UC-NRLF LIBRARY OF THE UNIVERSITY OF CALIFORNIA GIFT OK Received ) and "weight," (W). D. 56, 57. 2.* Distinguish three classes of levers, stating which of the three forces is in the middle in each case. D. 56, 57. 3.* In which class of levers is the weight always greater than the power ? always less ? sometimes greater and sometimes less ? 4.* Which force in each class of levers is equal to the sum of the other two ? 5.* State the law of levers. Q. XVI. 6.f Show that each of the three forces in equilibrium in the lever is proportional to the distance between the lines of action of the other two. 7.^ Describe one or more experiments with the arithmetical lever, and state what points each illustrates. D. 25. 8.* What is meant by a bent lever? D. 57. 9. Distinguish the "arms " of a bent lever from the " arms " of the couples which are in equilibrium. D. 57. 10* When are the forces acting on a bent lever in equili- brium ? Q. XVI. n.J Describe one or more experiments (with E. H. HALL'S apparatus) illustrating the equality of moments in the bent lever. 23 XVIII. COMPOSITION AND EQUILIBRIUM OF COUPLES IN PARALLEL PLANES. i.f Describe the condition of the rod of a torsion apparatus during torsion. 2.f Show that a rod subject to torsion can offset the action of a couple. 3. When a rod is twisted by a couple acting at one end of it, what is the nature of its reaction against the agent through which the couple is produced ? (Axiom). 4. Show (by considerations of symmetry) that a uniform free rod, subject to torsion, must react equally at both ends against agents producing the torsion. (Axiom). 5.t What is meant by a " shaft " or " axle," and when is it in equilibrium ? 6.f Show that couples, like forces, exist under four aspects, and obey the same laws of action and reaction. 7.f Prove that a couple is transmitted without loss at right- angles to its plane, from one section of a body to another, and hence from one end of the body to the other. 8. What is meant by the transmissibility of couples along their axes ? Prob. 9.f Show that the plane or axis of a couple has no definite position in space, but is determined only by parallelism to a given plane or axis. Prob. io.f Show that the resultant of any number of couples with parallel planes is a single couple, with plane parallel to those of the components, and with a moment equal to the algebraic sum of the moments of the components. D. 27, Q. XIII. and XVIII. 1 1. 1 Describe one or more experiments (due to E. H. HALL) illustrating the conditions of equilibrium between couples in parallel planes. i2.f Show that a couple is completely specified when its moment and general direction of rotation are known. Prob. 24 XIX. COMPOSITION AND RESOLUTION OF COUPLES IN NON-PARALLEL PLANES. i.f Find the locus of all antipodal points in a given spherical surface at which tangential forces can be applied so as to produce a couple equivalent to a given couple. 2.f Find two antipodal points in a given spherical surface where tangential forces can be applied so as to produce two couples equivalent to any two given couples in non-parallel planes. 3.f Show that the parallelogram of forces can be applied to the composition of couples in non-parallel planes. 4-f Prove that the resultant of two couples in non-parallel planes is a single couple in some intermediate plane. 5. Apply the Pythagorean proposition to the relation between the moments of two component couples in planes at right-angles, and the moment of their resultant. Prob. 6.f Show that a couple can be resolved into components, the first parallel to any plane, and the second making a right-angle (or any other angle) with the first. 7. Find the trigonometric relation between a couple and its (rectangular) component in any plane, making the angle A with the plane of the couple. Prob. 8,f Find the component of a couple in any plane whose normal makes an angle, A, with the axis of the couple. Prob. 9-t What is meant by the moment of a force about a given axis? and what distinction is made between positive and negative moments ? io.f Show that the algebraic sum of the moments of two equal and opposite forces about any axis is equal to the component of the couple in a plane at right- angles to this axis. n.f Show that, in a state of equilibrium, the algebraic sum of the components of all the forces along any axis, and also the algebraic sum of their moments about this axis must be zero. XX. WRENCHES. 1. Show that every force acting upon a body (whether part of a couple or not) can be resolved into a force with line of action passing through any given point, P, and a couple. D. 28, 29 ; Q. XIV. i. 2. What is the nature of the resultant in the last question, of all the forces passing through a single point, /*? Of all the couples ? D. 29 ; Q. VII., XVIII., XIX. 3t Show that all possible forces acting on a body can be resolved and recompounded into a single force with line of action passing through a given point and a single couple. D. 29; Q. 2. 4.f Show that the couple in the last question can be resolved into two components, one at right-angles to the force, the other parallel to it, and state the result of combining one of these components with the force. D. 29, Q. XIX. 6, XIV. 4. 5.f Show that all possible forces acting on a body (whether parts of couples or not), are equivalent to a single force with definite line of application, and a single couple at right-angles with this force. D. 29, Q. 6. What name is given to a force combined with a couple at right-angles to it ? D. 29, 30. 7. Define the word "wrench" as used in (theoretical) mechanics. D. 29, 30. 8. Show that all possible combinations of forces are reducible to a wrench. D. 29, 30 ; Q. 9-f Distinguish right and left-handed wrenches, according to the direction of rotation, as looked at in the direction of the force. 26 WRENCHES. [XX. io.f Show that the reaction against a right or left-handed wrench, being equal and opposite in respect to both force and couple, is a wrench of the same kind, as concerns the relation between linear and rotatory displacement. n.f Distinguish wrenches farther into two classes, according to whether extension or compression is produced. I2.f Discuss the wrenches exerted in certain familiar mechan- ical processes (such as driving a screw). 13-f Show that extension in one part of an elastic system implies compression in some other part. 14. In what respect are the wrenches exerted by two different parts of an elastic system on each other similar, and in what respect dissimilar ? Q. i5.f Discuss the longitudinal and torsional effects of mutual wrenches in the ordinary spring curtain-roller. 1 6. State the law of action and reaction as applied to wrenches. D. 30. 2 7 XXI. ELASTICITY OF SOLID BODIES. i.f Name three effects of a wrench upon an elastic body. 2.* Distinguish stretching, bending, twisting, and com- pression. 3. What is meant by elasticity ? by limits of (perfect) elasticity? by a .set ? or permanent strain ? D. 126. 4. Define a stress, a strain, and a coefficient or modulus of elasticity. D. 128. 5. 1 Describe one or more experiments illustrating effects of longitudinal forces upon wires or rods. 6.f State the law connecting the stretching of a rod with the force applied, with the length of the rod, and with its area of cross-section. 7. What is meant by YOUNG'S modulus of elasticity? D. 128. 8.f Give some idea of the relative magnitudes of the forces necessary to produce a given stretch in rods of different materials, but of the same dimensions. g.f State the usual effect of stretching on the diameter of a rod, and what is meant by " POISSON'S ratio." io.f What connection (if any) exists between the relative magnitudes of the forces necessary to produce a given amount of stretching, and the relative magnitudes of those required to produce a given amount of bending in bodies of given dimensions ? 1 1. 1 Describe one or more experiments relating to the laws of bending. 28 ELASTICITY OF SOLID BODIES. [XXI. 12. f State the laws connecting the bending of a. beam with the load, and with the length, breadth, and thickness of a beam. 134 Describe one or more experiments relating to the laws of torsion. i4-t State the laws connecting the twisting of a rod with its length, with its breadth, and with the magnitude of the couple applied. 15. f Give some idea of the relative magnitudes of the couples necessary to produce a given amount of twisting in bodies of the same dimensions, but different materials. i6.f What is meant by a modulus or coefficient of torsion ? 17. What is meant by elasticity of volume? by resistance to compression? by compressibility ? D. 129. i8.f Explain how the compressibility of solids (as well as liquids) can be demonstrated by an (OERSTED'S) piezometer. 19. f What connection (if any) exists between YOUNG'S modulus, the modulus of torsion, and the resistance to compres- sion for a given material ? 20. Show that HOOKE'S law applies to cases of stretching, bending, twisting, and compression. XXII. CENTRE OF GRAVITY. i.f When is a force applied to a rigid body said to have a definite point of application ? (D. 34). 2.f What is meant by a centre of force? by the centre of two or more forces ? 3-t Distinguish between a true centre of force, as for instance an ideal atom, and a virtual centre of force, as for instance the centre of Saturn's rings. 4. Show that the centre of two parallel forces (not equal and opposite) having definite points of application lies in the line joining these points of application. D. 34. 5. Where is the point of application of the resultant of two parallel forces (not equal and opposite) ? D. 34. 6. Show that if any number of parallel forces (whose algebraic sum is not zero) have definite points of application, they give rise to a resultant with a definite centre or point of application. D. 34. 7. Show that the forces which the earth's gravity exerts upon the molecules of a small body at the earth's surface, having definite points of application, and being (practically) parallel, must have a definite centre or point of application. D. 33, 34. 8. What name is given to the centie or ] cint of i f pl.'catic n of the (parallel) forces which gia\ ity exerts i yon the different elementary particles of a body ? D. 34. 9. Define the term " centre of gravity," and state whether it is applicable to solids only, or also to liquids and gases. D- 34. 35- io.f Show that if masses are measured by the forces which gravity exerts upon them, the centre of gravity coincides (by definition) with the "centre of mass." 30 CENTRE OF GRAVITY. [XXII. n.f Would the statement in the last question hold if the masses were not measured by the forces which gravity exerts upon them ? or if the forces in question were not parallel ? 12. Under what circumstances does the centre of (the forces due to) attraction coincide with the centre of mass ? Q. 13. Under what conditions does the resultant of two or more forces have a definite centre or point of application ? Q. 14. Where is the centre of gravity of two equal masses? of two unequal masses ? D. 34. 15. f Show that the centre of gravity of two, three, or four equal masses is at the geometrical centre of this line, triangle, or pyramid which they define. (D. 37-39). i6.f Show that three or more unequal masses must have a fixed centre of gravity, independent of the order in which they are considered. (D. 37). 17. Show (by geometry) that in a system of mass, M, consisting of two masses, m^ and m 2 , at the heights, /i, and // 2 , the height, H, of the centre of the gravity is such that MH=m^ h^ -j- m 2 h 2 . D. 22, 45. 1 8. Extend the principle of the last question to the case of any number of masses. D. 23, 45. 19. Show that the principle of the last two questions applies not only to the heights, H, h^ h z , etc., of the centre of gravity and separate masses composing a system, but also to their distances, D, d^ d 2 , etc., from any plane. D. 22, 23. XXIII. CENTRE OF GRAVITY OF GEOMETRICAL FIGURES. 1. What assumption (as to the distribution of weight) is made in calculating the position of the centre of gravity of bodies approximating to geometrical figures ? D. 35. 2. What is meant by the centre of gravity of a geometrical solid? surface? or line ? D. 35. 3.* Show that the centre of gravity of a unifoim thin rod (or line) is at its centre, and that it will balance about any axis passing through this centre. D. 36. 4.f Show that the centre of gravity of two equal triangles or pyramids, symmetrically situated with respect to their common apex, is at this apex. 5. Prove that if a figure can be cut up into strips, slices, pairs of triangles, pairs of pyramids, etc., each having its centre of gravity at or in a given point or axis, the centre of gravity of the whole figure must be at or in this point or axis. Prob. 6. Show that if any figure is symmetrical with respect to a centre (so that it can be cut up into equal and opposite pairs of pyramids, with common apex at this centre) its centre of gravity must lie at this centre. Prob. D. 36. Q. XVIII. 7. Show that if any figure is symmetrical with respect to -an axis, (so that it can be cut up into equal and opposite pairs of triangles with common apex on this axis) its centre of gravity must lie on this axis. Prob. (Q. XXIII.) 8. Show that if any figure is symmetrical with respect to a plane (so that it can be cut up into uniform thin strips, each with its centre of gravity in this plane), the centre of gravity of the figure must lie in this plane. Prob. (Q. XXIII.) 9. Where is the centre of gravity of a figure symmetrical with respect to two axes ? an axis and a plane ? two planes ? three planes ? Prob. 32 CENTRE OF GRAVITY OF GEOMETRICAL FIGURES. [XXIII. 10. Find the centre of gravity of a circle, square, ellipse, parallelogram, cube, block, parallelepiped, prism, cylinder, sphere, ellipsoid, etc. Prob. D. 36. 11. Show (by cutting up a triangle into strips parallel to the base, and considering the centre of gravity of each strip) that the centre of gravity of a triangle must lie in its medial line. D. 37. 12. Show that the centre of gravity of a triangle must lie at the point of intersection of its medial lines, and find (by geometry) the relative altitudes of the apex and centre of gravity. D. 37, Q. ii. 13. Show (by cutting up a pyramid into sections parallel to the base) that if a line be drawn from the apex to the centre of gravity of the base (so as to pass through the centre of gravity of each section), this line will contain the centre of gravity of the pyramid. D. 38. 14. Show that the centre of gravity of a pyramid must lie at the point of intersection of two or more lines drawn as in the last question, and find (by geometry) the relative altitudes of the centre of gravity and apex of a triangular (or any other) pyramid. D. 38. 15. Find the centre of gravity of a cone by the same method as in the last question. D. 38. 33 XXIV. CENTRE OF GRAVITY OF SUSPENDED BODIES. i.f Find the moment of the force exerted by gravity on a body of weight, W, about a horizontal axis in a vertical plane at the distance, D, from the centre of gravity. (D. 58). 2.f A body of weight, W, and centre of gravity at the horizontal distance, D, from a fulcrum (or axis) is balanced by a force, P, acting on a horizontal arm, A. Find the equal and opposite moments. (D. 58). 3. In the last question, express each of the four quantities, W, Z>, P, and A, in terms of the other three. Prob. Q. 3. 4.J How can you find the weight of a body by balancing it on an axis by means of a known weight ? Q. 5-J How can you locate the centre of gravity of a body of known weight when balanced on an axis with another body of known weight ? Q. 6.J Show how the weight of any object, great or small, can be found by a graduated lever of known weight. Q. 7-1 Show how the position of a bullet of known weight in a gun-barrel of known weight can be found from the displacement of the centre of gravity. Q. 8. * Describe one or more experiments with levers showing that their weight acts always as if concentrated at their centre of gravity. 9.* Explain and describe one cr mure experiments illustrating the behavior of bodies suspended at their centre of gravity. 10. Show that a body is in equilibrium if suspended at any point in a vertical line passing through the centre of gravity. D. 41. IT.* What is the result of suspending a body at a point not contained in the vertical line passing through the centre of gravity ? 34 CENTRE OF GRAVITY OF SUSPENDED BODIES. [XXIV. 12. State the condition of equilibrium in a body suspended at a point. D. 41. 13.* Describe the experimental location of the centre of gravity of a body by the method of suspension. D. 43, 44. 14. J Describe and explain one or more experiments in which a body is suspended upon a horizontal axis in a vertical plane containining the centre of gravity, and state how the centre of gravity may be located by such experiments. 15.* What is the result of suspending a body upon a horizontal axis not containing the centre of gravity ? 1 6. What is the result of suspending a body upon an oblique axis containing the centre of gravity ? not containing it? Prob. 17.* Describe the condition of a body suspended about a vertical axis. 1 8. State the conditions of equilibrium of a body suspended upon an axis, in terms of the position of the centre of gravity with respect to this axis. D. 41. 19. Show that in all the cases above, where a body is not in equilibrium, the action of gravity is such as to cause the centre of gravity to descend. D. 16. 20. Show that a heavy body is in equilibrium when, and only when, the nature of its restraints is such as to prevent the centre of gravity from descending. Prob. 2 1. 1 Explain the action of a double-pointed cone in rolling up two inclined, but diverging rails. 22. Explain the action of a ball in rolling into the deepest portion of a hollow. D. 46. 23. Show that, in general, the centre of gravity tends to descend. D. 46. 35 XXV. STABILITY, AND ITS RELATIONS TO CENTRE OF GRAVITY. 1. Show that a body, in which two points are fixed, is equivalent to one suspended upon an axis passing through these points. Prob. 2. How many points in a body must be fixed to prevent movement of any kind ? Prob. 3.f Three points in the same horizontal plane are fixed. Find the direction of the force on each, according to the position of the centre of gravity. 4. Describe the forces by which a tripod is usually held in place, stating the direction of these forces, and whether they vary uniformly or " per saltum " under displacement. D. 40, 52. 5. State the conditions under which a body may be fixed by the upward reaction of three forces. Q. 6.f What name is given to the area of a horizontal section included between the forces in the last question ? 7.* Define, in general, the "area of support" of a bod) T . Q. 8.f Show that a body is stable if its centre of gravity lies above its area of support. (D. 40). 9. What is meant by the limit of stability in a body supported at three or more points ? D. 54. 10. Show that a body, displaced within its limits of stability, tends to return to its original position. Prob. 11. When is a body said to be " practically stable ?" D. 54. 12. What, in general, is meant by stability? D. 42. 13. Distinguish stable, unstable, and neutral equilibrium, according to the tendency of a body when displaced to return to its original position of equilibrium. D. 42. 36 STABILITY AND CENTRE OF GRAVITY. [XXV. 14. When is a heavy body, suspended by a point or by an axis, in stable equilibrium ? in unstable equilibrium ? in neutral equilibrium? State the position of the centre of gravity relatively to the point or axis of suspension in each case. D. 41, 42. 15. Explain the toy called " the balancer." D. 53. 1 6. Show that, in a condition of stability, the height of the centre of gravity is at a minimum, in a condition of instability at a maximum. D. 46. 17. State the conditions which determine whether equilibrium exists, and whether it is stable or unstable, in terms of the path of the centre of gravity consequent upon a given displacement of the body. D. 46. 1 8. Find the state of equilibrium in a sphere or cylinder rolling on a flat, on a concave, and on a convex surface. Prob. (D. 46). 19. Describe and explain the action of a toy known as " the tumbler," or any other similar device. D. 53. 37 XXVI. SENSITIVENESS OF A BALANCE AS RELATED TO CENTRE OF GRAVITY. i. Name the essential parts of a balance. D. 70. 2.J Describe the beam of an ordinary balance, the knife- edges and their bearings. D. 75. 3.J Describe the pointer or index of a balance, the scale-pans, and the means for arresting the loads, or lifting them from the knife-edges. 4. Explain the importance of preserving the sharpness of the knife-edges, and the ordinary precautions taken to this end. (D. 75). 5. Where is the centre of gravity of a balance-beam, and how is the position of the centre of gravity adjusted ? D. 73. 6. Show that the weights act as if concentrated at the knife- edges. Prob. Q. XXII. i. 7. Show that if the three knife-edges be in the same straight line, the centre of gravity is not disturbed by loading the scale- pans so as to produce equilibrium. Prob. Q. XXII. 5. 8. Show that in a straight-arm balance, we have for the angle, a, of deflection due to excess of weight, p, on aim of length, /, of the beam of weight, w, with its centre of gravity at a distance, d, from the axis, tan a pi -j- wd. D. 73. 9-f What additional terms must be considered if the three knife-edges are not in the same straight line ? D. 74. 10. Show that if the outer knife-edge be above the middle knife-edge, the equilibrium of the balance may become unstable from loading. D. 74. 11. Show that if the outer knife-edges be below the central knife-edge, the equilibrium of the balance will become more stable on loading. Prob. D. 74. 12. Distinguish three types of balance, according to align- ment of the knife-edges. Q. 13. Show that a balance beam may belong, successively, to the three several types named in the last question, through bending under an increasing load. Prob. i4.f What is meant by the sensitiveness of a balance ? 38 SENSITIVENESS OF A BALANCE. [XXVI. 15. State certain conveniences in the use of a balance of known sensitiveness. D. 74. i6.f Describe the method of weighing under a constant load, and state some of the advantages and disadvantages of this method. D. 74. ly.J How is the sensitiveness of a balance under a given load ordinarily determined? i8.f Explain the use and construction of a table showing the sensitiveness of a balance under different loads. ig.J Describe the use of a " rider," and show that in weighing with a rider, the sensitiveness of a balance need not be kngwn. 20. J Explain the method of weighing by "oscillations" (or by "vibrations "), and state some of its advantages. 2i.f Show that friction, within certain limits, does not necessarily affect the sensitiveness of a balance, if the method of oscillations is employed. 22. f State what advantages are to be gained by using a balance with rapid oscillations, to what extent sensitiveness should be sacrificed to this end, and to what extent sensitiveness is consistent with rapidity. D. 71, 74. 23. f Show that it is convenient, but not necessary, that the pointer should indicate how much the loads differ. 24. Correct DESCHANEL'S statement (73) that the centre of gravity and axis of a balance " must not coincide." Q. 25. What qualities are sought for in a balance? D. 71, 73. Q. XXVI. 1-26. 26.* Describe the ordinary "steelyard," andshowthat equality of the arms of balance is not necessary in order that the true weight of a body may be found. D. 76. 27. Describe (BORDA'S method of) weighing by substitution, (or double weighing, according to DESCHANEL and other French writers). D. 72. 28. J Describe (what is meant by most English writers by) the method of "double weighing," (or GAUSS'S method, involving an interchange of the loads). 29. f Show that the method of double weighing (by inter- change) is twice as accurate as weighing by substitution. 30. J Describe, in detail, the processes actually employed in very accurate weighings. "-.' Qg THE 39 XXVII. WORK, AND ITS RELATIONS TO CENTRE OF GRAVITY. i.* What name is given to the product of a force (when constant in magnitude and in direction) and the displacement of its point of application (in the same direction) ? 2.* Define work, and name some of the units in which it is measured. 3.* Define the foot-pound and kilogram-metre. 4.f Define the foot-poundal, and erg (or dyne-centimetre). 5.* Find the work, W, necessary to raise a weight, w, through the vertical distance, d. D. 45, 47. 6.* Distinguish between the work done by a force upon a weight or against gravity, and that done by gravity or by the weight against the force. D. 47. 7. How is work measured when the directions of the force and displacement are nut the <-ame ? D. 45, 48. 8. The point of application, A, of a force, AB, moves to C. What kind of work is done by the force if BAC is acute? if BAG is obtuse ? if BAC is a right-angle ? D. 48. 9. Find the work done by a force, AB, in producing the displacement, AC, in terms of AB, AC, and a function of the angle, BAC. D. 48. 10. Show that the work done by a force in producing a given displacement may be measured (i) by the product of the displace- ment and the component of the force in the direction of the displacement, or (2) by the product of the force and the component of the displacement in the direction of the force. Prob. Q. 40 WORK AND CENTRE OF GRAVITY. [XXVII. 11. Show that the work necessary to move a weight in any direction is the product of the weight and the vertical component of the displacement. D. 48. 12. A body of weight, W, remains fixed, while another body, of weight, w, moves through the distance, D. Find (by geometry) the displacement, d y of the centre of gravity of the two weights. Prob. 13. Compare, in the last question, the components of the two displacements in the vertical (or any other) direction. Prob. i4.f Show that the product of a weight, forming part of a system, by its vertical displacement is equal to the product of the weight of the whole system by the vertical displacement of its centre of gravity. Q. i5-t What is meant by the work done upon the centre of gravity of a system ? i6.f Show that the algebraic sum of the quantities of work done upon the separate weights of a system (by considering these one by one) is equal to the work done upon the centre of gravity of the system. Prob. 17. State the "principle of work " as applied to the centre of gravity of a system of weights. D. 45. 1 8. Bxplain how the calculation of work done upon a body can be simplified by the principle of the last question. D. 45. 19. Find the work, W, necessary to pile a given weight, w, of fragments into a rectangular, into a triangular, or into a pyramidal heap of height, h, above the (mean) level of the fragments. D. 45. 20. Modify the last question so as to apply to the case of filling cisterns ot different shapes with liquids, or exhausting wells of different depths and shapes. Prob. XXVIII. WORK, AS A CRITERION OF STABILITY. 1. Find the work, W, necessary to turn a block of length, /, breadth, b, thickness, /, and weight, w, from its position of maximum stability into its position of medium or minimum stability. Prob. D. 45. 2. Modify the last question so as to apply to an ellipsoid. Prob. 3. A barrel of weight, w, rests upon a floor in stable equilibrium, because its centre of gravity is at a distance, d, below the axis : find the work, W, necessary to start the barrel rolling. Prob. Q. 4.* State some of the most important properties of the centre of gravity. Q. XXII.-XXVII. 5. Show that work must be done upon a body to displace it from a position of stable equilibrium. D. 50. 6. Show that work is done by a body when it is displaced from a position of unstable equilibrium. D. 50. 7. Show that no work is done upon or by a body when it is displaced from a position of neutral equilibrium. D. 50. 8. State the criterion of the stability of equilibrium in terms of the total work done upon or by a body when displaced from its position of equilibrium. D. 50. 9f What condition of equilibrium is assumed to exist in an ideal lever ? 10. t State, in general, what kind of equilibrium exists in various ideal machines or " mechanical powers." XXIX. PRINCIPLE OF WORK APPLIED TO MECHANICAL POWERS. , by geometry, that the work spent by a force acting upon an ideal lever is equal to the work utilized in the weight raised. . 2.* State the " principle of work " (or " virtual velocities ") as applied to the lever. D. 49. 3.* What assumptions (in regard to loss of work by friction, etc.) are made in the investigation of ideal machines? (D. 49). 4. What distinction is made between the force known as the "power" and the "weight" in various machines? D. 59. 5. What is meant by mechanical advantage ? D. 59. 6. Name some of the most important " mechanical powers." D. 55- 7. Show that the wheel and axle may be regarded as an endless lever. D. 60. 8. Find (by the principle of work) the relation between the power, weight, and radii of a wheel and axle. Prob. 9. How would the results obtainable with a wheel and axle be modified by the thickness of the ropes, if these were perfectly flexible ? D. 60. ro.f In what way would the stiffness of the ropes of a wheel and axle modify the result if the ropes were perfectly elastic, so that no work need be spent in bending them ? ii. Show that a pulley may be regarded as an endless lever. D. 61. i2.f What assumption as to the tension of the cords passing round one or more pulley-wheels is made in calculating the mechanical advantage of ideal pulleys ? 43 MECHANICAL POWERS. [XXIX. 13.* What mechanical advantage is gained by a single fixed pulley ? D. 61. i4.f What allowance must be made for the weight of a movable pulley ? for the weight of the cords ? 154 How is the influence of the weight of pulleys and cords eliminated in experimental demonstrations ? 1 6.* State the mechanical advantage of a single movable pulley, to which the weight is attached, assuming the cords parallel. D. 61. 17.* State in general the mechanical advantage of a pulley niOv'ed by x cords with uniform tension in the same direction. Prob. D. 62. 1 8.* Show that the principle of work applies to the case of a pulley with x cords, as in the last question. Prob. D. 61. 19. Find the mechanical advantage of a series of x single pulleys, each attached to the power cord of the next. Prob. D. 63. 20. f What is meant by an ideal smooth surface or plane? and what limitations exist in the direction of the force which such a surface is capable of exerting upon a body ? 21% Describe one or more experiments with inclined planes approximating to the ideal conditions. 22. Explain the resolution of forces in a body resting on a perfectly smooth inclined plane. D. 64. 23. A body of weight, w, rests on an inclined plane of length, AC, base, AB, and height, BC: find the force by which it is held in place, (i) if this force is tangential, (2) if the force is horizontal. Prob. Q. 24. Show that the principle of work applies to the forces known as the " power " and the " weight " in an inclined plane. D. 65, 66. 25. Show that a wedge may be regarded as a special case of inclined plane. D. 67. 44 MECHANICAL POWERS. [XXIX. 26. Find the mechanical advantage of an ideally smooth wedge of length, /, and thickness, /. Prob. Q. 27.* Show that a screw can be treated as a special case of inclined plane. D. 68. 28.* Find the mechanical advantage of a screw of circum- ference, c, and distance, d, between threads measured parallel to the axis of the screw, if the force is applied at the circumference, at right-angles to the axis. Prob. Q. 29.* By what means is the mechanical advantage of a screw greatly increased in ordinary screw presses ? D. 69. 30.* Find the mechanical advantage of a screw press with arms of length, /, and threads, at the distance, d. D. 69. 3i.*fKstimate, approximately, the force which can be brought to bear in an ordinary screw press, neglecting friction. 32.* Show that the principle of work applies to the screw press. 33. Show that all the mechanical powers are reducible to two types (the lever and the inclined plane). 34.* State, in general, the application of the principle of work to mechanical powers. D. 49. 35.* What is meant by the statement that " what is gained in power is lost in speed ?" Prob. Q. 36.* An ideal machine is actuated by a force, /% acting through a distance, D, and produces a force,/", acting through a distance, d ; express each of the four quantities, /% /" D, and d, in terms of the other three. 37.* In the last question, state what must be, and what need not be known about the construction of the machine, in order that the principle of work may be applied. Q. 45 XXX. CONSERVATION OF WORK. i.f A weight, w, is moved through the distances, AB, BC, . . . YZ, all lying in the same vertical line. Show that the work done is the same (w X AZ}, whatever may be the position of the intermediate points, B, C, etc. 2. Prove that if a weight, w, is moved along any broken path, A" B" C" . . . Z" , the work done is the same as along the projection of this path, A' B' C' . . . Z 1 ', on a vertical line, and hence, equal to w X A'Z'. Prob. 3. Show that in returning from Z" to A", the weight gives out the same amount of work that it received between A" and Z". Prob. D. 123. 4. Show that work done against constant forces, like those exerted by gravity, depends only upon the initial and final positions of a body, and is independent of the path of the body. Q. (D. 123). 5. Show that when a body returns by any " closed path " to its original position, the total work done upon it, or by it, is zero. Q. (D. 123). 6.f Discuss, from the point of view of the last question, the possibility of various proposed forms of "perpetual motion." (D. 49)- 7. State the relation between the work received by a body in one part of a closed path, and that given out by it in the remainder of its path. Q. 8.*fWhat is meant by the "conservation of work" in -mechanics ? XXXI. LOSS OF WORK BY FRICTION. i.* What element, entering into the working of all practical machines, causes a departure from ideal conditions ? D. 49, 125, 131. 2.* Define (kinetical) friction (D. 131), and state the peculiarity of sign in the work to which it gives rise. 3-t Describe and explain one or more experiments showing a great discrepancy between the force required to start a body sliding (no matter how slowly) over a surface, and that required to maintain the motion. (D. 131, 132). 4. | State the results of one or more experiments showing whether the force required to drag a body over a surface is or is .not affected by the velocity. (D. 131). 1 5.f Why is it necessary, in measuring forces due to friction, that bodies should be moved with uniform velocity ? 6. 1 State the results of one or more experiments showing the relation between the force required to drag a body over a surface, and the (normal) force urging the body and the surface together. D. 131. 7. J What is the result of varying the ?rea of rubbing surfaces, without varying either their nature or the force urging them together ? 8. What is meant by a ' ' coefficient of friction ?" D. 131, 132. 9-J Describe one or more experiments in which the coefficient of friction between two horizontal surfaces is determined. 10. A body of weight, W, requires a force, w, to drag it with uniform velocity, along a horizontal surface ; find the coefficient of friction between the body and the surface. Prob. Q. 47 LOSS OF WORK BY FRICTION. [XXXI. ii. What is meant by the "limiting angle of friction " for two surf aces? D. 132. 1 2. J Explain the experimental determination of coefficents of friction by means of an inclined plane. D. 133. 13. A body slides with uniform velocity down an inclined plane with base, AB, and height, BC: find the coefficient of friction. D. 133. 14. f Give some idea of the magnitude of coefficients of friction between wooden or metallic surfaces, and the effect of grease or oil upon these coefficients. 15. t Show that the mechanical advantage of a wedge cr screw, no matter how fine the pitch, cannot exceed the reciprocal of the coefficient of friction. i6.f Find a relation between the weight, W, of a truck, the diameters, D, and d, of the wheel and axle, the coefficient of friction,/, of the (loose) bearing upon the axle, and the force, F, required to pull the truck. Illustrate by a numerical example. ry.f Find the difference between the forces, F' and F" , trans- mitted by parallel cords to and from a pulley of diameter, Z?, with axle of diameter, d, and coefficient of friction, /, on its loose bearing, (calling the force borne by the pulley F' -\- F"). i8.f Trace out the effect in a double block (with forr cords) of a loss of 10 per cent., 20 per cent., etc., in the tension of each cord passing round each wheel, on the force required to lift a given weight by the block, remembering that the weight raised is given by the sum of the tensions on the cords. ig.f Find, conversely, the weight on the block necessary to overcome a given force at one end of the cord. 20. f Explain the enormous loss of work in long trains of clock-work, and the surprising effects of oil upon these. XXXII. EFFICIENCY. i. J Describe one or more experiments showing that the work spent upon the cord of a tackle in raising a weight is greater than that utilized through the pull upon this cord exerted by the descending weight. 2.t What name is given to the ratio of the work spent to the work utilized in a given case ? 3. Define "efficiency," as used in mechanics. Q. 2. 4.f Distinguish between the efficiency of a tackle as a machine for raising weights, its efficiency as a machine for utilizing work done by the descent of a weight, and its efficiency as a machine for storing useful work. 5.f State a certain necessary relation between the three efficiencies in the last question. 6. A man slides a box up into a wagon along a board inclined at the limiting angle of friction : (Q. XXXI.) find the efficiency of his combination. Prob. 7. Show that the efficiency of a screw, irreversible through friction, cannot exceed 50 per cent. Prob. 8.f Show, from a point of view of efficiency, the disadvantage of .a screw with too fine a thread (or of a differential screw). 49 XXXIII. POWER. i. J Describe one or more experiments with an ergometer (or friction-brake), showing how it is possible to measure the work spent by a given agent in a given time. 2.J Describe a "transmission dynamometer," and its uses. 3.f What name is given to the quotient of work by time? to the product of force and velocity ? 4.f Distinguish power, in its technical sense, from power as a name for one of the forces in a machine. 5f Define pow r er in two different ways, and show that these definitions are identical. Q. 3. 6.f Define "horse-power" (English or French), the erg per second, and the "watt." 7-t Show that the horse-power varies slightly in different latitudes, but that the erg and the watt are constant. 8.f Find the horse-power of a locomotive with two cylinders 100 square inches in section, and double two-foot stroke, making 125 revolutions per minute under a mean pressure of 33 Ibs. of steam per square inch, in both the forward and backward stroke, making no allowance for friction. g.f Find the power, P, spent upon a water-motor with piston of area, a (sq. cm.), making n double strokes of length, /, (cm.) in every second, under a pressure, p (dynes per sq. cm.), and state in what units the result is expressed. io.f Show that the power, P, spent upon a motor (by an incompressible liquid) is equal to the product of the pressure, p, per unit of area, and current, c, in units of volume per unit of time. ii.t Show that (as a consequence of the principle of the last question), if pressure is transmitted by an incompressible fluid without loss, power must also be transmitted without loss. XXXIV. SOLIDS AND FLUIDS, DISTINGUISHED. 1. 1 Describe an experiment (with OERSTED'S piezometer), showing that liquids are compressible. D. 130. 2.* Distinguish fluids from solids (in respect to their relative resistance to, and limits of recovery from, changes of form or shape unaccompanied by changes of bulk or volume). (D. 129). 3.f Distinguish fluids from solids in respect to their rates of yielding to tangential or transverse forces, and state whether this distinction is one of kind or of degree. 4.f What name is given to that property in solids which is connected with their slow yielding under transverse stresses ? 5.f When is a solid, and when is a fluid said to be especially viscous? (D. 135). 6.f Show that viscosity is used in two opposite senses in its applications to solids and fluids. y.f Distinguish frictional forces within the body of a fluid (whether due to viscosity or not) from forces due to the rubbing of two solid surfaces, in respect to their dependence upon (i) the compression, (2) the area, and (3) the relative velocity of the moving parts. 8.* State some of the characteristic distinctions between solids and fluids. Q. XXXV. LIQUIDS AND GASES DISTINGUISHED. 1. 1 Describe an experiment (with a gas-bag and air-pump) illustrating the indefinite expansibility of a gas. D. 194. 2.* Distinguish fluids into two classes, according to their tendency toward uniform distribution throughout any space within which they are confined. D. 194. 3.* How do liquids compare, as a class, with gases, in respect to density ? 4.* How do liquids compare, as a class, with gases, in respect to compressibility ? 5.f What is meant by the " free surface " of a fluid, and what kind of fluids alone present such surfaces under ordinary conditions ? 6.J Describe one or more experiments (e.g., with floating needles), showing that the surfaces of liquids resist deformation to a slight extent. D. 159. 7.J Describe one or more experiments (with camphor and water, oil films on water, and the effects of local heat or alcohol) showing that slight tangential forces are exerted by the surfaces of liquids. D. 192. 8. J Describe one or more experiments (with PLATEAU'S films) showing that the surface of a film is a minimum consistent with its boundary. (D. 186). 9. Explain the spherical shape of rain-drops or bubbles. D. 189. 10. J Describe one or more experiments showing that films of liquid, when free to do so, cause their boundaries to contract. ii. State the influence of the length, the breadth, and the thickness of a film on the force with which it contracts longitudinally. D. 185. 52 LIQUIDS AND GASES DISTINGUISHED. [XXXV. 12. What is meant by ".surface tension ?" D. 185. 13. How many surfaces is a (soap-bubble) film considered to possess ? and how is the surface tension of such a film measured ? D. 188. 14.* Describe one or more experiments illustrating adhesion between a liquid and a solid which it wets. 15.* Describe the surfaces of liquids of two different kinds, near the edges of the vessels containing them. D. 182. 1 6.* What is meant by a capillary tube ? by capillary forces? and by capillarity in general ? 17.* Describe the rise and fall of liquids caused by capillary tubes. D. 182. 1 8.* When will a liquid rise in a capillary tube, and when will it be depressed by it ? D. 182. 19. Describe the " meniscus " (or curved surface) of a liquid accompanying (i) capillary ascensions and (2) capillary depres- sions. D. 182, 190. 20. Kxplain the rise of liquids in capillary tubes which they wet, and their depression by capillary tubes which they do not wet. D. 1 86. 21. State some of the conditions which influence the amount of the rise or fall of a liquid in a capillary tube. D. 183, 184. 22.f What is meant by the height of the meniscus (or sagitta) ? 23. What is meant by the angle of contact between a liquid and a solid? D. 184, 185. 24. Find the height, /i, of a column of liquid of density, d, sustained by a surface tension (in gravitation units) t, in a tube of radius, r, assuming that the tube is wet by a film of liquid tangent to its surface. D. 186. 25. State the law of diameters governing capillary ascensions and depressions. D. 184. 26. J Describe one or more experiments (e.g., with inclined plates) illustrating the law of diameters. 53 LIQUIDS AND GASES DISTINGUISHED. [XXXV. 27. f How high are liquids known to rise by capillary action in certain vegetable structures ? 28. f Give reasons for supposing that a practical limit exists in the height attainable by capillary action. 29. J Describe an experiment with an air-pump, showing that capillary ascensions and depressions are not due to atmospheric pressure. (D. 185). 30.J Describe one or more experiments illustrating the tensile strength under certain specified conditions, of columns of liquid of considerable cross section. 31.* In what class of fluids, alone, are phenomena due to capillarity, cohesion, or surface tension perceptible ? Q. 32.* State some of the characteristic distinctions between liquids and gases. Q. 33. 1 Describe one or more experiments illustrating effects of endosmose (or diffusion through a diaphragm) and endosmotic pressure. D. 193. 34. Distinguish solutions into two classes with respect to the facility with which they pass through a diaphragm. D. 193. 35. Describe the process of separating colloids and crystalloids by "dialysis." D. 193. 36. Describe one or more experiments illustrating the diffusion of gases, and the pressures to which such diffusion may give rise. D. 193, 227. 54 XXXVI. FUNDAMENTAL ASSUMPTIONS IN HYDROSTATICS. i.f What assumptions are usually made in the solution of problems in hydrostatics, with respect to viscosity, capillarity, osmosis, etc. ? 2. Why does not viscosity enter into problems is hydrostatics? D. 135- 3-t State reasons for assuming that capillary forces may be neglected in the treatment of bodies of liquid of considerable size, or in the case of solids of considerable size immersed in these liquids. 4.J Give experimental grounds for the assumption that forces or pressures due to osmosis or to diffusion are not perceptible in the wide channels usually employed in hydrostatics. 5-t Show that, in the absence of viscosity and capillarity, normal forces and pressures are the only ones which need be taken into account. (D. 135). 6. Distinguish hydrostatics from hydrokinetics, treating both as branches of hydrodynamics. D. 134. 7. Distinguish pneumatics from hydrostatics. D. 3. 8.f To what extent are the properties of gases studied under the head of hydrostatics ? Q.f State certain obvious experimental evidence of the fact that fluids, when undisturbed for a long time, fall (practically) into a state of equilibrium. io.f What general principle would lead to the anticipation of the result stated in the last question ? u.f What hypothesis (as to the equilibrium of fluids) lies at the foundation of hydrostatics ? 55 XXXVII. CONDITIONS OF EQUILIBRIUM IN FLUIDS. i.f Show (by considerations relating to the centre of gravity) that a bubble, contained in a liquid within any enclosure, seeks the highest possible point. 2. Describe the construction and use of an ordinary spirit- level, its attachment to telescopes with cross-hairs, the conditions of, and one or more processes of testing its sensitiveness, and the method of eliminating errors of adjustment. D. 180, 181. 3. State the condition of equilibrium in the free surface of a liquid. D. 140. 4. Show, by the resolution of forces, that if the free surface of a fluid is not horizontal, it is not in equilibrium. D. 140. 5.f Show, by considerations relating to the centre of gravity, that the bounding surface between any two fluids of unequal density is in equilibrium only when horizontal. (D. 145). 6. 1 Describe one or more experiments illustrating the equili- brium of bounding surfaces between different fluids. D. 145. 7. Show that the equilibrium of a fluid would not be disturbed if any portion of it should become rigid, or should be replaced by a fixed rigid body. D. 153. 8.f Show that it is possible, by imagining rigid supports substituted for certain portions of a body of fluid, to treat the remainder as if contained in a /-tube, or in communicating vessels of any shape, without any change of equilibrium. 9.* State the condition of equilibrium of a liquid contained in communicating vessels. D. 178. i o.*J Describe one or more experiments illustrating the fact that a liquid stands at the same level in communicating vessels. D. 178. 56 CONDITIONS OF EQUILIBRIUM IN FLUIDS. [xXXVII. ii. Explain the construction and use of a water-level. D. 179. i2.f Show (by the principle of action and reaction) that the resultant of all the forces exerted by a fluid upon the walls of its enclosure must be equal and opposite to the resultant of the forces exerted by the walls of the enclosure upon the fluid, and hence (by the fundamental principles of equilibrium) equal to the weight of the fluid. (D. 148). i3-t Show that the weight of any portion of a fluid, of whatever shape, is equal and opposite to the resultant of the forces exerted upon this portion by the body of fluid which surrounds it. 14. Show that the difference between the forces exerted upon the top and bottom of a column of fluid with vertical sides (from which the column is supposed to derive no support) must be equal to the weight of the column. D. 139. 15. Show that the resultant forces upon the two ends of a fluid prism with horizontal axis must be equal and opposite. D. 138. 1 6. Find by the triangle of forces, the relation between the resultant forces upon the three rectangular faces of a triangular fluid prism of negligible weight, and show (by similar triangles) that the three forces are proportional to the areas of the three faces in question. D. 137. 57 XXXVIII. PRESSURE. i.f What is meant by intensity of pressure? and in what units is it expressed ? D. 136. 2.f In what sense is the word " pressure," when unqualified, used by most modern writers ? and how 7 does this differ from another sense in w 7 hich it was sometimes employed by earlier writers? D. 136. 3-f Find the (intensity of) pressure, p, (expressed in gravi- tation units) exerted by a weight, w, upon a horizontal surface of area, a. 4. A weight, w, of fluid is contained in a tube with vertical sides, and with an area of cross-section, a. Show that if the upper surface is free from force, the weight of the fluid is borne by the bottom of the tube, alone ; and find the pressure due to the weight of the fluid upon the bottom of the tube, supposing it to be horizontal. Prob. Q. XXXVII.-XXXVIII. 5. Find the weight, w, of fluid of density, d, standing at a height, h, in a tube with vertical sides and horizontal base of area, a ; find also the pressure upon the bottom of the tube. Prob. Q. XXXVII.-XXXVIII. 6. Find, as in the last question, the pressure, p, at the bottom of a vertical column of fluid of density, d, and depth, h, when in equilibrium with the surrounding fluid. Prob. Q. 7. Show (by the principle of action and reaction) that the upward pressure beneath any horizontal area, a, in the body of a fluid, must be equal to the downward pressure on this area, due to the fluid above it. Prob. 8. \ Describe one more experiments showing that the upward pressure at a given depth in a liquid is equal to the downward pressure due to a column of liquid of the same depth and density. D. 144. 5 8 PRESSURE. [XXXVIII. 9. Prove (by the conditions of equilibrium of the pressures on the faces of a small triangular prism, with axis horizontal) that the pressure upon any surface of given area at a given depth in a fluid is the same as upon a horizontal surface of the same area, and at the same depth. D. 137. Q. XXXVII. 15. 10.* Explain the statement that the pressure of a fluid is the same in all directions. D. 137. n. Prove (by considering the conditions of equilibrium of the forces on the ends of a prism with horizontal axis) that the pressure at two points at the same level in a fluid must be the same. D. 138. 12. Prove (by considering the conditions of equilibrium of a column of fluid with vertical sides) that in descending through the vertical distance, h, in a liquid of density, d, the pressure (measured in gravitation units) increases by the amount, hd, whether the starting point is in the free surface of the liquid or not. D. 139. I3.| Describe one or more experiments (with communicating tubes) illustrating the equality of pressure between columns of liquid of given vertical height in tubes having different inclinations. 14. Show (by a zigzag of horizontal and vertical prisms) that the pressure at two points on the same le\el in communicating vessels must be the same. Prob. D. 139, 147. 15. Show (as in the last question) that the pressure of a fluid of density, d, at a depth, 7z, below a given surface is greater than at this surface by the amount, hd, regardless of the shape of the vessel. D. 139, 147. i6.*J Describe one or more experiments illustrating the fact that the force exerted on a given area by a liquid of given depth and density is independent of the shape of the vessel containing the liquid. D. 147. I7.J Describe one or more experiments {e.g. , with E. H. HAUL'S pressure-gauge) illustrating the fact that the pressure 59 PRESSURE. [XXXVIII. in a liquid increases with the depth, and is the same in all directions. 1 8. Discuss the applicability of the principles of hydrostatic pressure to the case of liquids in capillary tubes. D. 190. 19. Show that the pressure of a liquid just within its curved surface in a capillary tube must differ considerably from that just outside of it, and show that the difference is explainable by surface tension. D. 190, 191. 20. Find the sign of the pressure of a liquid raised in vacuo by a capillary tube, on the assumption that its pressure outside of this tube, is zero. D. 190. 21. Explain the attraction between two parallel plates, or between two floating bodies of which (i) both are wet, or (2) neither is wet by the liquid. D. 190. 6o XXXIX. BALANCING COLUMNS. i. *J Describe an experiment in which water is introduced into one branch of a /-tube containing some mercury. (D. 146). 2.* If the water in the last question reaches from the level, a, to the level, b, and the mercury reaches from the level, b, down to the bottom of the /-tube and up again to the level,