THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES GIFT OF John S. KEY AND SUPPLEMENT TO ELEMENTARY MECHANICS. BY DE VOLSON WOOD, PROCESSOR MATHEMATICS AND MECHANICS IN BTETEN8 INSTITUTE OF TECHNOLOGY. FIRST EDITION, JOHN S. PRELL Gtiil & Mechanical Engineer* SAN FRAJS CISCO, CAL, FIRST THOUSAND. NEW YORK: JOHN WILEY & SONS, PUBLISHERS, 53 EAST TENTH STREET. 1894. COPYRIGHT, 1882, BY DE VOLSON WOOD. MH or j. j. LITTII 4 co. ( it 10 M.ASIO* rLACE, NE rear. Engineering Library NOTICE. THIS work contains not only solutions of the exam* pies and answers to the Exercises of the Elementary Mechanics of the author ; but also additions to the text, and other new matter intended to interest the general student of mechanics, and be of service to teachers of the science in connection with any other text book. Valuable assistance has been rendered in correcting proofs by Professor H. A. Howe, of the University of Denver. DEYOLSON WOOD. 737390 Engineering Library KEY AND SUPPLEMENT TO ELEMENTARY MECHANICS. PAGE 1, ARTICLE 1. Motion is a change of position. Motion is determined by the relative position of bodies at different times. If bodies retain the same relative position during successive times, they are said to have no motion in reference to each other; in other words, they are said to be at rest in reference to each other. All bodies of which we have any knowledge are in motion ; hence all motion is, so far as we know, rel- ative. Absolute motion implies reference to a point absolutely at rest, but as no such point is known, such motion has only an ideal existence. PAGE 2, ART. 6. No definition of space will give a bet- ter idea than that obtained by experience. Metaphysi- cians have indulged in speculations in regard to its na- ture, but they are able to assert with certainty only that it has the property of extension. Descartes taught that the properties of extension, known as length, breadth, and thickness, were solely properties of mat- ter, and hence when a body was removed no space remained in the place formerly occupied by it. So far as we know, no space exists which is perfectly de- ll KEY AND SUPPLEMENT void of matter. Perfectly void space is an ideality ; still modern philosophy distinguishes between the thing contained, and that which contains it between matter and the place occupied by matter. It abstracts (so to speak) space from matter, and, in a measure, matter from space. It seems impossible to conceive of matter not occupying space, but it is not difficult to conceive of a given quantity of matter as occupying a very small or a very large space. We are able to con- sider matter in the abstract without considering the dimensions of the space it occupies ; and also we may consider space in the abstract as not including matter. The latter is called absolute space; it is conceived as remaining always similar to itself and immovable. Time is duration. We gain a knowledge of it by the order of events. Every event has its place in time and space, and by means of memory we gain a knowledge of the order in which events occur. "With- out memory we would gain % experience no knowl- edge of time. Sir Isaac Newton considered mathe- matical time as flowing at a uniform rate, unaffected by the motions of material things. This idea induced him to call his new ealcnlns fluxions. Rate refers to some unit as a standard. Thus, to illustrate, rate of interest is a certain amount of money paid for the use of one dollar ; passenger rates refer to the amount paid for one passenger ; rate of shipping per ton is the amount paid for carrying one ton ; rate of motion is the space passed over in one second, one minute, one hour, one day, or one of any other unit. The term velocity is simply the equivalent of the rate of motion. Angular velocity is rate of angular motion TO ELEMENTARY MECHANICS. 3 (p. 4, Art. 12). Acceleration is the rate of change of velocity, being the amount of change in the velocity for one second, or one of any other unit (p. 10, Art. 22). Mechanical power is the rate of doing work (p. 55, Art. 99). Rates may involve two units. Thus, rate per ton- mile implies a certain amount paid for one ton for one mile ; passenger rates are often an amount for one person for one mile ; mechanical power, or rate of doing work, is the amount of work done in one foot for one second, or by one pound for one sec- ond, etc. Hate is a thing used for measuring quantities, as a yard-stick is used for measuring cloth, a chain for measuring land, the pound for weighing groceries, ton for measuring merchandise, etc. PAGE 4, ART. 12. The definition here given for rotary motion is applicable to the case where the motion is in a curved path not circular, as CD. But the analysis given in the text is not applicable to this case. PAGE 6, ART. 14. Just after Fig. 5, for If two ve- locities, etc., read If two concurrent simultaneous velocities, etc. PAGE 8, ART. 20. Speaking of the rotation of the moon, suggests an interesting question in practical mechanics. If the wheel rolls around on the cir- cumference of an equal wheel A, will the former turn once or twice on its own axis? Mark the point a which, initially, is in contact with KEY AND SUPPLEMENT the wheel A, and roll B half around A ; it will be observed that the marked point will then be at the left of the centre of B, as it was at the start. Con- tinuing the rolling, the point a will again be at the left of the centre when B has gone completely around A ; hence it is sometimes asserted that the wheel B has turned twice on its axis. But we wish to show that it has turned but once on its own axis, and the whole wheel has been rotated once about the axis of the wheel A. Let the axis of the wheels be at right angles with each other, then it will be evident, from mere inspection, that when B has turned once on its axis it will have gone once around A. Thus B will have gone once around the axis of A, and once about its own axis. Next, incline the axis of B upward, so as to ap- proach a parallelism to A, and the same result will be seen from mere inspection, and it will continue to remain evident as it be- comes nearer and nearer parallel, and when they become actually parallel, the same condition will hold true. Hence, in the former figure, the wheel B will turn but once on its own axis in rolling once around A. The same result may be shown in an- other way. Let a block be placed at a facing a mark on the axis of B, and conceive this axis to be rigidly connected with the axis of A while the wheel B is TO ELEMENTARY MECHANICS. 5 free to turn on its own axis. In this way the axis of B will be carried bodily about A. When B has rolled half around A, it will be found that the block will face the same direction in space say towards the east but that it will not face the mark on the axis, for the mark will be on the opposite side of the axis. Continuing the rotation, it will be found that the block will face the mark only once at each revolution about A. Similarly, the moon turns but once on its own axis in one revolution about the earth, but the rotation about the two centres are not exactly coincident ; for it is found by observation, that in some parts of the orbit more of the surface of the moon is seen on the eastern (or western) side than in other parts of the orbit ; thus showing that the rotation about the earth is sometimes faster, and at other times slower than the rotation of the moon about its own axis. This phe- nomenon is called Libration. EXERCISES. PAGE 9. 1. 4^- miles. 2. The former. 3. 66 feet per second. 4. 17 feet ; Vln feet. 40 x 5280 5 " 4 * 60x60 = ^ SCCOnds ' 200 x 2zr 200 x 360 10 _ A ^ = 6|7r m arc ; or ^ degrees. 7. V& + 2" = Vl3~= 3-605 + miles per hour. 6 KEF AND SUPPLEMENT PAGE 10. 8. 1/15' 4- = ^3961 = 20-98 ft. per sec- ond. 9. 0-0009J. 10. See Article 14. v = A/5' + 10" + 100 cos 60 = Vl25 + 50 = V175 = 13-22 + feet ; hence the distance between them in two seconds will be 2 x 13-22 + = 26-44 + feet. PAGE 10, ART. 22. Observe that acceleration is not the rate of change of motion, but the rate of change of the rate of motion. It is the rate of change of a rate. Tlifi rate of change is usually measured in the same units as the rate of motion. If one is in feet per sec- ond, the other is also. It is possible to conceive of mixed units. Thus in the case of falling bodies, the velocity at the end of the first second is 16-^ feet, and the acceleration is 643| yards per minute. Strive to get a clear conception of the meaning of acceleration and of its measure. It is one of the ele- ments of the absolute measure of force. PAGE 13, ART. 26. The expression " The locus of these ,\ points will be a parabola," means that if any number of points in the path be determined in the same manner, they will all be in the arc of a parabola. A parabola is a curve which may be cut from a right cone by a plane parallel to one of its elements. (See Author's Coordinate Geometry.) TO ELEMENTARY MECHANICS. 7 EXAMPLES. 1. Space 250 feet ; velocity 50 feet per second. PAGE 14. 2. / = 25 feet ; * = 12 feet. 3. During 4 seconds the space will be s = % of 32 x 4" (Art. 14), and during 3 seconds the space will be \ of 32 x 3* ; hence during the 4th second the space will be J x 32 (4 a - 3 s ) = 112 feet. 4. By means of equation (1), Art. 25, find 6 feet per second. 5. t= Vl2^5 = 3-533 + seconds. 6. /= (20 -T- 120) 3-28 = 0-54f feet per second. 7. 32| -5- 3-28 = 9-8 metres per second. PAGE 15. Matter and force are two grand realities of the external world, and of these we know nothing directly. Our knowledge of the former is confined to its properties, and of the other to its laws of action. But we have no reason to believe that one exists in- dependently of the other. In our earlier experiences matter is conceived to be hard, gross, and unyielding ; but later we find that it is yielding, and that many solids, as iron and lead, may be changed to liquids by heat, and that liquids may be changed to gases so that matter is proved to be more or less viscid or at- tenuated. Solids are porous. Changes of form are effected by forces, so that some metaphysicians have reasoned that there may possibly be no gross matter, but, instead thereof, those things which we consider as bodies are only aggregations of forces. On the other hand, all investigations in mechanics proceed on the hypothesis that matter is in no sense a force, 8 KEY AND SUPPLEMENT or an aggregation of forces, but that it is something distinct from force, something npori which force acts. In the case of attraction, the matter in two bodies may remain constant, while the force exerted by each upon the other will depend upon the distance be- tween them. It is true, however, that we gain a knowledge of matter only through the action of forces. Every avenue to the mind through the senses is an agent for transmitting the result of the action of cer- tain forces, and the very act of transmission brings into play certain forces. PAGE 16, ART. 28. Mathematics applied to the laws of physical science enables us to determine magnitudes which far transcend the powers of accurate measure- ment or even of conception. Sir Win. Thompson gives four methods for ascer- taining the mean distance between molecules. Optical dynamics. Contact electricity of metals. Capillary attraction. Kinetic theory of gases. " Optical dynamics leaves no alternative but to admit that the diameter of a molecule, or the distance from the centre of a molecule to the centre of a contiguous molecule in glass, water, or any other of our transpa- rent liquids and solids, exceeds one ten-thousandth of the wave-length of light, or a two-hundred-millionth of a centimetre " (sooooVooooo of a metre). " However difficult it may be even to imagine what kind of thing the molecule is, we may regard it as an established truth of science that a gas consists of mov- ing molecules disturbed from rectilineal paths and TO ELEMENTARY MECHANICS. 9 constant velocities by collisions or mutual influences, so rare that the mean length of proximately rectilineal portions of the path of each molecule is many times greater than the average distance from the centre of each molecule to the centre of the molecule nearest it at any time. If for a moment we suppose the mole- cules to be hard elastic globes all of one size, influenc- ing one another only through actual contact, we have for each molecule simply a zigzag path composed of rectilineal portions, with abrupt changes of direction." " If the particles were hard elastic globes, the aver- age time from collision to collision would be inversely as the average velocity of the particle. But Max- well's experiments on the variation of the viscosities of gases with change of temperature prove that the mean time from collision to collision is independent of the velocity, if we give the name collision to those mutual actions only which produce something more than a certain specified degree of deflection of the line of motion. This law could be fulfilled by soft elastic particles (globular or not globular), but not by hard elastic globes." " By Joule, Maxwell, and Clau- sius we know that the average velocity of the mole- cules of oxygen, or nitrogen, or common air, at or- dinary atmospheric temperature and pressure, is about 50,000 centimetres per second (500 metres per sec- ond, or about 1,600 feet per second), and the average time from collision to collision a five-thousand- millionth of a second (goooo^oooo)- Hence the aver- age length of path of each molecule between collis- ions is about Yo-oVoir f a centimetre " (iTnnnnnnr ^ a metre). 10 KEY AND SUPPLEMENT " The experiments of Cagniard de la Tour, Faraday, Regnault, and Andrews on the condensation of gases do not allow us to believe that any of the ordinary gases could be made forty thousand times denser than at ordinary atmospheric pressure and temperature without reducing the whole volume to something less than the sum of the volumes of the gaseous molecules, as now defined. " Hence, according to Clausius, the average length of path from collision to collision cannot be more than five thousand times the diameter of the gaseous molecule ; and the number of molecules in unit of volume cannot exceed 25,000,000 divided by the vol- ume of a globe whose radius is that average length of path. Taking now the estimated y^Vur of a centime- tre for the average length of path from collision to collision wo conclude that the diameter of the gaseous molecules cannot be less than sooooVopir f a centime- tre Cnnnnf crznnnnr f a metre) ; nor the number of molecules in a cubic centimetre of the gas (at ordi- nary density) greater than 6,000,000,000,000,000,000,- 000." "The densities of known liquids and solids are from five hundred to sixteen thousand times that of atmospheric air at ordinary pressure and tempera- ture : and, therefore, the number of molecules in a cubic centimetre may be from 3 x 10** to 10 s 6 (that is, from three million million million million, to a hundred million million million million). From this the distance from center to nearest center in solids and liquids may be estimated at from ^OTJOTTFITU to iinnnhnnnr of a centimetre (rnn.uVu.nnn; to TO ELEMENTARY MECHANICS. H 46060000UUU of a metre). The four lines of argu- ment lead all to substantially the same estimate of the dimensions of molecular structure. Jointly they establish with what we cannot but regard as a very high degree of probability the conclusion that, in any ordinary liquid, transparent, solid, or seemingly opaque solid, the mean distance between the centers of contiguous molecules is less than the hundred- millionth, and greater than the two thousand-millionth of a centimetre. To form some conception of the coarse-grainedness indicated by this conclusion, im- agine a rain drop, or a globe of glass as large as a pea, to be magnified up to the size of the earth, each con- stituent molecule being magnified in the same pro- portion. The magnified structure would be coarser- grained than a heap of small shot, but probably less coarse-grained than a heap of cricket-balls." (Ex- tracts from a paper by Prof. Sir "Wm. Thomson on the size of Atoms, Am. Jour, of Science and Art. 18TO, vol. ii., pp. 38-45.) Mr. 1ST. D. C. Hodges, in an article on the size of molecules (Phil. Mag. and Jour, of Science, 1S79, vol. ii., p. 74), says : " If we consider unit mass of water, the expenditure on it of an amount of energy equivalent to 636.7 units of heat will convert it from water at zero into steam at 100. I am going to con- sider this conversion into steam as a breaking-up of the water into a large number of small parts, the total surface of which will be much greater than that of the water originally. To increase the surface of a quantity of water by one square centimetre requires the use of .000825 metre gramme of work. The total 12 KEY AND SUPPLEMENT superficial area of all the parts, supposing them spher- ical, "will be 4 7t r*N) the number of parts being W. The work done in dividing the water will be 4 it r* N x .000825. For the volume of all the parts we have f n r*N. This volume is, in accordance with the requirements of the kinetic theory of gases, about 3,000 of the total volume of the water. The volume of the steam is 1,752 times the original unit volume of water. Hence i nr*N 3000 = 1752 4 n r* ^.000825 = G36.7423. One unit of heat equals 423 units of work (in French units); solving these equations for '/ and JV, we get r = .000000005 centimetre (or diameter = .00000001 centimetre = nnnnrWinnnF metre), a quantity closely corresponding with the previous results of Sir "Wm. Thomson, Maxwell, and others ; and N equals 9000 (million) 8 , or for the number in one cubic centime- tre 5 to 6 (million) 3 ." The extreme tenuity of a gas is further shown by the following extract taken from the Beiblatter zu den Annalen der Phymk und Chemie, 1879, No. 2, p. 59. "At 0C. and IQOmni pressure a cubic cen- timetre (.061 cubic inch) holds nearly one hundred trillions of gas molecules. Under these conditions the molecules themselves fill nearly the -j-^ of the space occupied by the gas. The absolute weight of a 15 hydrogen molecule is represented by -^ or soootr f a milli- metre ; t and it makes this movement in 10oo6 1 u - u6o66 of a second. Now the wave length of a chemical ray is about ^-^ f a millimetre, hence we find that the molecule of air travels through a distance which is one-fourth as long as the length of this particular wave in this fraction of a second. J According to Pouillet, the mechanical energy of a cubic mile of sun light at the earth equals 12,050 ft. Ibs. * Clausius's limit is 6,000 times this amount. f Clausius's estimate is one-half this value. \ Mr. E. H. Cook, in an article on the Existence of the Lumi- niferous Ether (Phil. Mag. and Jour, of Science, 1879, vol. I, p. 235), after quoting the above figures from Mr. Stoney's paper, adds : "Then in one moment the particle travels roTJooTHT f a metre in TuoHootfiToocT of a second;" also in his deductions he adds : " hence we find that the molecule of air travels through a distance which is more than twice as long as the length of this particular wave in this fraction of a second." The reader can easily see that this deduction is erroneous. KEY AND SUPPLEMENT (Phil. Mag., 1855, vol. ix., p. 39) ; accepting which Sir Wm. Thomson calculated the weight in pounds of a cubic foot of the ether of space (that which is instrumental in transmitting light, and called ether) by the formula : where g = 32 (acceleration per second of gravity), F=the velocity of light per second, being about 192,000 miles per second, and n = -g^, being the ratio of the greatest velocity of a rotating particle to the velocity of light. Herr 1 Glan asserts that n is not constant, that he found n = ^V in one case, and y^Vs" in another. (Am. Jour, of Arts and Sciences, 1879, vol. xviii., 404. Annalen der Physilt und Chemie, No. 8, 1879, p. 584.) Assuming n = 7 V> we find that a cubic foot would weigh about w _ 83 x 32| x 5Q2 ~ (52bO x 192000) 8 1-5 ,, = ioTo ^.nearly; and for the weight of a cubic mile - - pound. The weight of a volume of the size of the earth would be about 240 pounds. Admitting this result, it follows that a sphere equal in diameter to the diameter of the earth's orbit (or say 190,000,000 miles) will contain an amount of ethereal matter nearly equal to j^Vo- of that of the mass of the earth. Probable tension of the ether of space. The above TO ELEMENTARY MECHANICS. 15 results combined with one or two other plausible assumptions enable us to find a probable value of the tension of the light ether. As stated above, the velocity of the particles of air producing a pressure of 15 Ibs. per square inch, is about 1,600 feet per second, which is about 50 per cent, more than the velocity of sound in air. Assuming now that the normal velocity of the ether particles is somewhat more than the velocity of light or, more definitely, that it is 195,000 miles per second, and that the tension is directly as the masses, and also as the square of the velocities of the particles ; also that the weight of a cubic mile of the ether is -^- lb., as found above, and observing that 100 cubic inches of ordinary air weigh 31 grains, and 7,000 grains make a pound, we have for the pressure P in pounds of the ether upon a square inch J_ iK /195000 x 5280\ 8 10 9 P = 15 x ( x 1600 J 31 1728 100 - = 0-00000185 = lb. per inch of section. A Mr. Preston, an English writer, in his work on the Physics of Ether, estimates, or rather assumes, 500 tons per square inch as a probable inferior limit of the pressure (p. 18), and, with this as a basis, he finds the weight of a cubic mile of ether to be about 220 Ibs. (p. 120). But as he has used 56-5 grains for the weight of a cubic foot of air when it is nearly ten IQ KEY AND SUPPLEMENT times this amount, he should have found for the weight of a cubic mile of ether nearly one ton. His assumption, however, in regard to pressure is quite arbitrary, and does not seem to be well founded. It seems improbable that there should be so much mass in the ether of space. Even 240 Ibs. for a volume equal to that of the earth seems a high value when \\Q consider the amount that must be displaced by the planets while moving about their orbits. Temperature of the Sun. The following may be interesting, although it does not fall under the article above referred to. " The effective temperature of the sun may be defined as that temperature which an incandescent body of the same size placed at the same distance ought to have in order to produce the same thermal effect if it had the maximum emissive power. If we consider the surrounding temperature during the observation to have been about 240 we obtain . . . for the effective tempera- ture in degrees centigrade 9,905.4 ... I think, then, that I may fairly conclude that the temperature of the sun is not very dif- ferent from its effective temperature, and that it is not less than 10,000, nor much more than 20,000' centigrade." Phil. Mag., 1879, vol. ii., pp. 548-550. See also Am. Jour. Sc. and Arts, 1870, vol. ii., p. 68. Sir Win. Thomson calculates the mechanical energy of the solar rays falling annually on a square foot of land in latitude 50 to equal 530,000,000 foot pounds, or 396 H. P. per yard per day. He finds tha*, the heat alone hourly given out by each square yard of the solar surface is equivalent to 63,000 horse power, and - would require then the hourly combustion of 13,500 Ibs. of coal. Appleton's Cyclopaedia, 1868, vol. ix. p. 23. PAGE 16, AET. 29. A better definition is Force is an action between bodies for this form of the statement recognizes the existence of at least two bodies in every action. A force never acts upon one body without producing an equal opposite action upon an- TO ELEMENTARY MECHANICS. 17 other body. We speak of the action of a force upon a body because in most cases the second body is so large, relatively, that the force produces little or no perceptible effect upon it. But according to the law of Universal Gravitation each particle in the universe is attracted by every other particle with a force which depends upon their masses and the distances between them ; hence, in a highly refined sense, it may be said that every force producing motion involves every body in the universe. The entire universe of matter is bound together by a something an action which we call Force. H/very phenomenon which we witness in the physical world is the result of force, acting through space, or during a certain time. Force alone is stress. In other words stress is force abstracted from time and space. The science of Stress is the science of Statics. Stress is always measured in pounds or its equivalent. If a force pro- duces motion, that part of the phenomenon which is abstracted from time and space is stress, so that the attractive force between the earth and moon, or be- tween the earth and sun, measured in pounds, is stress. When force is compounded with time or space the result is work, or energy, or momentum, as will here- after be shown. The following are some definitions of force as given by different authors : La Place says : " The nature of that singular mod- ification, by means of which a body is transported from one place to another, is now, and always will be, unknown ; it is denoted by the name of Force. 18 KEY AND SUPPLEMENT We can only ascertain its effects, and the laws of its action." (Mecanique Celeste, p. 1). " .Force is an action between bodies, causing or tending to cause change in their relative rest or mo- tion " (Rankine's Applied Mechanics, p. 15). " Force is that principle of which, considered sim- ply as a mechanical agent, we know but little more than that when it is imparted, that is, put into, a body, it produces either motion alone ; or strain, with or without motion." " What is called overcoming inertia, is simply putting in force" (Trautwine's Engineer's Pocket Book, p. 445 and p. 447). We consider this as a misuse of terms. We cannot safely say that force is put into a body. When force acts upon a body free to move, energy is put into the body, when force and space are involved in the result, and time is abstracted (see text, p. 66) ; or taoinentum when the elements involved are force and time, space being abstracted (see Chap. Y ). Although it is too early in the text for a full discussion, we add a re- mark for the benefit of those who have some knowl- edge of the subject. When a constant force, F, acts through a space, s, it does the work represented by the product of F and s, or Fs. If the body upon which it acts is wholly free, the entire work will be stored in the body, and is then called energy, the measure of which is % Mv*; hence Fs = Mv*. Elimi- nating s by means of Eq. (2), p. 12, of the text, gives Ft = Mv, which is the measure of the effect of force combined with time, and the second member is the measure of the momentum (text p. 78). It would be better to say that force and time, or force and space, TO ELEMENTARY MECHANICS. 19 are put into a body than that force alone is put into it. " Nothing but force can resist force." "Matter, in itself, cannot resist force " (ib. p. 445). By re- sistance is here understood to be such a condition of things as that motion will not result, and, in this sense, these statements are correct. "Force put into a body " is, properly speaking, put- ting it under stress, and the body is said to be strained, but no amount of internal stress will produce motion of the body. " Whatever changes the state of a body or the ele- ments of a body, with respect to rest and motion, is called force" (Bartlett's Analyt. Mech., p. 17). " Force is defined as that which changes or tends to change a body's state of rest, or motion, and any given force may be measured bj r the acceleration it imparts to a gramme." (Cumraing's Theory of Elec- tricity, p. 5). " Force is whatever changes or tends to change the motion of a body by altering either its dimension or its magnitude ; and a force acting on a body is meas- ured by the momentum it produces in its own direc- tion in a unit of time." (Maxwell's Theory of Heat, p. 83). It will appear that the momentum produced in a unit of time is the same as the acceleration, and hence the two last definitions are equivalent. "We, however, deem it advisable to avoid the expression momentum produced because it is liable to be confounded with the actual momentum of a body, although it is not in- 20 KEY AND SUPPLEMENT tended to even imply the latter. Acceleration is specific and correct. " Force is matter in motion, nothing more, nothing less ; the abstract idea of force without matter is a nonentity." (Nystrom On tlie Force of Falling Bodies and Dynamics of Matter, p. 20). The preceding remarks will show that this defini- tion contains a misapplication of terms. Matter in motion is either Energy or Momentum, according as time or space is abstracted in considering the ele- ments which enter into the combination. " Force is a mere name, but the product of a force into the displacement of its point of application lias an objective existence." " Force is the rate at which an agent does work per unit of length." " The mere rate of transference of energy per unit of length of that motion is, in the present state of science, very conveniently called force." (Lecture by Prof. G. P. Tait, Nature, 1876, vol. xiv. p. 462). In regard to these views, we observe that the name applied to anything is a mere name. In a certain sense it is an ideality, but generally the name stands for a reality. In this case, if force is a mere name, what is the sense of the remainder of the sentence ? How can a force have a poi,nt of application if the force is a mere name? And granting that it may have, how can the product have an objective exist- ence? In regard to the second definition, it is analytically correct, but we consider it rather as a deduction than TO ELEMENTARY MECHANICS. 21 as a fundamental definition. It is shown on pages 52 and 67 that or, in terras of the calculus, Fds = d'(lMv*) = $Md(tf) = dK. From the former we have and from the latter ~~ds > hence, generally, force is the rate of doing work per unit of length. But is force merely a rate ? What shall be said of force as a stress, where no transfer- ence of energy takes place ? That this definition is not elementary, but a mere deduction, is not only evi- dent, but may be more forcibly shown by means of other deductions. Thus, from the former equation we have s = hence, space is the rate of doing work per unit of force (per pound). Or again, 22 KEY AND SUPPLEMENT hence velocity is the square root of twice the rate of doing work per unit of mass ! Or, again, in regard to momentum, page 78, Ft = Mv ; ' t ' hence, force is the rate of producing momentum per unit of time. Or, again, _ M v Tjr. hence, time is the rate of producing momentum per unit offeree ! Now all of these are correct deductions ; but the fundamental equations are established on the hypothe- sis that all, except velocity, are substantial quantities. The value of J^is fundamentally measured in pounds. Indirectly it may be measured in a variety of ways as shown above, and as will be still further shown in Article 86 of the text. PAGE 17, ARTS. 31 AND 32. Terrestrial Gravitation as a force causes, or tends to cause, bodies to move to- wards the earth, and when a spring balance or other weighing machine is interposed to prevent any move- ment, the intensity of the impelling force may be de- termined in pounds or an equivalent. PAGE 19, ART. 35. The fundamental idea of a point of application of a force is that of a definite attachment, like the attachment of a rope, or chain, or rod of iron, to a body ; but it is certain, in regard to the forces of nature as gravity, chemical forces, etc. that the conception is erroneous, for there is no attachment. TO ELEMENTARY MECHANICS. 23 Still it may be conceived that the force acts upon a particle which may still be considered as the point of application, and thus the old term, with its gross as- sociations, is useful in the most refined sense. PAGE 20, ART. 36. Inertia is a name merely to express the fact that matter has not of itself power to put itself in motion, or being in motion to bring itself to rest, or even to change its rate of motion. Yet some writers call Inertia a force, and others, with little if any more propriety, speak of the/bra? of inertia. M. Morin, a French physicist, attempted to prove that inertia is a force. He took a prism standing on its base, and by a sudden pull or push applied to its base caused the prism to fall backwards. He argued that the falling over of the prism indicated the action of a force. A force might have been applied at the top of the prism which would have overturned it directly, and Morin argued, that when it overturned by a sud- den action at the base, there must have been a force equivalent to one at the top, and this he called the force of inertia. The fact is, any force applied to a body, not acting in a line through its centre, tends to rotate the body ; and in all cases where tho body is free, will cause it to actually rotate. If the prism referred to, standing on its base, be acted upon by a force applied at any point above the base, it will not be overturned unless the moment of the force exceeds the moment of the weight in reference to the point about which it tends to turn. If the force applied at the base be so intense as to produce a rotary mo- ment exceeding the moment of the weight above re- ferred to, it will cause the prism to fall by rotating 24 KEY AND SUPPLEMENT backwards ; but if the force be less intense, it will simply cause the body to slide on the plane. Again, inertia does not fulfill any of the conditions of a force. It is not an action between bodies. It is not an action in any sense. It cannot be measured by pounds. It is a negation an entire lack of some- thing a lack of force. We repeat, it is not a force. EX ERCISES. PAGK 22. 1. The least force. One object of some of these exercises is to enable the student to get a correct idea of the relation between force and the result- ant motions of bodies by the inductive method. The student who has not correct notions of these relations will doubtless insist that it requires more force to move a large body than a small one ; and he may go so far as to say that it will take 10 pounds of pull or push to move a body weighing 10 pounds. But the fact must be perceived that the smallest force (10 Ibs. of push, for instance) will just as certainly move 100 Ibs. of matter free to move in the direction of the push, as it will one pound. It will not move the former as rapidly as the latter or, in other words, it will not move it as far in the same time. In this way we get an idea of the fact that the visible effect of a force depends conjointly upon the mass moved, and the space through which it is moved in a given time. If necessary make an experiment by suspending different weights with equally long strings, and pull them sidewise by a string attached to the body passing over a pulley or edge of the TO ELEMENTARY MECHANICS. 25 table and holding at the suspended end a small weight. It will be found that any weight which is sufficient to overcome the friction of the string on the pulle} 7 or edge of the table will pull the heaviest weight sidewise. Observe that the experiment is simply to show actual movement, and not the amount of movement. 2. The least force. Also the least force would de- flect it from its course. This shows that a force will have the same effect upon a moving body as upon one at rest. PAGE 23. 3. Because it is opposed by an equal oppo- site force. 4. 100 pounds. A man once pulled a spring bal- ance so as to indicate, say, 100 pounds. Another gen- tleman asserted that he could pull two balances at- tached end to end so that each would indicate 100 pounds. This he did to the astonishment of the ob- servers, but their astonishment ceased when they found that it was no more difficult to pull two in that way than one. 5. Yes. The fact that the boat is in motion, does not affect the result. Hence we have the principle that " action and reaction are equal and opposite." 6. No. To show it (should there be doubt) assume that a string passes from each sled to the hand, then will the tension on each be less than 10 pounds, but on both strings it will be just 10 Ibs. Conceive that the strings become one back to the first sled the ten- sion on the one will be 10 Ibs. but on the part back of the first, it will be the same as before which was evidently less than 10 Ibs. If the two sleds are 26 KEY AND SUPPLEMENT of equal weight the tension on the connecting cord will be 5 Ibs. I have heard students assert that no tension could be produced unless there were a re- sisting force showing that they had not yet a correct conception of the relation between forces and masses. It requires force to move a free body. In such cases I have asked them to conceive that a cord were at- tached to the moon, and that they pulled upon it ; when they will severally admit that a pull of ten or more pounds may be easily exerted. If the sleds were not very heavy, it would not be possible for the boy to run sufficiently fast to maintain a constant pull of 10 pounds for a long distance ; but if it can be done for a few feet only, it will answer the purpose of the illustration. T. No tension. PAGE 23, ART. 48. The so-called Three Laws of Motion did not spring suddenly into philosophy. There was a long period of darkness succeeded by twilight and dawn before the truth shone out clearly. The prin- ciples of the three laws were known, and to a consid- erable extent realized, before Newton's time, but per- haps they had not been so clearly and sharply defined by any preceding writer, and much less had they been made the foundation of mechanical science ; hence there is a certain propriety in calling them Newton's Laws. Correct notions in regard to these principles date from the time of Galileo. Prior to his day, the leading philosophy was Aristotelian. Aristotle flour- ished between 300 and 400 years before Christ. In his philosophy there was no distinct idea of /0m? as a cause, much less any idea of a relation between the TO ELEMENTARY MECHANICS. 27 cause of motion and the momentum produced. He taught that a heavy body would fall faster than a lighter one that when a body is thrown by the hand it ou<^ht to cease to move as soon as it left the hand o were there no surrounding impulses, but that it con- tinued to move because the hand sets in motion the air about the body, and that the air acted afterwards in impelling the body. He divided motions into Natural and Violent ; the former of which is illustra- ted by a falling body, in which the motion is constantly increasing, and the latter by a body moving on the ground, where the motion is constantly decreasing. It must not, however, be inferred that philosophers had no idea of cause and effect. Some general notions of this kind have always been entertained. Between the period of Aristotle and Galileo, many important principles were established. Archimedes (born 287 B. c.) developed some important properties of the Centre of Gravity, established the principle of the Lever, and some of the principles of Hydrostatics. History seems to show that the advanced position se- cured by this eminent philosopher was not main- tained, and that little or no advance was made until the time of Stevin or Stevainus, as commonly writ- ten (1548-1620). His determination of the condi- tions of equilibrium of the inclined plane is so in- genious, it is worth repeating. Consider two inclined planes having a common vertex and horizontal base. Conceive a uniform long chain to be placed on them, and joined underneath so as to hang freely. He showed that it would hang at rest without friction, because any motion would only bring it into the same 28 KEY AND SUPPLEMENT condition in which it was at first. The part hanging below would evidently be in equilibrium by itself, hence if that part be cut off the remaining part will be at rest ; hence the condition of equilibrium is the weights on each part must l)e exactly proportional to the lengths of the planes. If one side becomes verti- cal the same proportion holds true. Galileo forms the grand connecting link between the philosophers of the ancient and modern physical sciences. He was born at Pisa, February 18th, 1564, 39 years before the death of Michael Angelo, and 21 years after the death of Copernicus, and died on the 8th of January, 1642, the year in which Sir Isaac Newton was born. The science of motion began with him. He taught that motion was due to force that all bodies in a vacuum would fall with equal velocities that inertia of matter implies persistence of condi- tion he gave a satisfactory definition to momentum also stated with approximate precision the princi- ple that " action and reaction are equal " also estab- lished the principle of " virtual velocities," which was made by Lagrange to include all of mechanical science in one expression. He made a mathematical analysis of the strength of beams, of projectiles, of the pendulum, of floating bodies, and of the inclined plane. For his investigations in other fields of sci- ence see some biographical sketch. PAGE 23, ART. 49. First Law of Motion. Says "Wliew- ell, in his History of the Inductive Sciences : "It may be difficult to point out who first announced this Law in a general form." We have already seen that the facts involved in it were recognized by Galileo. TO ELEMENTARY MECHANICS. 29 It is equivalent to saying that every change is due to a cause, and yet, to cover the entire ground, this state- ment needs modification. Motion is due to a cause, but when the cause ceases, the motion of a free body does not cease, it simply becomes uniform. Change of position is not then necessarily due to a coexisting cause, but may be due to a cause remote in time. Change of condition, however, requires a present act- ing cause ; and the latter will produce either a change in the rate of motion, or of the direction of motion or of both. The law cannot be proved by direct experiment, for it is practically impossible to remove from the body all acting forces, and hence uniform motion un- der the action of no forces is not realized by experi- ment. It may, however, be observed that the less the resistance the more nearly uniform will be the motion, and hence we are led to infer that if all resistance could be removed, the motion would be strictly uni- form. Similarly, it is observed that a body projected on a very smooth plane moves so nearly in a straight line that we are led to infer that if there were no de- flecting causes the path would be exactly straight. The law is the result of induction rather than of proof, but it appears so perfectly reasonable that we assent to it as soon as it is properly illustrated. The strongest proof of its correctness lies in the fact that deductions founded on this hypothesis agree with the results of observation. The process of induction consists in conceiving clearly the law, and in perceiving the subordination of facts to it. 30 KEY AND SUPPLEMENT PAGE 23, ART. 50. Second Law. Change of motion is in proportion to tJie acting force. If all bodies were of equal size it would only be necessary to consider their relative velocities in determining the effect of forces. But a larger body requires more force than a smaller one of the same substance to produce the same velocity under the same circumstances ; in other words mass and velocity are both involved and the term motion here means momentum. This also agrees with Newton's explanation of the term. It is better, therefore, to word the law thus : Change of momen- tum is in proportion to the acting force. It must be particularly observed that the law does not assert that momentum varies as the force y but that it is a change of the momentum that varies as the force. This law was clearly perceived by Galileo, and by means of it and the first law, he determined that the path of a projectile in a vacuum was a parabola. The law, however, was not considered fully established until the theory in regard to the motion of the earth, involving both this law and the law of universal grav- itation, was realized. The triumph of both laws was complete at the same time. PAGE 24, ART. 51. Third Law. Action and Reaction are equal and opposite. The first and second laws refer to one body only ; this involves two bodies. It asserts that an action between two bodies is of the same intensity upon each, but that the direction of action upon one is directly opposed to that upon the other. Every action implies an equal opposite action one being called a reaction in reference to the other. No force, acting in one direction only, exists ; TO ELEMENTARY MECHANICS. 31 it is always accompanied by an equal but opposite ac- tion. No force is known to exist without the pres- ence of matter. Force does not act in curved lines ; the action and reaction between two particles is in the right line adjoining them. Newton gave three examples illustrating this law : If any one presses a stone with his finger, his fin- ger is also pressed by the stone. If a horse draws a stone fastened to a rope, the' horse is drawn backward, so to speak, equally towards the stone. If one body impinges on another, changing the motion of the other body, its own motion experiences an equal change in the opposite direction. It does not seem rational that the stone will pnsli the finger in the same sense that the finger presses the stone. The finger appears to be an active agent while the stone is inert. In the strictest sense we should say that, in the attempt to press a stone a force is developed between the finger and the stone, which force acts equally in opposite directions ; in one direction against the finger, in the other against the stone. Similarly, in regard to the horse and stone. In the former example the condition is statical, but in this the horse is supposed to move the stone. The horse, evidently, is not actually pulled backward, although there is an actual backward pull upon the horse by the rope. The fact is, that, in the effort to draw the stone, a force is developed which produces tension in the rope, which tension acts to pull the stone one way, and the horse the opposite way. As the horse 32 KEY AND SUPPLEMENT is able to take a footing on the earth, he is able to exert a force on the rope equal to, or exceeding, that necessary to overcome the friction of the stone, and thus move the stone. The pressure between the horse and the earth also acts in opposite directions. In the third illustration another idea is presented. Motion is used in the sense of momentum, as before stated, and it should read, the change of momentum in two impinging bodies is the same in both bodies but in opposite directions. Tin's is a necessary result of the second and third laws. The forces bcinsj o measured by the change in the momentum, and the pressures being equal between the impinging bodies, it follows that the changes in their momenta must be equal. Hence if one loses momentum the other must gain, and the loss in one case must equal the gain in the other. Thus if the body whose mass is J!/i impinges upon another whose mass is Jf 2 , Vi and v z , their respect- ive velocities before impact, and v\ and v' z their re- spective velocities at any instant after the first contact, and assuming, as we will in establishing the formula, that the velocities are in the same direction, in which case it will only be necessary to change the sign of one of the velocities if they move in opposite directions ; then will the momentum of MI before impact be and after impact and as M^ is supposed to be the impinging body it will lose velocity, and we have for the momentum lost by MI TO ELEMENTARY MECHANICS. 33 Similarly, the momentum gained by M z will be which, according to the third law, must equal that lost by the former body ; hence for all stages of the mo- tion after the first contact, not only during compres- sion, but also at and after separation, we have If J/2 be the impinging body, we have which easily reduces to the former equation. If they move in opposite directions before impact make i\ or v 2 negative, the impact is here supposed to be direct and central, for which case the equations are true whether the bodies be elastic or non-elastic. Several cases are discussed on pp. 85-90 of the text. (It appears that Whewell, in his History of the Inductive Sciences, has not drawn a sufficiently clear distinction between the second and third laws. He appears to hold that the third law gives a measure of the pressure or force, whereas the sec- ond law is the only one of the three that gives it). PAGE 24, ART. 52. We are not informed who first gave the laws for the composition and resolution of forces ; but Galileo was one of the first to make use of them in explaining curvilinear motions. The method was systematized by Descartes by the aid of the systems of rectilinear axes. KEY AND SUPPLEMENT EXERCISES. PAGE 26. 1. Because the force of gravitation draws it from the rectilineal path in which it was projected. 2. The least force. 3. 500 pounds. PAGE 27. 4. Midway between their initial positions. 5. He must aim to walk southwesterly. To find the direction, draw a line, AB, of any A B length to represent the one mile due east, and a line, AC, perpendic- ular to AB, and of such length that the hypothenuse, BC, will be three times as long as AB / then will the angle AB C represent the required direction. We have cos B = - ; .-. ABC = 40 32. 6. Make AB = 3, the angle DBG = 45, and BC equal to 8; join AC, then will AC represent the resultant direc- tion. A numerical solution gives DA C = 33 10' and the course will be S. 56 50' E. TO ELEMENTARY MECHANICS. 35 BD = DC = 8 sin 45 = 8 x 4 A/2 = 4 V2 ^4j9 = 3 + 4 V2 n^ 4V1 32 - 12 A/2 tan. .ZM C = 7= = - = 3 + 4 V2 23 0-6535; .-. DAC= 33 10'. The velocity will be 10-34 miles per hour. 7. It will be 10 x cos 45 = $ A/2 x 10 = 5 x 1.4142 + =7.0710+. 8. Yes, and will move towards the rear end of the car (First Law). Because it tends to preserve the velocity which it had just before the col- lision. 9. In the first edition F z was 30 Ibs. It was in- tended to be 20 Ibs. so as to show that the re- sultant of two velocities may be the same as those producing it. In this case the triangle of velocities will be equilateral, and hence each of the angles will be 60. If 20 and 30 pounds be used, we have for the diagonal of the parallelogram ,# = A/900 + 400+2 x 20 x 30 x 0-5 = 43-59 Ibs. ; and for the angles 36 35' 12", and 23 24' 48" respectively. PAGE 27, ARTS. 59, 60. The intensity of a force is known only by its results. PAGE 29. ART. 63. The case of a force acting normal to the path of a body was a difficult one with the philoso- 36 KEY AND SUPPLEMENT phers living about Galileo's time, and is not the easiest to explain by elementary processes at the present time. It will be considered in Chap, xvi., p. 220 of the text. ART. 64. We say that gravity tends to draw bodies towards each other ; but we only know that it causes them to move towards each other when free. It is quite as proper to speak of its pushing as of drawing them towards each other. Some attempts have been made to explain the essential nature or the cause of gravitation. One of the most celebrated of these theories was given by one Le Sage. According to his theory, lines of force acted in all conceivable directions through space, and as two bodies intercept those lines which would pass through both bodies, there would be more force to drive them towards than from each, other. (See Theories of Gravitation by "Win. JB. Taylor, of the Smithsonian Institute, Washington, D. C.) But no theory thus far suggested is con- sidered sufficient to account for the fact of gravi- tation. PAGE 29, ART. 65. The law of universal gravitation is one of the discoveries which aided in immortalizing the name of Sir Isaac Newton. He first conceived the nature of the law, and then proceeded to prove it. He assumed that if the law was correct it ought to explain the circular motion of the moon about the earth in other words, that the pulling force (so to speak) of gravity at the moon would be just sufficient to draw it the required amount, from a tangent to the orbit TO ELEMENTARY MECHANICS. 37 If g be the acceleration of a pulling body at the sur- face of the earth, at the moon it will be g ~- D 2 where D is the number of diameters of the earth be- tween the center of the earth and center of the moon, and is about 60-3612 ; and as g = 32-246 ft., we would have at the moon, the acceleration 32-246 -4- (60-3612) 2 = 0-0088 + . The force at the moon which would produce this acceleration equals the cen- trifugal force, and is given by the last equation of Art. 315 of the text, and is 4**p Force = m -=- where r is the radius of the orbit, T the time of a complete revolution, and m the mass of the body. But the force divided by the mass equals the accel- eration as is shown by the last of the equations of Article 86 ; 7 , . Force 4:7t z r .*. accelerat^on = = f^- , in T* which applied to the moon, and reduced, gives 0-0089 + (see Art. 319). The two results should agree if the law and data are both correct. It will be seen that the value of the radius of the earth enters into the computation, the correct value of which was not known to Newton at the time of his first investigations. Some fifteen years after he began the investigation, while attending a lecture in London, he obtained the correct value of the radius, with which he proved the truth of his proposition. 38 KEY AND SUPPLEMENT The analysis by which the above result is reached is anticipated, but it will enable the reader to under- stand why an error in the true value of the radius of the earth vitiated the first result. It is questionable whether the story that Newton conceived the law of gravitation by seeing an apple fall from a tree so often taught to juveniles, is not purely fictitious. It is certain that he did not con- sider the law established for fifteen years after he first conceived it, during which time, it is said, he re- viewed his work many times. PAGE 30, ART. 67. A history of pendulum experiments would furnish material for a book. The 'mathematical pendulum is an ideality, but a very useful one in dis- cussing the subject. Compound pendulums are necessarily used in making experiments. The most practical method of determining the acceleration due to gravity is by means of a pendulum ; and some of the results thus found are given on p. 244 of the text. The length of the seconds' pendulum has also been used for determining the standard of linear measure. Thus, the English law requires that the length of the yard shall be to the length of the simple pendulum vibrating seconds at the Tower of London reduced to the level of the sea as 36 to 39-13908. (See Art. 328.) PAGE 31, AET. 68. The formula g = 32-1726 - 0-08238 cos. 2Z, is given in The Mecanique Celeste, Tome iii. v., 42, [2,049]. TO ELEMENTARY MECHANICS. 39 EXAMPLES. PAGE 35. ,08 _ 1. h = 2j .'. v = V2gh = V2 x 32| x 100 = 80-20 feet. v 300 2. rf = - = 77P = 9-3 seconds. ff S2 i 3. t = = 3-109 seconds. 9 h = v t- $g? = 100 x 3409 - $ x 32 x (3-109) 2 = 1554 feet. PAGE 36. 4. Acceleration = 32| ft. per sec. x 60 = 1,930 ft. per minute. 5. Acceleration per second = 32-16666 feet, = 32- 16666 -4- 3-28 = 9-807 + metres. For 4 sec- onds, 9-807 x 4 = 39-228 metres. 6. h = lfffi + v t, (Eq. (6) p. 34.); .-. 120 = | x - fl + 25*. D Solving for t we find t = 2-0627 + sec. Also (Eq. (11) p. 13.) v =v +gt 40 KEV AND SUPPLEMENT 7. = 25 + x 2-0627 D = 91-35 feet. iqq - 15 A i/ ~ ^ 6 + V 1800 x 193 + 22500 193 193 2 -150+608-19 193 = 2-37 seconds. 8. For the falling body (Eq. (2) p. 34) and for the body projected (Eq. (8) p. 34) h-vt- \g$. Eliminating h gives v = gt; but < = ^; V hence eliminating t gives 9. The sound will be -~^r seconds in returning; hence the time of falling will be TO ELEMENTARY MECHANICS. 41 h ' 1130 ~ or .*. A = 331-6+ feet 10. Let x = the distance upward from the lower point to the point of meeting ; then will the point of meeting be a x from the upper point. If t be the time of meeting, we have for the falling body a x = vt + and for the body projected upward x= Vt-\(j&. Adding we have a=(V+v)t; V+ v' which in the second equation gives 42 KEY AND SUPPLEMENT The distance below the highest point will be a> f TT- \ a x= -77 ( V + lq -TT ) V + v \ V + vj EXERCISES. PAGE 36. 1. He will. To draw it at a uniform veloc- ity he must overcome the friction only, but at an increasing velocity he not only overcomes the friction but also exerts additional force to overcome the inertia of the mass (see Second Law). 2. Attraction is more. 3. Because the air resists less. 4. Bounce. PAGE 37. 5. 10 inches. 6. More, for the force of gravity is less at the equa- tor than at the poles. 7. It will neither gain nor lose in weight if weighed with the same beam scales. It will lose in weight if weighed with the same spring bal- ance, for the resistance of the spring will re- main constant, while the force of gravity will be less. 8. 9-807 + metres per second. PAGE 38, ART. 78. In the last two lines of the page it is assumed that the attraction of a sphere upon an external particle varies inversely as the square of the distance from the center of the sphere. We here submit a proof of the truth of the statement. New- TO ELEMENTARY MECHANICS. 43 ton, in his Principia, proved the proposition of the statement geometrically ; we now use the calculus. Let ABD be a spherical shell, center (7, radius a, P the posi- tion of an ex- ternal parti- cle. C o n- c e i v e two consecutive radial lines drawn from P, cutting the shell in the points A and B. Proof. Let ds be an element of length of the circle at A, PA = r, PC = c, CB = a, k = thickness of shell, 8 = den- sity. Conceive a line joining Pand C, Q = APC, andy = per- pendicular from A upon PC. Then 2 Ttykds = volume of shell generated hy the revolution of A about PC, 2 nySkdii = mass thus generated . The attraction upon the particle will be 2 ndkyds p which resolved along the axis PC gives 2 itSkyds Let p = CE = the perpendicular from C upon PB, then p = c sin 6 ; . . dp = e cos Q d Q ; also r* 2rc cos + c* = a* ; dr re sin 9 and ' dO r e cos ds ar dO r c cos 6 ' (for ds* = dr* + rW). 44 KEY AND SUPPLEMENT Hence 2 itSkyds 2 TtSky cos ardO 5 COS O = 5 . r r r c cos 2 itSkyadp ~ cr (r c cos 6) x _ pdp Ya* _ p* which gives the attraction for a circular element of the shell ; hence for the entire shell we have pdp 4:Tt8Jca_ f -c*- J hence for a constant radius a, the attraction varies inversely as the square of the distance of the particle from the center of the shell, which was to be proved. If c = a, we have hence the attraction of a spherical shell upon a particle in its surface is independent of the dimensions of the shell. To find the attraction of a homogeneous sphere upon an exter- nal particle, make k = da and integrate, and we have 4 itS f a , , 4 xSa 3 (fda = 3 r Jo S = volume X -5 . c PAGE 39. ART. 79. Weight, to an uneducated person, is con- ceived to be essential to matter. Such an one would doubtless assert that a body falls to the earth because it is heavy ; but to the student of mechanics, weight TO ELEMENTARY MECHANICS. 45 is nothing but the measure of a force, the magnitude of which depends upon the quantity of matter con- stituting the body. PAGE 41. ART. 83. The unit of mass might be the piece of platinum which is used as the standard pound (see Art. 33), but as we have occasion to compare the force of gravity at different places, and as the force of gravity at London is assumed to be 32-J- feet, we have chosen to consider the unit of mass as about of the weight at that place. 62% PAGE 42. ART. 85. Density is sometimes used in the sense of specific gravity, and if the density of water be taken as unity, the specific gravity of any substance (compared with water) will equal its density. We prefer to make the definition conform to the sense used in mechanics. The following are the definitions given by several authors : By the density of a body is meant its mass or quantity of mat- ter compared with the mass or quantity of matter of an equal vol- ume of some standard body arbitrarily chosen. TOWNE'S Ele- mentary Chem., p. 29. Density is a term employed to denote the degree of proximity of the atoms of a body. Its measure is the ratio arising from dividing the number of atoms the body contains by the number contained in an equal volume of some standard substance whose density is assumed as unity. The standard substance usually taken is distilled water at the temperature of 38. 75 F. BART- LETT'S Elements of Analytical Mechanic*, p. 29. Enfin si 1'on represente par Z> la masse, sous 1'unite de volume du corps que Ton considere, D sera ce qu'on nomine la Densite 46 KEY AND SUPPLEMENT de ce corps. On prend commandment pour unite de densite celle de 1'eau destillee a cette derniere temperature (4 du thennome- tre centigrade). POISSON, Traite de Mecanique, page 108. The quantity of matter in a body does not depend on the size of the body only, but also on the closeness with which the par- ticles are packed. This difference is denned as a difference of density. Thus there is more matter in a cubic inch of lead than in a cubic inch of oak, and this is expressed by saying that the density of lead is greater than the density of oak. MAGNUS, Les- sons in Elementary Mechanics, p. 60. Experiment shows that the weight of a certain volume of one substance is not necessarily the same as the weight of an equal volume of another substance. Thus seven cubic inches of iron weigh about as much as five cubic inches of lead. We say then that lead is denser than iron, and we adopt the following definitions. When the weight of any portion of a body is proportional to the volume of that portion, the body is said to be of uniform density. And the densities of two bodies of uniform density are proportional to the weights of equal volumes of the bodies. TODHUNTER, Mechanics for Beginners, p. 7. The density of a body is the mass comprised under a unit of volume. SILLIMAN, First Principles of Philosophy, p. 67. Density "1.3 the quantity of matter contained in a unit volume ; the absolut-j density or the closeness with which the particles are packed being uniform throughout that unit volume. This def- inition is directly applicable if a body is homogeneous ; but if it is heterogeneous, and the density varies from point to point, the density at any point is the quantity of matter contained in a unit volume throughout which the density is the same as that at the point. Density is usually measured by means of comparison with some substance the density of which is assumed to be the unit-density. PRICE, Infinitesimal Calculus, p. 164. The quantity of matter in a body, or as we now call it, the mass of a body, is proportional, according to Newton, to the vol- ume and the density conjointly. In reality the definition gives us the meaning of density rather than of mass, for it shows us TO ELEMENTARY MECHANICS. 47 that if twice the original quantity of matter, air for example, be forced into a vessel of given capacity, the density will be doubled and so on. But it also shows us that, of matter of uniform den- sity, the mass or quantity is proportional to the volume or space it occupies. THOMSON AND TAIT, Treatise on Natural Philoso- phy, p. 162. Heaviness (Fr., densite, Ger.. dichtigkeit) is the intensity with which matter fills space. The heavier a body is, the more mat- ter is contained in the space it occupies. The natural measure of heaviness is that quantity of matter (the mass) which fills the unity of volume; but since matter can only be measured by weight, the weight of a unit volume, e. g. of a cubic meter or of a cubic foot of another matter, must be employed as a meas- ure of its heaviness. The product of the volume and the heaviness is the weight. The heaviness of a body is uniform or variable according as equal portions of the volume have equal or different weights. WEISBACH, Mechanics of Engineering, p. 160. The density of a, body is the degree of closeness between its particles. The term depends upon the hypothesis that the ulti- mate particles of matter have weight, and therefore mass pro- portional to their bulk. It coincides with specific gravity. PBOP. NICHOL, Cyclopcedia of the Physical Sciences, page 177. On sait que le poids d'un corps varie avec 1'intensite de la pesanteurmaisque sa masse ne varie pas. Sous 1'influence de la meme pesanteur, par exemple en un meme lieu du globe, le poids est evidemment proportionnel a la masse et le rapport des poids de deux substances sous le meme volume sera precisement celui de leurs masses sous le meme volume ; de la, la synonymie, qui existe entre les mots poids specifique et densite, qui ex- prime ses rapports. WUBTZ, Dictionnaire de Chimie. Sous Den- eite. The density of a body is the ratio of its mass to its volume. SMITH, Elementary Treatise of Meclianics, p. 44. 48 KEY AND SUPPLEMENT EXERCISES. PAGE 43. 1. The matter outside of one-half the radius would produce no effect ; and that within would at- tract as if it were all at the centre. The sphere of one-half the radius will contain one-eighth the matter, and the inverse square of the dis- tance will be 4 ; hence the weight will be x 4 x 10 = 5 Ibs. We get this result more directly by saying, as in Art. 78, that the at- traction will be directly as the distance from the centre. 2. 10 -r- 2 2 = 2* Ibs. 3. Nothing. 4. At a distance from the outside somewhat less than half the thickness of the shell. The ex- act distance cannot be found unless the thick- ness of the shell be given. (If R be the out- side radius, r v the inside, and r the required distance from the centre ; then 10 r 2 I? or which is a cubic equation, and may be solved by Cardan's method.) PAGE 44. 5. Yes for the first answer ; No for the second. 6. On the opposite side. He could not stop at the TO ELEMENTARY MECHANICS. 49 centre by a mere effort of the will. Some entertain the idea that the will of a person, causing a movement of parts of the body, could, in a measure, control the movement of the body as a whole ; but, as a fact, the matter of the body is subject to the action of forces, like any other matter. No part of the body can be moved except in accordance with the three laws of motion. If an arm is moved in one direction, some other part of the body will be moved in an opposite direction (the body being free). While passing across the hollow referred to in the Exercise, the person might throw his arms about, or kick, or perform the evolutions of swimming, or rowing ; but as there is supposed to be no matter, or no body, for him to act against, he could not change his rate nor direction of movement. In drawing his feet forward he would necessarily pull some part of the body backward. It is shown, by higher analysis, that the centre of gravity of a moving system is unaffected by the mutual ac- tions of internal forces so, in this case, the motion of the centre of gravity of the person would be en- tirely unaffected by any contortions the person might make. 7. At the centre of the sphere. At the same point. Uniform. PAGE 43. 8. The ball. If the ball were very small compared with the person, the movement of the person might be neglected compared with that of the ball ; just as the motion of the sun is neglected compared with that of the planets. In the case of the planets, the pulling force is dependent upon their masses and dis- 3 50 KEY AND SUPPLEMENT tances, but in the case of the person this force is de- pendent upon his muscular exertion. 9. He could not. In the effort to throw the ball away, a force is developed between the hand and ball, which acts equally between the ball and hand, but in opposite directions in accordance with the third law ; and hence the ball would move one way and the per- son the opposite way ; and both would move in straight lines in accordance with the first law, and their relative velocities will be inversely as their masses in accordance with the second law. If the person were placed at rest in any position in the hollow and unable to reach anything, he could not turn over, nor change ends ; that is, if his head were towards the north and his feet towards the south, he could not so change as to have his head towards the o south and his feet towards the north. Should he at- tempt to so turn as to bring his head towards the south, he would cause his f jet to approach his head, and they would meet about half way. He might suc- ceed in kicking his own head, or, if he were very flexible, the head and feet might pass each other ; but the body could not turn over so as to change ends. Neitlier could he roll over. If one end of the body turns one way, the other end necessarily turns the opposite way. If a cat be held with her back clown two feet or so from the floor, and dropped, she will strike on her feet: how does ehe do it? According to the 7 O principles of mechanics, if there were nothing for her to act against it would be impossible for her to turn, and she would necessarily strike on her back. TO ELEMENTARY MECHANICS. 51 While experimenting, I was surprised to find how near the floor the cat might be held, and often apparently perfectly unwatchful when dropped, and yet alight on her feet. Now the movement of the body should be accounted for on mechanical principles. Instinct operates quicker than reason, and it appears to be certain that the cat, instinctively, initiates a rotation of her body at the instant she is dropped. "While it is difficult to see how muscular action can be quick enough to produce this result, yet I see no other way of accounting for the rotation. Suspend the cat with a string at each foot, then suddenly cut the strings, and she will rotate herself. It is also an interesting fact that she will strike on her feet if let fall several feet, say six or eight feet. Now if she had the same initial rotation when falling six feet, as when falling two feet, why would she not turn too far in the former case and strike on her side ? To ex- plain this, we here state a principle not yet proved in this course. If a rotating Ijody self -contracts, it will rotate more rapidly r , l>ut if it self-expands it will ro- tate more slowly. The cat has the ability, within a limited range, of expanding or contracting the trans- verse dimensions of her -body, and to that extent of regulating the amount of her rotation. Consider still further the relations of the man and ball in the ninth exercise. He puts his hand in close contact with the ball, but without grasping it, and they move away from each other, until both strike the walls of the hollow. Then suppose that the man springs for the ball and seizes it, and then springs towards the centre of the sphere, but before reaching 52 KEY AND SUPPLEMENT the opposite side throws the ball in anger. If the ball goes in a direction perpendicular to the line of his body (or, generally, in any line not passing through the centre of his body), the man will be thrown into a rotary motion as well as a motion of translation, and he will inevitably perform somer- saults while backing away to the opposite side of the hollow. PAGE 44. 10. 100 -4- 32 = 3 T Vj = 3-109. -.., n .. Mass 200 x 6 11. Density = =-. -- = -^ = 3-109. Volume 193 x 2 10 -, f 2-2 B> x 5 x (5 12. Mass = - - = -342. = - 00581 ' 14. If the resistance of the air be considered it would. In the second case, it would not stop, but would go from surface to surface with the regularity of a pendulum. (See text, p. 249.) In the third case the velocity would be greatest at the centre, if the hole be a vacuum ; but if it be filled with air, the greatest velocity would be passed before the ball reached the centre. AKT. 86. The value of F= Mf is sometimes called the absolute measure of force, but nothing is gained by the term, except that it distinguishes it TO ELEMENTARY MECHANICS. 53 from the mere stress which it would produce if no motion resulted. Some English writers call the value of F when thus expressed The Poundal, but this term has not come largely into use. Observing that acceleration is the rate of increase of velocity (Key, p. 6), it follows that this value of 1 F is the same as that given by Newton's second law, that the force is proportional to the change of mo- mentum produced. But we are confident that the constant use of the term acceleration for rate of change of velocity possesses great advantage, since rate of change is liable to be considered the same as actual velocity. Indeed some text books assert that the momentum of a body is a measure of the force acting upon it and call it the second law. Now a body may have momentum when no force is acting upon it. It is the rate of change of mo- mentum that measures the force producing the change ; in other and better words, the mass into the acceleration. Observe that the establishment of this equation contains a very important principle. There is, strictly speaking, no relation between pounds and feet ; but the ratio of two weights may be the same as the ratio of two linear measures. A ratio is an abstract number, and often serves to connect con- crete quantities, forming an equation. Thus, in this case, F f -. = a ratio, a mere number, = . g The equation being established, it is operated upon 54 KEY AND SUPPLEMENT algebraically. This use of ratio has many appli- cations in physics. ANSWERS TO EXAMPLES. PAGE 47. 1. Velocity = y 2 x 2000 x 193 x 1> 6 x 100 1286-6G = 35-87 feet. 25 x 193 x 100 965 2 80-42 or SO 'feet. o TT 7 v // C* x (25-10)x 193x100^ 3. Velocity = \/ (- r, ^7^- \ Ox 500 / % 193 = 13-88 feet. 100 x 100 Q1 . _ 4. Force = ~-^ --^- = dl-2o IDS. 32 xlO SOLUTIONS OF PROBLEMS. PAGE 50. 4. The tension equals the weight, P, plus the force W P which will produce the acceleration. . p g is the acceleration when P is raised vertically. The mass multiplied by the acceleration is the moving force, or P W P W P f-. - ^ g : hence the tension is P + ^ f v P g W + P ' ' + P = ~* _ . Similarly, it equals W minus the accel- TO ELEMENTARY MECHANICS. FP 2TFP -* TTT _ a rr -* erating force, or I ^ + p ' W+P 5. The effective moving force is W T, hence from W Prob.2, W- T = --/. ff Substitute/= fa and W-T=iW; .: T-^W. If ascending, T-W = /, or T - W = I TF; i? T= W + the force which will produce the ac- W celeration = W + \g = -W. 9 EXAMPLES PAGE 50. 2. , = .^ = ixlti x 25 =134^ ft - g , 2 or P - - *'P+ W g ' ~ I x 25 + 2 x 10 = 5 -i|^ x 25 - 2 x 10 TF P 2 5 = * ^^' or P = 47 ' 57 fts * P- W 2.

W + hW be- comes li W, in which case the work equals that neces- sary to raise the weight through the height of the plane. SOLUTIONS OF EXAMPLES. PACK 64. 1. It will raise 50 X 33,000 x 60 ft. Ibs. in 1 hour, which divided by 500 x 62-5 will give the cubic ft. = 3,168. 2. The average height to which the material is raised will be 10 ft. Hence the work = 140 (i 7t x 3 2 x 20) x 10 = 197,920-8. Ibs. F + F' 1000 + 200 10 _ . 3 ' 8 = -W a= 2000- X 12 = 7 ' 2m - 4. 39-37 inches = 3 2808 ft. x 2-2 Ibs. = 7-217 + ft. Ibs. 5. 43,333 x 7-217 + ~ (32,808) 2 = 29,057 ft. Ibs. 6. Substituting in the answer to Prob. 2, p. 61, we } have H. P. = 0-9114 x 1 x 12 = 37-89. 7. Work of the fall = 2,000 x 8 = 16,000 ft. Ibs. Let x the distance driven, then 10,000a; = 16,000; .-. # = 1-6 ft. 8. Find the velocity in feet per minute. We have 64: KEY AND SUPPLEMENT 2 x 5280 rp, , v = - -- . I he horse-power = Jrv -f- 60 200 x 2 x 5280 33 ' 000 > r 60x33000 - 1 *- Q _ y^P -2(P + TT)s_ 1^x9x8-2(8 + 40)4 ^JF ii x 9 x 40 = 0-1668. PAGE 65. P- HW t 5-25x045 10. 6-= i.-^^tf.. 5+25 x x 25 = 1206-25 H- 72 = 16-75 ft. 6 ANSWERS TO EXERCISES. 1. One pound raised one foot. Work is a com- pound quantity, compounded of stress and space. 2. See preceding remarks, text, p. 56. 3. It is dependent upon time only implicitly, and in the sense that motion requires time. But, strictly, time should be abstracted. 4. Tang. 15 = 0-268. 5. 5 Ibs. 6. Mechanical power is not work, but rate of doing work just as velocity is not space, but rate at which space may be passed. Mechanical power involves a unit of time, but work does not. PAGE 66. ART. 41. The doctrine of energy is the grand, gen- eral principle of modern physics. All the changing TO ELEMENTARY MECHANICS. 65 phenomena of nature are but manifestations of the transmutation of energy. Its principles are not de- duced by any system of mathematics, but by a long series of inductions. We accept its general principles without attempting a general demonstration. Even work in a higher sense is but a means of transmitting energy. Thus, a horse works by draw- ing a load, but it is simply a means of transmitting the energy possessed by the horse, first into energy stored in the mass of the load, and second into heat by means of the friction overcome. Still, work is not only a convenient, but a useful term. Work done is one idea, energy produced as its equivalent is another force and space are the elements of the former, mass and velocity of the latter. DEFINITIONS OF WORK AND ENERGY. " Work is the overcoming of resistance continually recurring along some path." BARTLETT'S Elements of Analytical Mechanics, p. 26. " Work consists in moving against resistance. The work is said to be performed, and the resistance overcome." RANKINE'S Applied Mechanics, p. 477. " Work is the effect of strain and motion combined. -TRAUTWINE'S Engineer^ Pocket-Book, p. 445. Remarks on above from p. 446 same book : " Grav- ity acting in a body falling freely in a vacuum, and consequently unresisted, exerts no effort upon it, it neither goes before, and pulls it along, nor behind, and pushes it ; for there can be no pull or push except 66 KEY AND SUPPLEMENT when there is some force to pull or push against. But it simply, as it were, animates the body, or en- dows it with the power of locomotion. As the body falls, the force of gravity which gives it motion all remains unimpaired, and stored up in it, ready to ex- ert an effort against any other force which it may chance to meet with. Therefore a body falling unre- sistedly has no weight ; for gravity, which gives it weight alone while at rest, now gives it motion alone." (The word strain in the above definition is improp- erly used for stress. AUTHOR.) " A force is said to do work if its place of applica- tion has a positive component motion in its direction ; and the work done by it is measured by the product of its amount into this component motion." THOMSON AND TAIT, Nat. Philos., p. 176. " Work done on a body by a force is always shown by a corresponding increase of vis viva, or kinetic en- ergy, if no other forces act on the body which can do work or have work done against them. If work be done against any force, the increase of kinetic energy is less than in the former case by the amount of work so done. In virtue of this, however, the body pos- sesses an equivalent in the form of potential energy, if its physical conditions are such that these forces will act equally, and in the same directions, if the motion of the system is reversed." /&., p. 177. " An agent is said to do work when it causes the point of application of the force it exerts to move through a certain space. Motion is essential to TO ELEMENTARY MECHANICS. 67 work." TWISDEN, Elementary Introduction to Prac- tical Mechanics, p. 18. " Mechanical effect, or work done, is that effect which a force accomplishes in overcoming a resist- ance. It depends not only on the force, but also on the space during which it is in action, or during which it overcomes a resistance." WEISBACH'S Me- chanics of Engineering^ p. 168. " Work is the production of motion against resist- ance." TODHUNTER, Mechanics for Beginners, p. 337. "Work, same as before. According to this defini- tion, a man who merely supports a load does not work ; for here there is resistance without motion. Also while a free body moves uniformly no work is performed ; for here there is motion without resist- ance." TODHUNTEK, Nat. Philos. for Beginners. p. 255. " Whenever a body moves through any space in a direction opposite to that in which a force is acting on it, work is said to be performed. It is evident that the application of force is necessary to overcome resistance, and it is very often found convenient to measure the work done by the amount of force ex- pended, and the distance in the direction of the force through which it has been employed." MAGNUS'S Elementary Mechanics, p. 102. " Thus the increase of vis viva, which is also the work done by the acting forces on the body." PRICE'S Infinitesimal Calculus, vol. iii., p. 636. " Work is done when resistance is overcome, and 68 KEY AND SUPPLEMENT the quantity of work done is measured by the prod- uct of the resisting force and the distance through which that force is overcome." MAXWELL'S Theory of Heat, p. 87. " Work is the overcoming of mechanical resistance of any kind." NYSTKOM'S Pamphlet on Force of Falling Bodies, etc., p. 25. DEFINITIONS OF ENERGY AND VIS VIVA. " Energy expresses power to do work, or force stored and ready for use." McCuLLocn's Treatise on Me- chanical Theory of Heat, p. 40. " Vis viva (energy) is a quantity which varies as the product of the mass of a particle and the square of its velocity." PRICE'S Infinitesimal Calculus, vol. 3, p. 360. 2d Ed. Oxford. " Living force, or vis viva (or energy), is nothing more than an expression referring to the quantity of work (motion and strain combined) which the force in a body at any given instant could perform, if left to itself, without afterwards receiving any additional force." TRAUTWINE, Civil Eng. Pocket Book, p. 446, Ed. 1872. "Energy measures the quantity of working power of a moving body." BARTLETT, Elements of Anahjt. Mech., 9th Ed., p. 116. " The product of the mass of a body by the square of its velocity is called its living force or vis viva" Jlecanique Celeste, p. 99. TO ELEMENTARY MECHANICS. 69 " Energy means capacity for performing work." RANKINE, Applied Mechanics, p. 477. " Vis viva, or living force (or energy), is the power of a moving body to overcome resistance, or the measure of work which can be performed before the body is brought to a state of rest." SILLIMAN, Prin- ciples of Physics, 2d Ed., p. 78. "Energy is the capacity a body has, when in a given condition, for performing a certain measurable quantity of work." TODHUNTER, Natural Philosophy for Beginners, p. 264. "Energy of a body is power of doing work."- MAGNUS, Lessons in Elementary Mechanics, p. 110. " Energy of a body is the capacity which it has of doing work, and is measured by the quantity of work which it can do. The kinetic energy of a body is the energy which it has in virtue of being in motion." GUMMING, Theory of Electricity, p. 5. " Kinetic energy or vis viva is defined as half the product of the mass into the square of the velocity." Ibd., p. 13. " Energy is defined to be capacity for doing work." " It is of two kinds kinetic or actual when the body is in actual motion. Potential or latent when the body, in virtue of work done upon it occupies a position of advantage, so that the work can be at any time recovered by the return of the body to its old position." Ibd., p. 14. "Energy is the capacity of a body to perform 70 KEY AND SUPPLEMENT work. Energy is said to be stored when this capacity is increased, and to be restored when it is diminished. The units of work and of energy are the same." WEISBACII, Mech. and Eng. Translator's note, bottom of page 168. " If we adopt the same units of mass and velocity as before there is particular advantage in defining kinetic energy as half the product of the mass into the square of the velocity." ROUTE'S Rigid Dynam- ics. PAGE 67. AET. 112. If the body had an initial velocity v , the work done upon it in passing from a velocity V to V, or the work which would be given out by it in passing from the velocity v to v , will be This may be deduced in another way, if we antici- pate the equations for momentum. Thus, Eq. (2), p. 78, of the text is Ft = M(v - v ). The space over which F acts will be equivalent to the average velocity into the time, or s = \(v + vjt. Multiplying these equations, member by member, and canceling , gives Fs = TO ELEMENTARY MECHANICS. 71 as before. But we do not consider this method as good as the first, for when the velocity changes irreg- ularly, it is not so evident that the space equals the average of the extreme velocities into the time. O ART. 113. Potential energy is relative. Thus if a body whose weight is 10 Ibs. is 40 feet above the earth, its potential energy in reference to the earth is 10 x 40 = 400 ft. Ibs. ; but if it be over a well 20 feet deep, the potential energy in reference to the bottom of the well, will be 10 x 60 = 600 ft. Ibs. ; and in reference to its own position it is nothing. It is the work which the body may do in reference to some point or condition arbitrarily chosen. PAGE 69. ART. 115. For the mathematical theory of heat, see Poisson Traite des Cheleur, Fourier Theorie de la Cheleur, Rankine on The Steam Engine, Clausius on The Theory of Heat, Hirn's investigations, Max- well's Theory of Heat, McCulloch's Theory of Heat, Tait's History of Thermodynamics, etc., etc. PAGE 72. ART. 117. This is one of the most important phys- ical constants which has been determined in recent times. For a comprehensive and able review of the methods by which it has been determined, as well as for a description of that author's methods and results, see Mechanical Equivalent of Heat, by Professor II. A. Eowland. (Proceedings of the American Academy of Arts and Sciences, 1879, p. 45.) Prof. Rowland's results differ by only a small per cent, from those given by Joule. He states, 72 KEY AND SUPPLEMENT p. 44, that the value found by Joule at 14, agrees with his results at 18 C. The value 772 foot pounds still stands as a practically correct one. PAGE 73. ART. 118. In regard to energy generally, it appears that all of it or at least very nearly all originates with the sun. It was a beautiful remark of John Stephenson as he saw a railroad train winding its way through the country " that train is drawn by the heat of the sun." The heat and light of the sun caused the growth of vegetation ; that vegetation in time was gathered into great masses, which in time became coal in the mine. This coal was brought forth and used as fuel in the locomotive ; so that it originated in the action of the sun. All energy on the earth is due to the light and heat of the sun. Activity in the commercial world is directly depend- ent upon it ; for if the sun, on account of spots upon it, or from other causes, does not dispense with its usual heat, short crops will result, and thus affect all the business of a country, if not of the world. In this way the sun may be charged with causing more or less directly depressions in trade, or activity in com- merce, as the case may be ; and hence, in some cases at least, of producing sadness or cheerfulness in the home circle. Even the religious world is affected by the action of the sun. In earlier times, on account of the superstitions of the people, religious leaders appealed more or less effectually to the fears of their followers when the sun's rays were cut off by an eclipse ; and in modern times the feeling of depend- ence upon the Creator is too often modified by the TO ELEMENTARY MECHANICS. 73 prosperity or depression caused by the circumstances surrounding them, the conditions of which were de- pendent upon the elements of nature, and these caused, more or less directly, by the action of the sun. The sun appears to be dispensing its energy to his family of planets, and in this way wasting itself away. Sir Isaac Newton realized this condition of things and saw or thought he saw the necessity of the sun's being replenished in order to maintain its stock of energy ; and he conceived that this might be done by one comet after another falling into the sun. As the comets come from remote regions of space, they would possess a large amount of energy when they struck the sun, and by falling into it would pro- duce intense heat. No comet, however, has within historical times been known to fall into the sun ; but, on the other hand, the most critical examination of their orbits shows that their paths are nearly as well defined and nearly as fixed in position as any of the planets ; and no cause is known to exist that will cause them to fall into the sun. It was formerly sup- posed that the ether of space caused a resistance to the motion of all bodies in space; and if it did, not only would the comets, but also the planets, ultimately fall into the sun. However, nothing absolutely noth- ing is known in regard to the effect of this ether upon the motion of bodies in it. Comets and planets have moved in their orbits for untold ages, and, for aught we know, have maintained their relative posi- tions. If the comets were destined to fall into the sun, they must have been doing so for ages and ages, and hence must originally have been comparatively 4 74 KEY AND SUPPLEMENT very numerous. It would seem that sufficient time had elapsed since the existence of the solar system to have exhausted this stock of energy still, at the present time, comets are numerous. Similarly, in re- gard to the planets, the most refined observations, combined with the most refined analysis, have failed to detect any modification of motion due to the ether of space. More recently, the late Professor Benj. Pierce, of Cambridge, Mass., put forth the theory that the en- ergy of the sun was supplied by meteorolites tailing from an immense distance directly into the sun. The meteors are dark bodies, and it is assumed that there may be multitudes of them scattered through space. That there are many, is shown by the fact that they frequently fall upon the earth. Admitting the truth of this hypothesis, it seems inevitable that, in the course of time, the supply will be exhausted and then the question, What will follow? becomes a serious one to science. Space, which, in our younger days, we conceived to be void, is really filled with something, and there may be vastly more inert matter scattered through it than we have imagined. A mere contraction of the volume of the sun, caused by the mutual attraction of its own particles, will produce heat. The author has shown that if the earth contracted from twice its present diameter to its present size, and all the energy thus produced be changed into heat, and uniformly disseminated throughout the mass, the temperature would be raised 44,655 degrees F., if it had the specific heat of water, TO ELEMENTARY MECHANICS. 75 or 357,240 degrees F., if it had the specific heat of iron. (See Analytical Mechanics, p. 229, or Mathe- matical Visitor, 1880, p. 134.) We believe that the solar system is stable, that it is not made to run down, that it has the elements of self-preservation ; but we cannot prove it. Neither can those who entertain the opposite view prove their position. This problem is, at present, beyond the reach of science. SOLUTIONS OF EXAMPLES. PAGE 75. 25 1. - = the mass, and as 32| is the acceleration 32% in feet per second, the velocity should be in the same units ; hence v = W = | feet per second, and the work will be OK /S\2 'J|-(t) =m + f '- lbS - 2. The work will vary as the depth, and the energy as the square of the velocity ; hence, depth = 2(y) 2 = 18 feet. PAGE 76. 3. Reducing the tons to pounds, we have 60 tons = 60 x 2000 Ibs. = 120,000 Ibs. Similarly, 40 x 5280 ,, 40 miles per hour = ft. per second 60 x 60 KEY AND SUPPLEMENT The friction = 60 x 8 = 480 Ibs. Let x = the required distance in miles = 5280# feet, then the work will be 480 x 5280 x ; hence we have 480 x 5280, = i .12p(*L which solved gives 100 /:L Py= 2-53 miles. 4. Let x = the required number of pounds of wa- ter ; then the energy put into water in raising it from 32 F. to 212 will be 772 (212 - 32)aj = 772 x 180*. The kinetic energy stored in the train will be 200000 /30 x 5280V. 32 \ 60 x 60 / ' which, by the conditions of the problem, equals the heat energy to be put into the water; hence 200000 /88\ 2 772 x 180a; = \ x I "a" J > OA-g \ O / .-. x = 19-2 Ibs. PAGE 76. - f _P - W _ 4 - 0-2 x 10 193 _ 193 ^ = P + W g ~ 14 6 = 42 = 4-6; TO ELEMENTARY MECHANICS. 77 also for the velocity v* = 2/6- = 2 x 4-6 x 5 = 46. The work will be 10 x // x distance, and the energy will be .. 0-2 x 10 x distance = $ x x 46 ; j.yo /. distance = 3^ feet. 6. The friction = 200 x 0-2 = 40 Ibs. Let the velocity per minute be #, then the work per minute will be 40#, and for 3 minutes it will be 120a?. The energy of one pound of water raised one degree F. is 772 ft. Ibs., and of 5 Ibs. it will be 5 x 772, and for 50 degrees it will 50 x 5 x 772. Hence 120 = 50 x 5 x 772 ; /. x = 1,608 ft. per minute. ANSWERS TO EXERCISES. PAGE 76. 1. It is not ; force is only one of the elements pro- ducing energy. 2. Ability to do work. "Work has, however, been done. 3. It produces action of the stomach, thus in- 78 KEY AND SUPPLEMENT volving energy ; promotes action of the heart ; causes the growth of the bone, muscle, and flesh, and these enable the animal to move about to do work to swallow more food to lie down to get up, etc., etc. 4. Because, in the first place it is not as concentra- ted, and, in the second place, the heat is more quickly conducted away. 5. It will. It is due to this cause that meteors are visible. The meteorolites which fall upon the earth have the appearance of having been partly melted, and hence must have been subjected to great heat. This is due to the compression of the air in front of the meteor and of the friction of the air against its sides as it passes swiftly through the air, and the heat thus produced is so great that the meteor is heated to redness, and thus appears like a shooting star, as it really is. It is probable that the smaller meteorolites become so nearly consumed by the great heat that they could scarcely be found after they had fallen upon the earth, but larger ones have been found which struck the earth with such violence that they nearly or quite buried themselves. It has been sug- gested that the great iron deposit in the upper pen- insula of Michigan, a short distance west of Mar- quette, was probably a large meteor, and a similar suggestion has been made in regard to the iron mount- ain in the State of Missouri. PAGE 77. 6. The friction of the water produced by moving against the banks and bed of the stream produces heat, which, escaping into the surrounding air, modi- fies its temperature. TO ELEMENTARY MECHANICS. 79 7. First by the heat due to the friction, and second, by inducing a quicker circulation of the blood more heat is supplied to the parts. PAGE 77. 8. $2.50. Tliis exercise is intended to draw atten- tion to the fact that the value of wood for fuel de- pends upon its capacity for producing heat ; or, in other words, of its inherent heat energy. If the heat- ing power of a given weight of hickory be 100, it has been found that the heating power of the same quan- tity of oak will be about 80, and of maple about 50. The practical value, however, of fuel may be governed largely by other circumstances. Thus, when a fire is wanted for only a short time, as a kitchen fire in mid- summer, or where steam must be raised quickly, etc., the cheaper fuel may be quite as valuable as the more costly. PAGES 78, 79. ARTS. 122-125. Momentum, according to New- ton's definition, is strictly quantity of motion. He says (Principia B. 1, Def. II.) " The whole motion is the sum of all its parts, and therefore in a body, double in quantity, with equal velocity, the motion is double ; with twice the velocity it is quadruple." It is said in the text that quantity of motion does not fully express the desired meaning, but this is due sim- ply to the fact that quantity had not been defined. Including Newton's definition, it does express the meaning correctly and fully. If a force of constant intensity acts upon a free body moving from rest, the product of the force and 80 KE Y AND SUPPLEMENT time equals the momentum produced. Space is here entirely abstracted from force and time. Although the body cannot move without involving space, yet all considerations of space must be discarded. It is immaterial whether the space over which the body must move in acquiring the velocity v be great or small ; and hence, so far as the momentum is con- cerned, the space may vanish. n\ From the equation Ft Mv, we have F M , t v f where (or for a variable force, we have in the nota- t \ tion of the calculus ) is the rate of change of the dtj 1) velocity, and hence, M is the rate of change of mo- u mentum, which is, according to the second law, the measure of the force F. We thus reproduce the ex- pression for that law. The expression Ft is not the momentum, but sim- ply its equal under the restrictions given above. It has been proposed to give a special name to this prod- uct, just as 7% is called the measure of work, while its equal jfy 2 , in case of a free body, is called energy. Maxwell called Ft an impulse (Matter and Motion, p. 44), to which we do not object, since the effect will be the same whether it be produced in an impercep- tibly short time, or in a longer time. But the prod- uct has no meaning except where the force moves a body. If F be a mere stress, like the pressure of a stone upon the earth where no motion is involved, then Ft produces nothing. The time effect of a TO ELEMENTARY MECHANICS. gl stress, is its effect in moving a body producing velocity. Much has been written in regard to force, vis viva (now called energy) and momentum. For many years, in the earlier history of the science of mechanics, there were long and sharp discussions as to whether work or momentum was the proper measure of force ; but, as we have shown in what precedes, neither is the proper measure, and hence theirs was merely a war of words. One factor in the product Fs is the force ; and one factor in the product Ft is also force. The former placed equal to j-3/v 2 , and solved gives F = ? - : hence the measure of force is the rate of s change of energy per unit of space; and we have already shown that it is also the rate of change of mo- mentum per unit of time. Efforts are sometimes made to determine a relation between momentum and energy ; but no physical re- lation exists, and hence none can be found. In order that there shall be a ratio between them, they must have a common unit. Since one is compounded of force and time in which space is excluded, and the other of force and space in which time is excluded, they have not a common unit. PAGE 79. We here note the three following statements from different authors. First, In Yan Nostrand's Engineering Magazine for 1877 and 1878 is a series of articles on momentum, and Vis Viva by Prof. J. J. Skinner. On pp. 129, 4* 82 KEY AND SUPPLEMENT 130, 131, it is stated that momentum which equals MV represents the number of pounds pressure which the mass M with the velocity F is capable of exerting under the conditions that the pressure is con- stant and capable of bringing the body to rest in one second. This is numerically correct as a deduction, but in the articles referred to there is apparently a labored effort to show that momentum is pressure only, and not quantity of motion (see also p. 137 of the Eng. Mag.}. Also on page 132 it is stated that, " The unit of momentum, then, is a force of pressure equal to one pound " (see also p. 240). In this defi- nition, the element of time does not appear, but it is not proper to drop it simply because it is one second. The above-named writer corrected himself in a later article, page 501 of the same Magazine. To measure anything requires a unit of the same kind considered as a standard. Strictly speaking, the unit of momentum is the momentum of a unit of mass moving with a unit velocity. Momentum cannot be measured by pounds only. (See also article by the author, Eng. Mag., vol. xviii., 1878, p. 33.) Second, It is stated that Beaufoy determined that a body of one pound weight, with a velocity of one foot in a second, strikes with a pressure equal to 0-5003 Ib. ; and hence to find the pressure produced by the impact of any projectile, we have the general iorm\\\&, pressure = 0-5003 Wv* (Silliman's Physics). Now we assert that the formula is false. Admitting, for the sake of the argument, that he did find such a result when the projectile struck a hard body, like a piece of iron, it would have been very much less TO ELEMENTARY MECHANICS. 83 had the body struck been more yielding, like a gas- bag, or a sack of loose feathers, and so a great range of values might be found depending upon the character of the bodies. Third, Professor Tait, iu an interesting lecture upon force, delivered before the British Association, 1876, saj r s (see Nature, 1876, p. 462), " With a moderate exertion you can raise a hundred weight a few feet, and in its descent it might be employed to drive ma- chinery, or to do some other species of work. But tug as you please at a ton, you will not be able to lift it ; and, therefore, after all your exertion, it will not be capable of doing any work in descending again. " Thus, it appears, that force is a mere name, and that the product of the force into the displacement of its point of application has an objective existence." " Force is the rate at which an agent does work per unit of length" . . . These definitions have already been referred to on p. 20 of the Key, and the remarks there made will be more readily understood in this place, after having passed over energy and momentum. Force, funda- mentally, is a quantity instead of rate ; just as inter- est is a quantity an amount of money and not rate. It is true that the amount of money paid for the use of a hundred dollars is identical with the rate per cent., but every one readily distinguishes between rate per cent, and interest. Professor Tait made use of interest in an illustration of this principle, but, we think, used it improperly. Si KEY AND SUPPLEMENT PAGE 83. ART. 131. If A = Z, and k = 1, we have E = F y hence as a deduction the coefficient of elasticity may be defined as the force (stress) necessary to elongate a prismatic bar whose section is unity to double its length, provided the original conditions remain constant except that of length ; that is, the elasticity and the cross section must both remain con- stant. But these conditions are never realized, hence this definition is highly ideal. In fact, the coefficient of elasticity is constant for any material for an elonga- tion of only a very small fraction of the length ; but even with this limitation it is of untold importance in certain physical sciences. PAGE 90. ART. 138. Substituting from equations (1) and (2), page 89, we have Mv? + M' v\* = ~Mv + M'v eM' . , , N ( "- ,, r ( M + M ) 2 - ZeM^vv + ZeMMvv' + or (M+M 1 } TO ELEMENTARY MECHANICS. 85 ^vv' + JTV 2 ]. (2) Similarly, (3) Adding equations (2) and (3) we have + + 86 KEY AND SUPPLEMENT (M + M') where h is the height of fall, and h' the height of rebound. The sum of an infinite decreas- ing progression is _ first term, 1 ratio TO ELEMENTARY MECHANICS. 89 9. If = 1, and 6. We have P 2 = (P - F) 2 = P 2 + F* +2PF cos 0, , or, P 2 - 2PF+F* = F* + F 2 + 2PF cos ^; .-. cos = - 1 ; .-. = 10 D . 7. From the proportion on p. 98 of text we have 50 : F : : sin (F,R) : sin 115, 50 : E : : sin (F,R) : sin 35, TO ELEMENTARY MECHANICS. 93 F: E : : sin 115 : sin 35, P : 7? : : sin 30 : sin 35. From these we have sm 35 R = !! x 50 = 57.35 lbg> sin 30 / T-T r>\ "'J ^ sin oo ~ K ~~,v sin (JP,72) = = 0-5000 ; .-. angle (F,K) = 30. 8. The string will make a right angle at the point where the weight is applied, and the sides of the triangle representing the forces will be as 3 to 4 to 5 ; hence we have 5 : 4 : : 20 :x = 16; 5 : 3 : : 20 : x = 12. PAGE 100. 9. The parallelogram representing the forces will be a rectangle, of which the diagonal will be a diameter of the circle. ANSWERS TO EXEECISE8. 1. It will be the resultant of two forces acting away from C, one of which will equal GA, the other CD = AS. 94 KEY AND SUPPLEMENT 2. A line through A equal and parallel to a line joining J^and B. 3 They would not. 4. When acting upon the same particle, in opposite directions, and equal in magnitude. 5. "With 4, 5, and 9 they can, if 4 and 5 act opposite to 9. But forces 3, 4, and 8 cannot, since two of them, 3 and 4 together, do not equal the third. 6. It will not. The resultant takes the place of the other two. PAGE 102. ART. 158. It will be observed that, in Fig. 45, ft is the complement of a / hence cos /? = cos (90 a) sin a. hence the equations for ^Tand Y become X. = FI cos <*! + Ft cos af z + etc. = 0, Y = F l sin a z + FZ sin a 2 + etc. = ; from which we see that the equation for Y may be deduced directly from that of X by writing sin in place of cos in the first equa- tion. SOLUTIONS OF EXAMPLES. PAGE 103. 1. We have X = 20 cos 30 + 30 cos 90 + 40 cos 150 50 cos 180 = R cos a, TO ELEMENTARY MECHANICS. 95 Y= 20 sin 30 + 30 sin S0+ 40 sin 150 + 50 sin 180 = R sin a in which all the terms are written as positive, and their essential signs made to depend upon the trigonometrical functions. Reducing gives 20 x % V3 + - 40 x | V3 - 50 = R cos a, 10 + 30 + 20 + = R sin a; or 67-32 + = R cos or, 60 = R sin a. Dividing gives Rco&a _ 67-32 72 sin a: ~ 60 ' or cota = -1-122; .-. a = 138 17'. Squaring and adding gives ^(cos 2 + sin 2 *) = (6T-32) 2 -I- (60) 2 . But cos 2 a + sin 2 a 1 ; /. 7? = V8131-98 = 90-18 Ibs. 96 KEY AND SUPPLEMENT 2. 7? cos a = 20 cos 180 + 10 cos 270, R sin a = 20 sin 180 + 10 sin 270 ; hence It, cos a = 20, It sin or = 10. Squaring and adding gives R = V500 = 22-36. 3. We have ltcosa = P cos + P cos 90 + P cos 225 + P cos 270, E sin a = P sin + P sin 90 + P sin 225 C + P sin 270 ; hence R cos a = P (1 - = 20 42' 17". To find A C we have from the figure AC=AB&\nABC = J# sin (0 - cp) TO ELEMENTARY MECHANICS. 101 = AB sin 24 17' 43" = 04115 AB. Substituting in the first equation above gives Q.7071 x 50 04115 = 85-97 + Ibs. To get the compression on the bar, take the origin of moments at D, in which case the moment of the tension will be zero. The perpendicular from D upon AB prolonged will be AD sin #, and from D perpendicular upon the vertical through B will equal BE = AB sin 6 ; hence we have calling c the compression c.AD&hie = W.AB&inff; AS ^ ~ AD To find AD we have sin ABD : sin cp : : AD : AB ; /in _ 04115 .*. A.JJ AB 0-8592 + ' which substituted above gives c- 0-8592 W = 42-96 Ibs. 102 KEY AND SUPPLEMENT PAGE 116. 6. Let fall a perpendicular p from B upon AC, then p AB sin A. Let D be directly under W, then taking the origin of moments at 13 we have t.p = W.BD; BCcos CBD AB sin A W. l.lit = W, then AB sin BAG = BD, or BC will bisect the angle ACD. 8. Take the origin of moments at ^4. Let fall a perpendicular from ./4 upon CB produced; its length will be p = AB sin CBD Q = 6x = 6 x BG 8 48 feet. ~ 8-9442 Let c be the compression on BC, then the equation of moments becomes, c.p= W. AD; AD P _ 10 x 8-9442 48 = 931 -Tibs. 500 TO ELEMENTARY MECHANICS. 103 9. For equilibrium we have Ol = -^ ^- be = co; or there can be no equilibrium. 10. We will have be = P *~ Pl Ob _2 x Pi = 0; or the forces must act at the same point. PAGE 118, ART. 187. It is well to illustrate this article still further. If the forces constituting the couple be & equidistant from the cen- tre c of the body, it is suffi- ciently evident, without a thorough demonstration, that it will produce rotation only. But is it equally evident that, if the same couple act upon the same body in such a way that the

F\ hence Px sin

3/ - _ ~ - ', + Er + T+r Iir = i 5, then Ac = %R. Ifr= R, Ac =$R. 5. Let AF = AD a ; then the diagonal AE = a A/2, CE = i#A/2, and the equation of mo- ments will be .-. AS = &aV% =f iA/2 = $AC. PAGE 146! 6. The centre of gravity of the triangle ABC w\\\ be at \ of CFirom F\ of DCE, \ of CG from G; or from ^it will be FG + ^^ The tri- angle ^(7 will be to triangle DCE&s (FCf TO ELEMENTARY MECHANICS. is to (<7) 2 ; and ACS: ADEB = (FC)*: Taking the origin of moments at F we have, ADEB x Fg = ACS x $FC - DGE x (FG + ICG}. From the proportions, we have, (FC?-(QC? Substituting, ((FC'f - (GC}*)Fg = l(FCf - (CG)\FG From the figure we have CG _DE FG ~ AB ' and FO=CG + CG _DE . "CG + FG ~ AB' hence we find AT) _ AB-DE Substituting above gives AB \ 2 ( DE AB - BE) ' (AS - BE FG. * 17 v n- K ' x ~ 118 KEY AND SUPPLEMENT DE f AB V; wr* (AB-DE) (FG} " \AB-DE) Reducing gives -(DEY T (AB)* (AB - DE)* ? ' T i(AB - DE? (DE)\ZAB-WE)-] (AB - DE)* J , r(^g) 8 - 3AB.(DE)* + 2(Zl57n . ^L (AB-DE)* J' /. Fg = ( A B ) 3 - - DE)(AB - DE)(AB + DE) - ZAB.DE+ DE*)(AB +DE)_\ AB + WE AB+ DE' 7. The slant height will be A/V 2 + A 2 ? and the lat- eral area, 2?rr4vV 2 + A 2 ; and the centre of gravity of the lateral area will be in the axis at \li from the apex. Let x be the required distance, then + TrrVr* 4- A 2 ) # = o v ' + A 2 4- r , /. x ,.., A. TO ELEMENTARY MECHANICS. 119 PAGE 149, ART. 230. PROBLEM. To find the of gravity of a segment of a sphere. A segment of one base, GAH, is considered in the text. To find the volume of the segment AGH, we have from geometry centre "We put this under another form thus, in the right- angled triangle AlfC, we have (AH) 2 = (AC) 2 - (CH) Z = (CG) Z - (CG - GH? = 2CG.GIf- (Gil)* = GH(2CG - GH), which, substituted in the preceding expression, gives for the volume of the segment, which is the value used in the text. If r be the radius of the sphere, and h the altitude of the segment, the expression becomes rch\r - lh). For the volume of the spherical sector ACG, we have, from geometry, 120 KEY AND SUPPLEMENT x GH = f 7ZV' 2 A, which is also the value used in the text. The volume of the cone the radius of whose base is AH, and altitude UC, is n(AHJ- x ==*(/*- (r - A)*H(>- - A) - h)(r - li). These values in the first equation of the article gives r , - A) - 3(r* - (r - Jif)(r - " To put this under another form, let the angle ACG = 6, then h = r CH r r cos 9 /(! cos ff) ; hence 2r h = 2r (r r cos 0) = 2r r + r cos = r (1 + cos 6). TO ELEMENTARY MECHANICS. From trigonometry 2 cos*J0 = 1 + cos ; .-. (2r - /O 2 = ^ cos 4 \B. Also we find = r l(r rcosd) = ^r(2 + cos 6} ; and these substituted above, give 3 cos 4 iff If A = 0, we have 0=0, and the segment will vanish, and we have W, we will have e > 1, and the value of y will be imaginary ; hence W necessarily exceeds P. 7. They cannot, for the eccentricity would be less than unity. 8. If the weight W be at the upper extremity of the axis, at A, there will be equilibrium. 9. The length will equal twice the diameter of the bowl, and it will rest on the edge F. 10. Horizontal. 11. It would rest on the horizontal plane. PAGE 186. The expression for the moment in the 4th line of page 186 should be F^ cos A - y l cos a j), for reasons given in Article 176 of this Key. This is the same as F l fa sin ^ y l cos aj). The signs of all the other expressions on that page should also be changed from + to and to + . 136 KEY AND SUPPLEMENT SOLUTIONS OF EXAMPLES. PAGE 192. 1. t = *' . 5 100 = 90-5 Ibs. 2 x 0-bb33 2. Let AD = u, DC= v, t = the tension of AB, t = the tension of BC', then x + y = 10, (1) u + v = 5, (2) t = 2*! ; (3) and the last formula of Article 269 of the text gives sec BAD = 2 sec BCD, or - = 2^. (4) u v From the figure, ~ y \ / Eliminating between equations (1), (2), and (4) gives "== (6 > Equations (1), (2), and (5) give TO ELEMENTARY MECHANICS. 137 a? - u 2 = (10 - a;) 2 - (5 - u}\ or 4x - 2w = 15. (7) Combining (6) and (7) gives x = 447 + ft., u =1-44 + ft; .-. y 5-53 + ft, v = 3-56 + ft PAGE 193. 3. These conditions require that the points A and C shall not be in the same horizontal ; and in the values of t and # t , page 188 of the text, BAD = 0, and we have t= W cot BCD, t 1 = WcosecBCD W " sin BCD ' 4. The angle DAC will be 45, and the last formu- las on page 191 of the text give c cos 45 = t c sin 45 = 250 Ibs. ; = 250 Ibs. 138 KEY AND SUPPLEMENT 5. The last equation on page 191 of the text gives TFsin CAD = % TF; .-. sin CAD = ; hence CAD = 30, and the depth CD will be one-half the length of the rafter. 6. The equation = |TFcot CAD, page 191 of the text, gives W=$Wcoi CAD-, .'. cot CAD = 2 ; .-. CAD = 26 33' 54". ANSWERS TO EXERCISES. PAGE 193. 1. They will not. The two added forces R and R will form a couple. 2. They will not, for the resultant of F and f\ combined with R will constitute a couple. If the force equal to R were to act in a direc- tion opposite to the resultant, then, in Exercise 1, there will be a resultant equal and oppo- site to the fourth force, and in the 2d Exercise, after the 3d force be removed, there will be a resultant equal to 27?. 3. No modification is needed, for the moment of TO ELEMENTARY MECHANICS. 139 the components will equal tlie moment of the single force. 4. It cannot to do so would require an infinite tension. 5. If the weight at B is free to adjust itself, the tension on each will be equal, and each equal toTF. 6. The tension will remain the same. The tension is dependent only upon the weight and slope of the parts, as shown by the equations on p. 188 of the text, 7. An ellipse for the sum of the distances from the fixed points A and C will constantly equal the length of the string. 8. Decreased and the thrust at the lower ends will be diminished. See the last equations of Art. 271 in the text. 9. The thrust and stresses on the braces will both be increased. 10. If the strut supports the weight, there will be no stress on the rafters. PAGE 197. Galileo was the first writer, of whom we have any knowledge, who established formulas for the strength of beams. His work was published at Bologna in 1656. Although the hypotheses upon which the for- mulas were founded were false, yet the law of variation of strength which he deduced for rectangular beams was correct. This law is the strength varies directly 140 KEY AND SUPPLEMENT as the first power of the breadth and the square of the depth jointly, and inversely as the first power of the length of the beam. But the factor used by him for determining the value of the strength was three times too large. SOLUTIONS OF EXAMPLES. PAGE 201. 1. Let w = the load per unit of length of the beam, I = the length of the beam, then will W= wl We have from the last equation on page 200 of the text, *K

- s = * ltb - 3 8000 x 12 x 12 2 x 10000 = 86-4; .-. d = 9-29 inches. 4. From the same equation as the preceding, we have _ 3 20000 x 10 x 12 4x9x9 = 11111-1 + Ibs. 5. The required stress will be the value of J2 found from the equation above, 142 KEY AND SUPPLEMENT 20000 x lj x (3j) 2 10000 = 24-5 inches. 6. Problem 4, p. 200 of the text, gives I 12000 x 6 x 144 15 x 12 = 76,800 Ibs. 7. The load will be uniform, and will equal the weight of the beam. We have W=2 x 2 x I x 4- and the formula of problem 2, p. 199 of the text, gives W 30000 x 2 x 4 . /.P = 80,000; and Z = 282-8 inches = 23 feet 6-8 inches. TO ELEMENTARY MECHANICS. 143 PAGE 205. The straight line of quickest descent is not the line of quickest descent. Curves of quickest descent are called Brachistochrones. Their form depends upon the conditions assumed. The forces may be assumed to vary according to the inverse squares, or directly as the distance, or inversely as the distance, or ac- cording to some other law, and they may be assumed to act in parallel lines or radiate from a point. If they are constant and parallel, as in the case of terres- trial gravitation, the curve will be a cycloid. It was problems of this character that gave rise to the Cal- culus of Variations a very high order of analysis. SOLUTIONS OF EXAMPLES. PAGE 208. 1. From the 3d equation, p. 203 of the text, we have 2* 200 ~ 32* x 25 = 2487; .-.

/. * = 50* - loY 2 x = 50* - 144 KEY AND SUPPLEMENT Hence, in 3 seconds s = 150 - 72|\/2 = 47-65 feet. At the end of 5 seconds s = 250 - 201&V2 = - 34-3 ft., that is, it will have ceased to ascend the plane, and returning, will, at the end of 5 seconds, be 34-3 feet below the starting-point. At the end of 10 seconds s = 500 - 804|V2"= - 637 3 feet below the starting-point. 3. The required velocity will be the same as that acquired by the body in sliding down half the length of the plane ; hence the required veloc- ity will be v = -s sn cp x 100 x T 2 / h - 17-6 328 x 2640 324 146 KEY AND SUPPLEMENT PAGE 209. - 1634 sec. = 2-72 minutes. For this time on the horizontal, we have from the 4th and 5th equations 4858 4-6 = 303-3 seconds, = 5-05 4- minutes. 8. Equation (5) will give the velocity which it must acquire in moving down the plane. We have = 4~T _1000 ~ 4-6 = 217-3 feet per second. The required height will be given by equation (2) ; we have h = 81 x 217-3 + 17-6 x 1200 1200 = 32-27 feet. TO ELEMENTARY MECHANICS. 9. Equation (5), page 208 of the text, gives = 13-19 ft. per sec. Equation (2) gives Sis' " h - 17-6 _ 81 x 173-91 7-4 = 1903-64 ft The author once had occasion to use the prin ciples of the last example in constructing the approach to an ore dock at Marquette, Mich. soLrnoxs OF EXAMPLES. PAGE 217. According to Article 299, the range will be 2A sin 2 = 904-69 feet. 4. From Article 299, we have ; * ~ 2 sin 2a ,.,=*/! 2 x 32i x 25000 2x1 = 89G-8 + feet From Art. 301, we have A = h sin 2 a 150 KEY AND SUPPLEMENT = 6,250 feet. From Article 300, we have sin a 2 x 897 x 1 = =. 89 + seconds. 5. We have from Articles 299 and 301, AS = 4CD, or 4A sin a cos = 4/i sin 2 tf, dividing by sin a cos tf, tan a 1 ; /. a - 45. 6. In equation (3) page 214, make y = 150, which is negative, because in Fig. 139 y is positive upwards, and the point where it will strike the plane, in this example, is below the point of starting, tan a = 1, and we have Q1 rZ 1 ~ c\ ~ 2(75)* x' which solved for x gives x = 271-5 feet. TO ELEMENTARY MECHANICS. 151 7. Iii this example a - - 30, y = - 25 feet. The velocity with which the body will leave the eaves will equal that of a body falling through the vertical height of the ridge above the eaves, and this value will be considered as the velocity of projection. We then have y 2 = 2#r x 7 = 14gr, and these substituted in equation (3), page 214, give - 25=- 0.57785, - which solved gives x = 17-6 + feet. 8. Substituting the values given in the example for a and y in equation (3) page 214 gives 60 = 400 tan a - 50 = 600 tan a - s a 2 32j(600) ! 2^ cos 2 a Multiplying the first by 9 and the second by 4, and subtracting the latter from the former, gives 3 10 = 1200 tan a; /. tan a = -jrj, and 152 KEY AND SUPPLEMENT a = 15 49' 9". Substituting this value in the first of the pre- ceding equations gives 60 = li x 400 32i(400)* _ 2 x (0-96213) 2 x 2 = 228-2 feet per second. EXERCISES. PAGE 219. 1. Zero. 2. The velocity of projection being the same, they will strike the sea at the same time, and their range from the point where the ship will be at that time will be the same ; but not the same if reckoned from the point of projection. 3. 15 miles per hour = Hu^H- f ee * per second = 22 feet per second ; hence the actual veloc- ity will be 11 feet per second in the direction of motion of the ship in reference to the point from which the projection is made. 4. In reference to the point on the earth, it will be the same ; but not in reference to the point in space from which the projection is made. 5. It will reach it in the same time. A horizontal motion does not affect the time of descent due to gravity. The projectile falls from the TO ELEMENTAKY MECHANICS. 153 highest point of its path (in a vacuum) in the same time that it would fall vertical down- wards. 6. They will ; for according to Article 304 of the text, we have for equal ranges the angle a and 90 - a. Let a = 45 - 6 ; then will the an- gles be 45 - <$ and 90 - (45 - 6) = 45 + since the motion in the arc is uniform ; and since the centrifugal force is constant, the space BE will be given by equation (2), p. 12 of the text, or BE = $v't, where v' is the velocity which would be produced by the centripetal force in passing over the space BE'm time t, if the force acted along the line BE. But since the times are equal, we eliminate t, between these equations, and find 1BE From the figure we have, since EC is a mean pro- portional between BE and EG, (Ed}* = BE(BG - BE), which ultimately becomes _ BG ' Substituting this value above gives , 2EC EC 156 KEY AND SUPPLEMENT since BG = 2r. But ultimately the velocity along BD or EC is that along the arc BC, and ultimately where v is the velocity along the arc. This substitu- ted, gives v ID t or, multiplying by m, mv' But the left member is, according to Article 122, the value of a constant force, hence the required result. PAGE 229, ARTICLES 319, 320. Sir Isaac Newton con- ceived the fact that if the attraction of gravitation varied as the inverse square of the distance from the centre of the force, it ought to account for the motion of the moon ; that is, the force of gravity ex- erted by the earth should just equal that necessary to cause the proper deviation of the moon from a tan- gent to its orbit. His first efforts to prove this law failed, due to the fact that an erroneous value of ./?, the radius of the earth, was used. Instead, however, of abandoning the idea, and attempting to account for the motion according to any other hypothesis, he TO ELEMENTARY MECHANICS. 157 returned to his calculation from time to time, but with no better results. Finally, while attending a lec- ture in London, he obtained a corrected value of the radius, which, when substituted in the equation he had so often reviewed, established his theory. He was so overcome by the grandeur of the problem as the final proof was becoming apparent, that he was unable to complete the numerical reduction, and called a friend to do it for him. It will be seen, in Article 319, that the radius of the earth enters the formula, in determining the dis- tance of the moon, in the expression 60-367?. The value of R which he at first used was too small by iV to iV f i* 8 true value. See also remarks on pp. 36 and 37 of this Key. The law of gravitation was not, at once, universally accepted. Several times, especially in the history of astronomy, certain phenomena appeared to conflict with this law, when it was called in question, and its truth assailed. But all opposition to it disappeared after Laplace, by his truly wonderful analysis, ex- plained all those paradoxes, and accounted for all the motions of the solar system, on the simple law of Universal Gravitation. It is now believed to be true, not only for the solar system, but for every particle of matter in the universe. Newton believed that the ether of space, whatever it might be, was more dense in the vicinity of the planets, than in remote space ; that, indeed, it might be only air extremely rarefied. To find the stress due to the attraction between the earth and moon. It equals the centrifugal force, the value of which is 158 KEY AND SUPPLEMENT The mass m of the moon is about 3 times that of a mass of water of equal volume, and as a cubic foot of water weighs 62 Ibs. when g = 32 feet, and the diameter of the moon is 2,160 miles, we have m = 3i x ITT (2160 x 5280) 3 X Ibs. The time of the revolution of the moon about the earth is about 27 days ; hence the angular velocity per second is 27T 6? = X 24 X 3600 ' The mean distance between the centres of the moon and earth is about 240,000 miles ; hence r = 240,000 X 5280 ; all the magnitudes being in feet, and all the times reduced to seconds. Hence we have _ 3 x |7T 8 x (2160) 3 X (5280) 4 x 240000 x 62j 32 x (27i X 24 x 3600) 2 which reduced gives, approximately, 44,000,000,000,000,000,000 Ibs. = 44 x 10 18 . A steel rod one square inch of section will sustain a pull of 120,000 Ibs. ; hence it would require (approx- imately) TO ELEMENTAKY MECHANICS. 159 370,000,000,000,000 = 37 X 10 13 square inches of steel to hold the moon in her orbit if substituted for the attraction between the earth and moon. In one square mile are 4,014,489,600 sq. inches ; which, divided into the above, gives, for the equiva- lent section in miles, 90,000 sq. miles nearly. Since the radius of the moon is 1,080 miles, the area of a great circle will be 3,660,000 sq. miles nearly. which, divided by the solid section of the steel rod, gives 40 + ; hence, if the rods were each one square inch in section, and the great circle of the moon be divided into inch-square spaces, the rods would cover one space in 40. The square of the diameter of the earth is nearly 15 times the square of the diameter of the moon, hence such a steel rod would cover about -^ of the merid- ian circle of the earth. If the material be iron instead of steel, and if 10,000 Ibs. be taken to represent the tenacity, a value quite commonly used in engineering structures, the rod or rods would cover more than one-fourth the cross-section of the moon, and about -fa of a great cir- cle of the earth. The same problem applied to the attraction between the sun and earth gives 163 KEY AND SUPPLEMENT x f 7r 3 (20500000) 3 x 5280 X 92500000 X 62j .. 32- x (365 x 24 X 3000) 2 where it is assumed that the mean density of tho earth is 5| times that of water, the distance between the centre of the sun and the earth 92,500,000 miles, the radius of the earth 20,500,000 feet, and the time of the revolution in the orbit 365 days. This reduced gives, approximately, 912 x 10 18 Ibs., or 912,000,000,000,000,000,000 Ibs., or more than 20 times that between the earth and moon. According to this result it would require a solid steel rod of a cross-section equal nearly to one- half the great circle of the moon, the tenacity being 120,000 Ibs. per square inch ; or if the rod be of iron, and 10,000 Ibs. be used for its tenacity, the section of the rod will be about f of the area of a great circle of the earth. These examples show the immense stress of gravitation when large masses are involved. The following examples will show that the same force, under certain circumstances, is comparatively weak. Required the time that it will take two spheres of the same material as the earth, each one foot in diam- eter, placed 12^ inches from centre to centre, to come together ly their mutual attractions, in void space. TO ELEMENTARY MECHANICS. 161 According to one of Newton's laws of attraction, the force varies as the mass. If the diameter of the earth be 41,700,000 feet (p. 33 of the text), then will the mass of the sphere 1 foot in diameter be and therefore the acceleration at the surface of the earth due to the attraction of such a sphere placed at the centre of the earth would be g (41700000) 3 ' an inappreciable quantity. According to another of Newton's laws, combined with the analysis on pp. 33 and 34 of this Key, the force varies inversely as the square of the distance from the centre of the sphere ; hence at the distance of one foot from the centre of the small sphere we have P: (20850000)- ^-p^^^:/; -p - 9 8 x 20850000 ' for tne acceleration of a particle one foot from the centre of the sphere. This will also be the accelera- tion of any uniform sphere whose centre is one foot from the first sphere, if the diameter of the second sphere does not exceed one foot ; or should it exceed that diameter, it will still be true for the mass of the sphere if reduced to a sphere whose diameter is less 162 KEY AND SUPPLEMENT PAGE 229. than one foot, and hence will be the acceleration pro- duced upon an equal sphere. If the first sphere were fixed in space, the second sphere would move the of an inch ; but as both are free each will move one- half the distance between them, or i of an inch. In order to simplify the problem, we will assume that the acceleration is uniform, while the spheres are moving the \ of an inch, and is that due to the at- traction at a distance of 12 inches from their centres ; then will equation (4), page 12 of the text, be applica- ble, and we have 2 x \ x T y __ _ _ 8 X 20850000 = V108031 = 328-7 seconds, nearly, or less than 5 minutes. This problem is in " The System of the World," by Sir Isaac Newton, p. 527 of our copy of the Principia. It is there stated that " the attraction of homogeneous spheres near their surfaces are (Prop. Ixxii.) as their diameters. Whence a sphere of one foot in diameter, and of a like nature to the earth, would attract a small body placed near its surface with a force 20,000,000 times less than the earth would do if placed near its surface, but so small a force could pro- duce no sensible effect. If two such spheres were TO ELEMENTARY MECHANICS. 163 distant but % of an inch, they would not, even in spaces void of resistance come together by the force of their mutual attraction in less than a month's time." "We have sought for the source of the error in the Principia, by determining the conditions necessary for giving his result, but have not satisfied ourselves. We observe that he made an error in saying that the force of attraction on the surface of the small sphere is 20,000,000 times less than on the earth ; for, ac- cording to his proposition considering the radius of the earth as 20,000,000 feet it should be 40,000,000 times less ; and according to the inverse squares, at the distance of one foot from the centre of the small sphere, it would be 160,000,000 times less. If now we assume that the particle is moved i of afoot, un- der the action of this force, we would have = which, according to the notation, will be the accelera- tion on the surface of the earth due to gravity ; hence E /i \ ff = t*jp, (1) and * = |^ (2) by means of which the numerical value of the unit of acceleration may be determined. Again, the acceleration produced by the attraction of the mass m upon one of the units of mass at dis- tance, unity will be and at a distance, x, the acceleration will be 166 KEY AND SUPPLEMENT winch will also be the acceleration produced upon m, by the attraction between m and m' at the distance x between them, since the result will be the same as if both masses were concentrated at their centres of gravity ; for all of m will exert the same force upon each particle of m' as upon each particle of the unit. But the pull in pounds will equal the mass into the acceleration (p. 44, Art. 86 of the text), or mm Similarly, considering the attraction of m' upon m, the acceleration produced upon m' will be eP and the pull in pounds will be m times this amount, or m'm , C v ^ IT ' which is the same as (4), as it should be. Substitut- ing /* from (2) in (4) or (6) gives for the pull in pounds (or their equivalent), of any two masses m and m', I 7~)9. - -I 3- CO The origin of the axis of x being at the centre of one of the masses and moving with it, and the total mass moved by the stress being m + m', we have TO ELEMENTARY MECHANICS. 167 JPx mm'JPy 1 , (rn + m}-^-- -5p.-r, (8) which integrated (Analyt, Mech., pp. 33, 34), observ- ing that for t = 0, x = 0, and v = 0, and that // in , mm'I&q . .,. the reference equals 7 -* in this case, gives (in + m}E [(m + m'}Ea~\ T, <= L a^'A J x L^-^ which for the limits gives If both spheres are of the same density, their masses will be as the cubes of their radii ; or r 3 and we have and if the spheres are equal, as in the problem, we have If a = 1 foot, ^ = 20,850,000, r = foot, ^ = we have 168 KEY AND SUPPLEMENT i = 3577 seconds, nearly = 59-6 minutes, nearly. PAGE 229. An exact solution of the former problem may be made by means of equation (9), by substituting in it a = 12i inches = 1A feet, and making x = a for one limit and 1 foot for the other. We would thus have = 384 seconds nearly, or less than 6 minutes. The following is the reduction of the preceding ex- pression. To find the value of ^"'(TOI) ' we log. 12 =. 1-079181 log. 12-25 = 1-088136 Dif. = 1-991045 Dividing by 2, 1-995523, and ' log. 0-98974 - f-995522, or log. cos 8 12' 46" = 9-995523. The length of arc will be 8 12' 46" 180 X 60 X 60 or 31416, TO ELEMENTARY MECHANICS. 169 29566 648000 which may be reduced as follows : log. 29566 = 4470892 log. 3-1416 = 0497150 ar. co. log. 648000 = 4-188425 subtracting 10, log. 0-14337 = 1-156467 adding log. 1-A, log. 1-02083 = 0-008951 gives log. 0-14636 = 1-165418 ; hence, U- cos-V--V = 0-14636. We also have (1& - 1) 1 = & = 0-02083 + which added to the preceding result gives 0-16719 + for the value in the brackets. For the first parenthesis, we have log. 1A- = 0-008951 log. 20850000 = 7-319106 log. 8 = 0-903090 log. 32|, ar. co. = 8-492594 subtracting 10 and dividing by 2, 6-723741 log. 2300-7 = 3-361870. Adding log. 0-16719 = 1-223209 gives log. 384-6 = 2-585079 ; 170 KEY AND SUPPLEMENT that is. the time will be 385 seconds nearly, or less than 6| minutes. To find the stress in pounds which would be ex- erted by the mutual action of two such spheres at a distance of one foot between their centres, we have from equations (7) and (10), since m = m', and x = 1, E The mass of the earth is 5 times an equal mass of water. The weight of a cubic foot of water is 62 Ibs. at the place where y = 32|, and the volume of the earth is 7tR? hence E = 5| x C2i X which substituted above gives for the stress %7t x 5 x 62| 20850000 x 32f ' = 0-00000215 + Ibs., or less than joirinnnjo- f a pound, a quantity inappre- ciably small. A would-be inventor once proposed to weigh the varying force of gravity by means of very delicate scales, and by using them on board a steamer, thus determine whether the water underneath were deep or shallow. Since the density of the solid earth ex- ceeds that of water, the force of gravity at the sur- face of deep water will be less than on shallow water, but it is evident that the rocking and heaving of the vessel would probably produce more effect upon such TO ELEMENTARY MECHANICS. 171 delicate mechanism than that due to the variations of the force of gravity. It is also stated that mariners have observed that two ships at rest in a quiet sea tend to approach each other, but it will be found that the gravitating stress due to their mutual attractions is so small that it might be more than neutralized by a very slight breeze, or by the beating of very small waves. To give some idea of the magnitude of this stress, assume that the vessels are of equal mass, and each equivalent to a sphere of the average mass of the earth, each 30 feet in diameter, and 200 feet between their centres. Conceive that the masses are reduced to their cen- tres, then since the mutual attraction of unit-spheres at distance unity between them is y-g^f f,nro f a pound, the attraction of the masses of the ships, reduced to their centres, at the same distance (one foot) will be TOrttmnr X 3 3 X 3( ' 3 = 14 ?1 P ound8 i and at the distance of 200 feet, it will be 1471 200 2 = 0.04 pound nearly. If the distance between them be 500 feet, the stress would be 1471 5QQ2 = T}TF of a pound, nearly. If all external forces, such as the wind and action of the sea were neutralized, this slight stress would be 172 KEY AND SUPPLEMENT sufficient to cause the ships in question to collide in a short time. SOLUTIONS OF EXAMPLES. PAGE 233. 1. We have from equation (5), page 226, v* W v 2 f = m - . r g r equal 2 W; W ?;* ...2TF=- -; g r or the velocity must be that acquired by a body falling freely through a distance equal to the radius of the circle (see eq. (3), p. 36 of the text). 2. If the tension is 3 W, we have W -u 2 3W= --; g >' Let n be the number of revolutions, then the velocity will be the space (2?mi) divided by the time, or 60 seconds, r must be reduced to feet and the time to seconds, for g is given in feet per second. Hence we have TO ELEMENTARY MECHANICS. 173 ftrn v = 60 30 ' r 30 3-1416 = 38-3. 3. The centrifugal force will equal the weight; hence tf W v> W = m - = - ; r a r Ttrn . ,. , x .*. v = *jgr = -K7T- (see preceding example) ; 30 u times the pressure due to the centrifugal force, and must equal the weight. Let n be the number of revolutions per minute ; then will the angular velocity per second be . n 60/> 60 and the centrifugal force will be (Art. 314) and the friction will be * ^WfT ' _ 30 ,/ 9 6. The body is assumed to be in a radial groove, and the string slightly elastic so as to allow the body to move slightly along the groove, and thus give the friction a chance to act. The angular velocity per minute will be 2* x 250 : and the centrifugal force will be TO ELEMENTARY MECHANICS. 175 f - "07500 *\ 2 30 7 : 7 V eo J ' 12 = iVjX *P (3-141 6) 2 ?F. The friction will be 0-15 of this amount, and the tension of the string 0-85 of the same; hence the tension will be T= 0-85 x T Vv X *| = 45-27 + pounds. PAGE 234. 7. The friction will be The centrifugal force will be (Art. 313), gR or 324- - 2500 10 = 89-675 feet per second = 61-14 miles per hour. 17G KEY AND SUPPLEMENT 8. According to Article 322, we have , v* I n = /3Q X 5280 \ 8 ' _ \ GO X 60 / 32 X 3000 0-09$ feet = 1-12 inches. 9. The weight will be to the centrifugal force as the length of the stri ment of the body ; or the length of the string is to the lateral raove- _ W To find/" we have equation (5), Art. 313, /40 X 5280 f _ W\ 60 X 60 / g 4000 which substituted above, gives _6 /40 x 5280X 8 . X 4000 V 3600 / ' x = 0460 ft. =. 1-92 inches. TO ELEMENTARY MECHANICS. 177 The value is independent of the weight. 10. To find the time of making one revolution, we have T = 60 -T- 100 = f. Then from p. 232 of the text, we have gT* cos cp = -j-f j^ 2 . AB X ) 8 ~ 5 x (3-1416) 2 = 0-23466 ; .-. tp = 76 25' 43". Or by logarithms log. 32 = 1.507406 log. (0.6) s = T.556303 adding, 1 063709. log. 5 = 0.698970 log. (3.1416) 2 = 0.994300 adding, 1.693270 which subtract- ed from the above gives log. cos q> = 1370439; .:

= 0, a relation will be es- tablished between AB and T, or we will have and if A B is assumed to be less than the value 8* 178 KEY AND SUPPLEMENT found by this formula, cos cp will exceed unity, and hence q> will be imaginary, or, in other words, the conditions will be impossible. We also have AC = AS cos

Q . . A 4 *** ~~ on c\f rr -^T" w w l 32 25 7 r w I 7/' 3. s = - -. In this case w l = ; .'. solving for ^ -8 x 60 48 0/m = =: :=: 4. A stone 5 ft. on each edge = 125 cu. ft. and will displace 125 x 62-5 = 7812-5 fts. water, x 2-3 = 17968-75 Bbs. = weight of stone. 5. 1= =f w u' 2 40-32 8 * = - T7J - TF F = !*" w w t 40 35 5 7 w + V& 35 + 18 53 _ c + ! '5+2 " 7 " -(*-*>!,. _ (10-5- 14)19-3 x 10 _ (* 2 -*i)V (10-5-19-3)14 -675-5 = 5 ' 483 ' 196 KEY AND SUPPLEMENT _ (s l - 8)% _ (19-3 - 14)10-5 x 10 _ ** ~ (*i - * 8 )* ' (19-3 - 10-5)14 556-5 o 1 - i _ P* + PI*I _ I _ 27 x 1 + 39 -4915 x 1-8 85 ' n = (o + Vfa ~ (27 + 39-4915)1-6321 _ ft, + 0-0013 (ft, - c) _ 14 + 0-0013(10 -7) flj + * 2 - c 14 + 10-7 14-0039 ANSWERS TO EXERCISES. PAGE 271. 1. It will not ; for water being more compressible than the solid, will be relatively more dense in the air than in a vacuum, and hence when in air will force the body upward more than when in a vacuum. 2. Because the smoke is lighter than the surround- ing air ; but if it be heavier it will fall in the air. 3. See answer to Exercise 1. 4. This assumes that the weight of the bag or some other cause causes the bag to sink, and if it sinks at all it will go to the bottom of the vessel. If now a pressure be exerted upon the surface of the liquid, it will cause the bag (or TO ELEMENTARY MECHANICS. 197 gas) to condense more than the liquid, and lience it will not rise. Toys have been made involving this principle. 5. Water is more compressible than iron for the same pressure, hence it seems possible, theoret- ically, for water to be subjected to such a pres- sure as to be as dense as iron at the same pres- sure. 6. If both are incompressible, their relative densi- ties will be unchanged by pressure, and hence the heavier body will sink indefinitely. If the body be compressible, it will become relatively more dense, and hence there will be no limit. 7. If the brine be sufficiently " strong " according to popular language it will float the egg. In order that it may float between the top and bottom, the brine must be more dense near the bottom than at- the top. 8. It will. It is related of Benjamin Franklin, that he asked a company of savants why a pail of water containing a fish would weigh no more than without the fish. Several reasons were given, and finally he was appealed to for the reason. He thus replied: "Are you sure it will weigh no more?" They had been trying to explain a false assumption. SOLUTIONS OF EXAMPLES. PAGE 2T6. f ? 1. The equation on page 273 gives us - = */ tan > 198 KEY AND SUPPLEMENT tan

. In this case the slope of the free surface will be 45, and tan 45 = 1 ; .\f=y. 4. Let A be the edge of the vessel. From A con- ceive a horizontal line drawn to XE, and mark the foot with the letter Z. The volume gen- erated by the revolution of the semi-parabola is one-half the product of the base and altitude (See Mensuration^ or works on the Integral Calculus). And as the volume of this parabo- loid is the unoccupied portion of the cylinder, the altitude ZE will be 2 x 3 = 6 inches. AZ is 12 inches. From a property of the parabola, we have EZ : AZ : : AZ : the parameter, (orx:y::y: 2/>) ; 12 2 .-. the parameter = -^ = 24 inches. This is known to be twice the subnormal DC] hence DC = 12 inches, which, as shown on p. 275 of the text, is g -r- or 2 , therefore . 12 TO ELEMENTARY MECHANICS. 199 Let n be the number of turns per minute sought. Then 2?r = will be the angular velocity per second, and we have 30 " 12 ' ..n = 11-77 turns. 5. The angular velocity will be f #2;r = ?r, which is the value of GO in the value of DO, p. 275 of the text ; hence DC ~s-. This is one-half 71 the parameter in the equation y* = hence the equation becomes y* 2 -^ x. SOLUTIONS OF EXAMPLES. PAGE 282. 1. To fulfill this condition we must have - A 2 ) A 2 = 4-5 h = 2-121 feet. 2. The area of each triangle will be % x 1-4 x 2-6 = 1-82. The centre of gravity of the triangle whose base will be in the surface will be | x 2-6 x sin 56 35' below the free surface; hence the pressure on it will be x 1-82 x i x 2-6 x sin 56 35' = 82-28 Ibs. ; and of the other triangle, 200 KEY A ND SUPPLEMENT 62| x 1-82 x f x 2-6 x sin 56 35' = 164-57 Ibs. 3. Pressure on concave surface = #&/i 2 , but b = .-.p = A. x 62-5 x 2 x 34416 x 9 = 1767-15 Ibs. Weight of liquid = TV* x A x 6 = 3-1416 x 3 x 62-5 =589-05 fts. Pressure on base = weight of liquid = 589-05 fts. 4. The weight of the liquid = ^TT^ = f x 62-5 x 3-1416 x 125 = 32725-0 fts. Normal pressure = 4#7r/- 3 = 4 x 62'5 x 3-1416 x 125 = 98175 fts. 5. Pressure on flood-gate = i(/i 2 3 ^'i 2 ) = i x 62-5 x 2 x (13 2 - TO 2 ) = 62-5 x 69 = 4312-5 fts. 6. Pressure on opposite side #5(7i' 2 A" 2 ) = ^ x 2 x 62-5(7 2 - 4 2 ) = 62-5 x 33 = 2062-5, and 4312-5 - 2062-5 = 2250 fts. SOLUTIONS OF EXAMPLES. PAGE 287. 1. By Art. 372, the centre of pressures of rectangle is at a distance from the top equal to f the height, .-. | of 3 = 2 feet. 2. By Art. 373, we have for the required depth, 35 1.5 /- + ION 35 5 19 1365 95 * 6 36 V + 5 ~~ 6 J8 13 ~ 234 234 TO ELEMENTARY MECHANICS. 201 = 5-427 ft; or -427ft. = 519 inches below the top of the flood-gate. 3. To resist overturning we have Stfh = 1 f 1 /V substituting the values given, and we have _ 125 x 512 ._ 64000 _ . 6 x 180 x 8 8640" " .-. & = V7-4074 = 2-72 ft. = 2 feet 8| inches. 4. In this case we have, p. 279 of the text, fibs. for the pressure of the liquid. Its mo- ment in reference to the edge of the wall will be x 31'|M 3 , and the wall must be capable of resisting twice this amount. The moment of the wall will be i x 4 x 8 x 120 x |- of 4 = 5120 Ibs. ; .-. | x 31$ JA 8 = 5120 ; /. h = 6-2 feet. 5. The pressures are proportional to the areas, and the areas are as (15) 2 -j- (1'5) 2 = 100 to 1 ; .-. total pressure = 500 x 100 = 50,000 Bbs. SOLUTIONS OF EXAMPLES. PAGE 310. 1 t = 2 ^_= 0-62 x 7r(- 4 \-)V64i x 3 9* 202 KEY AND SUPPLEMENT 2x1x3x2304 = - = 401-2 sec. = 6m. 41-2 sec. 0-62V193 2. Q = \mbt Vfyh 3 = | x 06-2 x 2 (45 x 60) x (f) 3 = 13618-8 cubic feet. 3. The equation on p. 309 of the text is x = -5V 4 - But h = 24 inches, and r = 3 inches, rp* " hence we have a; = y* = y 4 . To find the area of the orifice, we have on p. 309 of the text, nv*c h 2 1 z = I- ft., A = 2 ft. ; 3-1416 x x , sq. f t. 0-62 V64 x 2 = 0-01341 sq. in. 4. From the equation on p. 303 of the text we have 25 -0-025187-^ = 0-0006769 \k) or 25 - 32-64235^= 7895-36(^ + 0.0039360 ; . & 31-076 Q J>5_, 7928 V 7928' /. Q = 0-0542 en. ft. per sec. = 195-1 + cu. ft. per hour. TO ELEMENTARY MECHANICS. 203 ANSWERS TO EXERCISES. PAGE 310. 1. The time will be the same for each. 2. It will not. The time will be less, for the head producing the velocity will be equivalent to what it would be if for the weight of the water an equal weight of mercury be substi- tuted. 3. It will not. The flow of the water in this case will exceed that of the mercury in the preced- ing. 4. It will. It may be observed that when the pres- sure producing the flow of a liquid is the weight of the same liquid, the head equals the height of the liquid above the orifice. 5. It will not, but the depth of submergence will gradually increase. The block receives an ini- tial velocity downward which is being gradu- ally lessened as the surface of the liquid de- scends. 6. It will be lowest near the orifice. 7. It will be greater ; for the acceleration upward of the vessel will have the same effect as an increased pressure on the surface. 8. It will be greater ; for the head over the orifice will be greater. 9. It is not ; a part of the pressure is engaged in producing motion of the mass. 204 KEY AND SUPPLEMENT PAGES 326-329. The expression for the pressure (or rather the tension) of the air at any height, a?, above the earth, p. 326, reduces to _x