10. Hence, if the forces are inversely as the squares of the distances, the squares of the times are as the cubes of the disThat is, tances. if F :ƒ :: do : D2, then T2 : to :: D3 : d3. 11. The right line that joins a revolving body and its centre of attraction, called the radius vector, always describes equal areas in equal times, and the velocity of the body is inversely as the perpendicular drawn from the centre of attraction to the tangent of the curve at the place of the revolving body. 12. If a body revolve in an elliptic orbit by a force directed to one of the foci, the force is inversely as the square of the distance and the mean distances and the periodic times have the same relation as in art. 10. This comprehends the case of the planetary motions. 13. If the force which retains a body in a curve increase in the simple ratio as the distance increases, the body will still describe an ellipse; but the force will in this case be directed to the centre of the ellipse; and the body in each revolution will twice approach towards it, and again twice recede from that point. 14. On the principles of central forces depend the operation of a conical pendulum applied as a governor or regulator to steam engines, water mills, &c. This contrivance will be readily comprehended from the marginal figure, where a a is a vertical shaft capable of turning freely upon the sole a. C D, C F, are two bars which move freely upon the centre c, and carry P H at their lower extremities two equal weights P, Q; the bars c D, C F, are united, by a proper articulation, to the bars G, H, which latter are attached to a ring 1, capable of sliding up and down the vertical shaft ▲ a. When this shaft and connected apparatus are made to revolve, in virtue of the centrifugal force, the balls PQ fly out more and more from a ɑ, as the rotatory velocity increases; if, on the contrary, the rotatory velocity slackens, the balls descend and approach A a. The ring 1 ascends in the former case, descends in the latter and a lever connected with I may be made to correct appropriately the energy of the moving power. Thus, in the steam engine, the ring may be made to act on the valve by which the steam is admitted into the cylinder; to augment its opening when the motion is slackening, and reciprocally diminish it when the motion is accelerated. The construction is, often, so modified that the flying out of the balls causes the ring I to be depressed, and vice versa ; but the general principle is the same. Here, if the vertical distance of P or q below c, be denoted by d, the time of one rotation of the regulator by t, and 3.141593 by, the theory of central forces gives t = 2 x d 321/ = 1·10784 ✓ d. Hence, the periodic time varies as the square root of the altitude of the conic pendulum, let the radius of the base be what it may. Also, when I c Q=1 C P-45°, the centrifugal force of each ball is equal to its weight. Inquiries connected with Rotation and Central Forces. 1. Suppose the diameter of a grindstone to be 44 inches, and its weight half a ton; suppose also that it makes 326 revolutions in a minute. What will be the centrifugal force, or its tendency to burst? 2. If a fly wheel 12 feet diameter, and 3 tons in weight, revolve in 8 seconds: and another of the same weight revolves in 6 seconds: what must be the diameter of the last, when their centrifugal force is the same? By art. 8, Central Forces, F:ƒ:: D d Therefore, since F 3. If a fly of 12 feet diameter revolve in 8 seconds, and another of the same diameter in 6 seconds: what is the ratio of their weights when their central forces are equal? By art. 6, Central Forces, the forces are as the squares of the times inversely when the weights are equal:, therefore when the weights are unequal, they must be directly as the squares of the times, that the central forces may be equal. Hence w: w:: 36: 64 :: 1:17 That is, the weight of the more rapidly to that of the more slowly revolving fly, must be as 1 to 13, in the case proposed. 4. If a fly 2 tons weight and 16 feet diameter, is sufficient to regulate an engine when it revolves in 4 seconds; what must be the weight of another fly of 12 feet diameter revolving in 2 seconds, so that it may have the same power upon the engine Here, by art. 8, Central Forces, we must have therefore w= W D T3 w d ; 13 = 13 cwt., the Note. A fly should always be made to move rapidly. If it be intended for a mere regulator, it should be near the first mover. If it be intended to accumulate force in the working point, it must not be far separated from it. 5. Given the radius R of a wheel, and the radius r of its axle, the weight of both, w, and the distance of the centre of gyration from the axis of motion, g; also a given power P acting at the circumference of the wheel; to find the weight w raised by a cord folding about the axle, so that its momentum shall be a maximum. Here w= ✔(R1 P2 +-2 R3 P g2 w+g1 w2+P w R r ç2 + p2 R3 r) Cor. 1. When R = 7, as in the case of the single fixed pulley then w= √ (2 p2 R3 +2 R P ç2 w + W2 + P.W R 2) R Cor. 2. When the pulley is a cylinder of uniform matter R2, and the express, becomes w✔[R3 (2 P2 + P w 1⁄2 + & w2)] — § w — P. = = 6. Let a given power p be applied to the circumference of a wheel, its radius R, to raise a weight w at its axle, whose radius is r, it is required to find the ratio of R and r, when w is raised with the greatest momentum; the characters w and e denoting the same as in the last proposition. Cor. When the inertia of the machine is evanescent, with respect to that of P+w, then is r = R √(1+) — 1. W 7. If any machine whose motion accelerates, the weight will be moved with the greatest velocity when the velocity of the power is to that of the weight as 1+ ertia of the machine being disregarded. √(1+) to 1; the in W 8. If in any machine whose motion accelerates, the descent of one weight causes another to ascend, and the descending weight be given, the operation being supposed continually repeated, the effect will be greatest in a given time when the ascending weight is to the descending weight as 1 to 1618, in the case of equal heights; and in other cases when it is to the exact counterpoise in a ratio which is always between 1 to 14 and 1 to 2. 9. The following general proposition with regard to rotatory motion will be of use in the more recondite cases. If a system of bodies be connected together and supported at any point which is not the centre of gravity, and then left to descend by that part of their weight which is not supported, 2 g multiplied into the sum of all the products of each body into the space it has perpendicularly descended, will be equal to the sum of all the products of each body into the square of its velocity. Percussion or Collision. 1. DEFS. In the ordinary theory of percussion, or collision, bodies are regarded as either hard, soft, or elastic. A hard body is that whose parts do not yield to any stroke or percussion, but retains its figure unaltered. A soft body is that whose parts yield to any stroke or impression, without restoring themselves again, the shape of the body remaining altered. An elastic body is that whose parts yield to any stroke, but presently restore themselves again, so that the body regains the same figure as before the stroke. When bodies which have been subjected to a stroke or a pressure return only in part to their original form, the elasticity is then imperfect: but if they restore themselves entirely to their primitive shape, and employ just as much time in the restoration as was occupied in the compression, then is the elasticity perfect. It has been customary to treat only of the collision of bodies perfectly hard or perfectly elastic: but as there do not exist in nature any bodies (which we know) of either the one or the other of these kinds, the usual theories are but of little service in practical mechanics, except as they may suggest an extension to the actual circumstances of nature and art. 2. The general principle for determining the motions of bodies from percussion, and which belongs equally to both elastic and non-elastic bodies, is this: viz. that there exists in the bodies the same momentum, or quantity of motion, estimated in any one and the same direction, both before the stroke and after it. And this principle is the immediate result of the law of nature or motion; that reaction is equal to action, and in a contrary direction; from whence it happens, that whatever motion is communicated to one body by the action of another, exactly the same motion does this latter lose in the same direction, or exactly the same does the former communicate to the latter in the contrary direction. From this general principle too it results, that no alteration takes place in the common centre of gravity of bodies by their actions upon one another; but that the said common centre of gravity perseveres in the same state, whether of rest or of uniform motion, both before and after the impact. 3. If the impact of two perfectly hard bodies be direct, they will, after impact, either remain at rest, or move on uniformly together with different velocities, according to the circumstances under which they met. B Let в and b represent two perfectly hard bodies, and let the velocity of в be represented by v, and that of b by v, which may be taken either positive or negative, according as b moves in the same direction as B, or contrary to that direction, and it will be zero when b is at rest. This notation being understood, all the circumstances of the motions of the two bodies, after collision, will be expressed by the formula: which being accommodated to the three circumstances under which v may enter become These formulæ arise from the supposition of the bodies being perfectly hard, and consequently that the two after impact move on uniformly together as one mass, In cases |