(4) A particle moves in a plane under the action of any forces, whose resolved parts are P in the radius vector, and T perpendicular to the radius vector. Shew that the chord of 2v2 where is the exterior angle between the radius vector and tangent. curvature through the pole is P+Tcot (5) A catenary is freely described under the action of a force parallel to the axis; shew that the centrifugal force is constant. (6) A particle, projected from the origin along the axis of y, describes the curve y2 = 4ax under the action of a force uy parallel to y, and another parallel to x; shew that x2 = μ (1 + 2a2) (c* + 3°). (7) The curve y=(x) touches the axis of y at the origin, and is described freely by a particle under the action of forces Y parallel to y, and ƒ parallel to x; shew that (8) A particle describes the hyperbola x2=y2+a2, so that Find the forces. 1 as (9) The velocity of a particle in a central orbit varies Apply the principle of Least Action to find the orbit, and thence the law of force. (10) Shew that the amount of heat and light received by a planet in one revolution is inversely as the square root of the latus rectum of its orbit. (11) If P, P' be the central forces for an orbit and its hodograph, h'2 PP'=rr'. h2 (12) The hodograph is a circle about a point in its circumference, and if be the angle which the radius vector makes with the diameter, the angular velocity is given by shew that the path is a cycloid with its vertex upwards, and the velocity at any point, that due to a fall from the tangent at the vertex. (13) The hodograph for a particle moving in a vertical circle with the velocity due to the depth below the highest point, is (14) When the hodograph is a straight line described uniformly, the path is the trajectory of a projectile in vacuo. (15) When it is a straight line described with uniform angular velocity about a point, the path is the catenary of uniform strength, ky=sec kx. (16) The hodograph for a circle about a point in the circumference, is a parabola about the focus described with angular velocity proportional to the radius vector. (17) Determine the motion of a simple pendulum, oscillating in small arcs, when its point of suspension describes, uniformly, a horizontal circle. Explain the peculiarity of the solution when the time of rotation of the point of suspension is equal to that of a complete oscillation of the pendulum. (18) Apply the principle of Varying Action to the investigation of the motion of a simple pendulum, slightly disturbed in any manner from its position of equilibrium. (19) Find the form of the surfaces of equal action for particles projected horizontally from points of a vertical line, the velocity being due to the distance from a given horizontal plane. (20) Find a central orbit whose form and mode of description correspond with those of the hodograph of another central orbit. Shew that there is but one law of central force for which this is possible. § 265. (21) A particle is acted on by a repulsive force tending from a fixed point, and by another force parallel to a fixed line, and when the particle is at a distance r from the fixed point, the magnitudes of these forces are μ, a, c being constants; shew that if the particle be abandoned to the action of the forces at any point at which they are equal to each other, it will proceed to describe a parabola of which the fixed point is the focus. (22) A particle is acted on by a force the direction of which always meets an infinite straight line AB at right angles, and the intensity of which is inversely proportional to the cube of the distance of the particle from the line. The particle is projected with the velocity from infinity from a point P at a distance a from the nearest point 0 of the line in a direction perpendicular to OP, and inclined at the angle a to the plane AOP. Prove that the particle is always on the sphere of which O is the center; that it meets every meridian line through AB at the angle a; and that it reaches the line a2 AB in the time ,μ being the absolute force. √μ cos a CHAPTER X. IMPACT. 271. WE come next to the consideration of the effects of a class of forces which cannot be treated by the methods just employed. These are called Impulsive forces, and are such as arise in cases of collision; lasting for an indefinitely short time, and yet producing finite changes of momentum. Hence, in such questions, finite forces need not be considered. When two balls of glass or ivory impinge on one another, no doubt there goes on a very complicated operation during the brief interval of contact. First, the portions of the surfaces immediately in contact are disfigured and compressed until the molecular forces thus called into action are sufficient to resist farther distortion and compression. At this instant it is evident that the points in contact are moving with the same velocity. But, most substances being endowed with a certain degree of elasticity, the balls tend to recover their spherical form, and an additional pressure is generated; proportional, it is found by experiment, to that exerted during the compression. The coefficient of proportionality is a quantity determinable by experiment, and may be conveniently termed the Coefficient of Restitution. It is always less than unity. The method of treating questions involving forces of this nature will be best explained by taking as an example the case of direct impact of one spherical ball on another; first, when the balls are inelastic. Again, when their coefficient of restitution is given. And it is evident that in the case of direct impact of spheres we may consider them as mere particles, since everything is symmetrical about the line joining their centers. 272. Suppose that a sphere of mass M, moving with a velocity v, overtakes and impinges on another of mass M', moving in the same direction with velocity v'; and that at the instant when the mutual compression is completed, the spheres are moving with a common velocity V. If P be the common action between them at any time t during the compression, it must evidently be of the nature of a pressure exerted by each on the other; and we have, if r be the time during which compression takes place, M (v – V) = [* Pdt = R, suppose, M' (V — v') = ['" Pdt = R ; From these results we see that the whole momentum after impact is the same as before, and that the common velocity is that of the center of inertia before impact. Had the balls been moving in opposite directions, would have been negative, and (taking it positively) From this it appears that both balls will be reduced to rest if Mv = M'v'; that is, if their momenta were originally equal and opposite. This is the complete solution of the problem if the balls be inelastic, or have no tendency to recover their original form after compression. 273. If the balls be elastic, there will be generated, by their tendency to recover their original forms, an additional action proportional to R. Let e be the coefficient of restitution, v,, v,, the velocities of the balls when finally separated. Then, as before, |