been communicated to it by a similar blow if the rod had been free. In the subsequent motion show that the minimum angle which the rods make with one another is cos ' /j. 264. AB, BC, CD are three equal uniform rods lying in a straight line on a smooth horizontal plane, and freely jointed at /) and C ; a blow is applied at the midpoint of BC. Show that if ft) be the initial angular velocity of AB or CD, 6 the angle which they make with BC at time t, dO ^ ft) dt V^ I + sin'-^ e) 265. A lamina of any form lying on a smooth horizontal plane is struck a horizontal blow. Determine the point about which it will begin to turn, and prove that if c, c^ be the dis- tances from the centre of inertia of the lamina of this point and of the line of action of the blow respectively, rr' =/'^ where k is the radius of gyration of the lamina about the vertical line through its centre of inertia. 266. A circular lamina whose surface is rough is capable of revolution about a vertical axis through its centre perpendicular to its plane, and a particle whose mass is equal to that of the lamina is attached to the axis by an inelastic string and rests on the lamina. If the lamina be struck a blow in its own plane, determine the motion. 267. A bicycle whose wheels are equal and body horizontal is proceeding steadily along a level rough road. Obtain equa- tions for determining the instantaneous impulses on the machine when the front wheel is suddenly turned through a horizontal angle Q. 1: I r }' (i »J 1(1' I' 228 RICH) DYNAMICS. I 'i.i 1 '1 ■ 1 > ^^H| ^^1 I EH |li|Hi ^if 1 Show that the initial hoii/oiital angular velocity is pro[)()r- tional to the original velocity. 268. The radii of the portions of a horizontal differential axle of weight [Fare ti and /', and their lengths are /f and d. The suspended weight is also [/'. If the balancing power be re- moved and the weight be allowed to fall, show that in time it will fall through Or-/>f -A"^- 3(rt — />f + 2ab' 269. Show how to determine the angular velocities of a rotating mass by observations of the instantaneous direction cosines of points on its surface referred to three fixed rectangu- lar axes, and their time rates of increase ; i.e. — , etc. How dt many such observations are necessary ? 270. A sphere composed of an infinite number of infinitely thin concentric shells is rotating about a common axis under no forces. Assuming that the friction of any shell on the consecu- tive external one at any point varies as the square of the angular velocity and the distance of the point from the axis, obtain the equation /v- — rw , = 20)^ for the angular velocity at any time of shell of radius r, and show that the solution of this equation is r^a =/[ — + M' where/ is an arbitrary function. 271. An Q,g^ with its axis horizontal is rolling steadily round a rough vertical cone of semi-vertical angle «. The shape, weight, moment of inertia, etc., of the egg being known, find the friction acting, and the time of completing a circuit. 272. A vertical, double, elastic, wire helix is rigidly attached at one end to a horizontal bar, mass J/, and is constrained io retain the same radius a. When in equilibrium the tangent angle is a. An additional weight MgQ, or a torsion couple MgaQ^ can alter M into a-\- 6. If the bar be depressed, and consequently turned Misc'KLLANKOUS KXAMl'I.KS. 229 ial axle The he re- time it through ail angle, show that the time of a small oscillation will ; (cos a -f- sin a) . , where / and 2 fid are the lengths ot 'A'' 3 ' the helix and the bar respectively. 273. Four equal, smooth, inelastic, circular discs of radius d are p'aced in one plane with their centres at the four corners of a square of which each side =2 a. They attract one another with a force varying as the distance. A blow is given to one of them in the line of one of the diagonals of the square. Investigate the whole of the subsequent motion. 274. P and Q are two points in a uniform rod equidistant from its centre. The rod can move freely about a hinge at /'. The hinge is constrained to move up and down in a vertical line. If the motion be such that Q moves in a horizontal line, determine the velocity when the rod has any given inclination, the rod being supposed t start from rest in a horizontal position. In the ca.se in which the whole length of the rod =PQ^ I, show that the time of a comjilete oscillation is (2 7r)^(r |)"'f 275. A circular and a semicircular lamina of equal radii a are made of the same material, which is perfectly rough. Their centres are joined by a tight inelastic cord; also the centre of the circular lamina is joined to the highest point of the semi- circular lamina by a string of length ^j:V3. The semicircular lamina stands with its base on a perfectly rough, inelastic plane. The circular lamina rests on the top of the semicircular lamina and in the same vertical plane with it. It is disturbed from its position of equilibrium. Prove that just after it has struck the plane its angular velocity = — \\]- 276. A uniform rod, capable of free motion about one extrem- ity, has a particle attached to it at the other extremity by means of a string of length / and the system is abandoned freely to the action of gravity when the rod is inclined at an angle « to !30 RIGID DYNAMICS. the horizon and the strinir is vertical. Prove that the radius of i u I y I il: 't-5 curvature of the particle's initial path is 9/ )n-\-2p cos''^ a m sin«(2 — 3sin-«)' m and/ being the masses of ti.e rod and particle respectively. 277. To a smooth horizontal plane is fastened a hoop of radius r, which is rough inside, /u, being the coefficient of friction. In contact with this a disc of radius a is spun with initial angu- lar velocity ;/ and its centre is projected with velocity v in such a direction as to be most retarded by friction. Show that after an + V a time c~a B {P"!' ajiB the disc will roll on the inside of the hoop. 278. An elephant rolls a homogeneous sphere of diametf^r a inches and mass ^ directly up a perfectly rough plane inclined /3 to the horizon, by balancing himself at a point disiant « from the sphere's highest point at each instant. Show that, the elephant being conceived as without magnitude but of mass E, he will move the sphere through a space fi ,^ E &ma — {E-\- S)&\n ^ 2 a ii cos(«-}-/^) + ii+-|-5* where t is the time elapsed since the commencement of the motion. 279. A circular disc of mass M and radius r can move about a fixed point A in its circumference, and an endless fine string is wound round it carrying a particle of mass m, which is initially projected from the disc at the other end of the diameter through A, with a velocity v normally to the disc, which is then at rest. Show that the angular velocity of the string will vanish when the length of the string unwound is that which initially sub- tended at the point A an angle /3 given by the equation ( yS tan /8 + I ) cos'"^ yS -f ^— = o, 8 ;// and that the angular velocity of tlie disc is then -'—(2 -f-/3 tan /9)~'. MISCELLANEOUS EXAMPLES. 231 radius of ctively. hoop of : friction. ;ial angu- i' in such hat after inside of ameter a inclined it a from that, the ■ mass E, It of the wo about ne string s initially • through n at rest, ish when ially sub- 1 ftan/8)-i. 280. A uniform rod AB can turn freely about the end A, which is fixed, the end B being attached to the point C, distant c vertically above A, by an elastic string which would be stretched to double its length by a tension equal to the weight of the rod. If the rod be in equilibrium when horizontal and be slightly displaced in a vertical plane, prove that the period of its small oscillations is v("^)» where p is the stretched length of the string in equilibrium, 281. A hollow cylinder, of which the exterior and interior radii are a and b, is perfectly rough inside and outside, and has inside it a rough solid cylinder of radius c. When the two are in motion on a perfectly rough horizontal table, prove that where M and m are the masses of the hollow and the solid cylinder respectively, ^ the angle the hollow cylinder has turned through, and the angle which the plane containing their axes makes with the vertical after the time t. 282. A string of length Cy fixed at one end, is tied to a uniform lamina at a point distant b from the centre of inertia. The centre of inertia is initially at the greatest possible distance from the fixed point and has a velocity v given to it in the plane of the lamina and perpendicular to the string. Prove that when the angle between the string and line ^ is a maximum, the angular velocity of the lamina is - — and the tension of the string is 2vnh K''--2c{b + c) ""^^ , J/^2 being the greatest moment {b^cf K'-4c{b + c) of inertia of the lamina at the centre of inertia. 283. In a circular lamina which rests on a smooth horizontal table and which can turn freely about its centre, which is fixed, a circular groove is cut. If a heavy particle be projected along the groove, supposed rough, with given velocity, find the time in wnich the particle will make a complete revolution (i) in space, (ii) relatively in the groove. 232 RIGID DYNAMICS. It hi 284. Four equal rods, each of mass m and length /, are con- nected by smooth joints at their extremities so as to form a rhombus. A constant force mf is applied to each rod at its middle point, and perpendicular to its length, — each force tend- ing outwards. If the equilibrium of the system be slightly disturbed by pressing two opposite corners towards each other, and the system be then abandoned to the action of the forces, show that the time of a small oscillation in the form of the system is 27r^f — j. 285. A spherical shell of radius a and mass in rolls along a rough horizontal plane, whilst a smooth particle of mass P oscil- lates within the shell in the vertical plane in which the centre of the shell moves, the particle never being very far from the lowest point. Show that the time of its oscillation will be the same as that of a simple pendulum of length =■ ina{a^ ■\- Ic^) -^ \{in ^ P^c^ -\- inl^\, where k is the radius of gyration of the shell about a diameter. 286. A solid cylinder with projecting screw-thread is freely movable about its axis fixed vertically, and a hollow cylinder with a corresponding groove works freely about it without friction. Find the moment of the couple which must act on the solid cylinder in a plane perpendicular to its axis in order that the hollow cylinder may have no vertical motion. 287. A sphere rolls from rest down a given length / of a rough inclined plane, and then traverses a smooth part of the plane of length ml. Find the impulse which the sphere sustains when perfect rolling again commences, and show that the sub- sequent velocity is less than it would have been if the whole plane had been rough. In the particular case when m = 120, show that the velocity is less than it would other vvise have been in the ratio of ^J to "jy. 288. A rough sphere is placed upon a rough horizontal plane which revolves uniformly about a vertical axis ; tlie centre of MISCELLANEOUS EXAMPLES. 233 the sphere is attracted to a point in the axis of rotation, and in the same horizontal plane with itself by a force varying as the distance. Determine the motion. 289. A heavy uniform beam AB is capable of rotating in a vertical plane about a fixed axis passing through its middle point C, and is inclined to the vertical at an angle of 60°. If a perfectly elastic ball fall upon it from a given height, find how long a time will elapse before the ball strikes the beam again. 290. A sphere rests on a rough horizontal plane, half its weight being supported by an elastic string attached to the highest point of the sphere ; the natural length of the string is equal to the radius and the stretched length to the diameter of the sphere. If the sphere be slightly displaced parallel to a vertical plane, show that the time of an oscillation is '^\{-^\ 291. A uniform heavy rod, movable about its middle point A, has its extremities connected with a point B by elastic strings, the natural length of each of which is equal to the length AB. Find the period of its small oscillations. 292. A squirrel is in a cylindrical cage and oscillating with it about its axis, which is horizontal. At the instant when he is at the highest point of the oscillation, he leaps to the opposite extremity of the diameter and arrives there at the same instant as the point which he left. Determine his leap completely. 293. A perfectly rough sphere rolls on the internal surface of a fixed cone, whose axis is vertical and vertex downwards. Prove that the angular velocity about its vertical diameter is always the same and that the projection on a horizontal plane of the radius vector of its centre, measured from the axis, sweeps out areas proportional to the times. Show also that the polar equation to the projection on a horizontal plane of the path of the centre is (C^n , / , - • 9 N 5 iT sin « cos a 2 c^t cos a + (2 4- 5 sm2 a)n = ^■•^ ,,.,, 4 , d&^ IhP' h Jn||f j 234 RIGID DYNAMICS. m ^ i"\ fi ii' .1 where a is the semi-vertical angle, 7 the constant angular velocity about the vertical diameter, and // is half the area swept out by the radius vector in a unit of time. 294. A thin circular disc is set rotating on a smooth horizontal table, about a vertical axis through its centre perpendicular to its plane, with angular velocity w in a wind blowing with uniform horizontal velocity v. Supposing the frictional resistance on a small surface a at rest to be cvma, where in is the mass of a unit of area, show that the angle turned through in any time is - (i —e""), and that the centre of gravity moves through a space c vt — {i—e''*y Determine the same quantities for a frictional resistance = cv^ma. 295. A uniform rod of length 2a passes through a small fixed ring, its upper end being constrained to move in a horizontal straight groove. Show that if the rod be slightly displaced from the position of equilibrium, the length of the isochronous simple pendulum will be ^ ^, where b is the distance of the ring from the groove. 3^ 296. A homogeneous solid of revolution spins with great rapidity about its axis of figure, which is constrained to move in the meridian. Prove that the axis will oscillate isochronously, and determine its positions of stable and unstable equilibrium. 297. A wire in the form of the portion of the curve cut off by the initial line, rotates about the origin with angular velocity eu. Show that the tendency to break at a point = — is measured by ^^ V2 • mc^aP'^ where vi is the mass of a unit of length. 298. Show that in every centrobaric body the central ellipsoid of inertia is a sphere. Is the converse of this proposition true .-* MISCELLANEOUS EXAMPLES. 235 299. A uniform sphere is placed in contact with the exterior surface of a perfectly rough cone. Its centre is acted on by a force the direction of which always meets the axis of the cone at right angles and the intensity of which varies inversely as the cube of the distance from that axis. Prove that if the sphere be properly started the path described by its centre will meet every generating line of the cone on which it lies at the same angle. 300. A sphere of radius a is suspended from a fixed point by a string of length / and is made to rotate about a vertical axis with an angular velocity «. Prove that if the string make small oscillations about its mean position, the motion of the centre of gravity will be represented by a series of terms of the form L cos {kt-\-M\ where the several values of k are the roots of the equation {IB -g) [b - (ak - ^ = ^-^— . 301. A rigid body is attached to a fixed point by a weightless string of length /, which is connected with the body by a socket (permitting the body to rotate freely without twisting the string) at a point on its surface where an axis through its centre of inertia, about which the radius of gyration is a maximum or a minimum, = k, meets it. The body is set rotating with angular velocity eo about such axis placed vertically (the string, which is tight, making an angle a with the vertical), and being then let go, show that it will ultimately revolve with uniform angular velocity 302. Three equal uniform rods placed in a straight line are jointed to one another by hinges, and move with a velocity v perpendicular to their lengths. If the middle point of the middle rod become suddenly fixed, show that the extremities of the other two will meet in time - , a being the length of each rod. 1 i: r W.: ;i 1* i> ti I 236 RIGID DYNAMICS. 303. A top in the form of a surface of revolution, with a cir- cular plane end, is set spinning on a smooth horizontal plane about its axis of figure, winch is inclined at an angle a to the vertical. It is required to determine the motion and to show that the axis will begin to fall or to rise according as tan « > or < -, a where 6 is the radius of the circular plane end perpendicular to the axis, and a is the distance of the centre of inertia from this end. 304. A heavy uniform beam AB of length a is capable of freely turning about the point A, which is fixed; the end B is suspended from a fixed point 6" by a fine inextensible chain of length c. The system being at rest is slightly disturbed. Find the time of a small oscillation, the weight of the chain being neglected. Examine the case in which the line ACis vertical. 305. A perfectly rough sphere of radius a moves on the con- cave surface of a vertical cylinder of radius a + d, and the centre of the sphere initially has a velocity v in a horizontal direction. Show that the depth of its centre below the initial position after a time / is -^^d^i — cos;//), where n"^ = ~^. Show also that in order that perfect rolling may be main- tained the coefficient of friction must not be less than ^. 306. A heavy particle slides down the tube of an Archi- median screw, which is vertical and capable of turning about its axis. Determine the motion. ip^ B 11 ■ ' 1 '■' ' "1 ijh if ki i WORKS ON PHYSICS, ETC PUDLISIIED i;V MACMILLAN & CO. AIRY. Works by Sir G. B. AiUY, K.C.15., lornu'rly Astronomer- Royal. On Sound and Atmospheric Vibrations. With tiie Mathematical K'.ements of Music. I2mu. S2.50. Gravitation. An Elementary Explanation uf the Principal I'urturbatiuns in the Solar System. l2nio. #1.90. ALDIS: Geometrical Optics. An Elementary Treatise. By W. 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