(19) If the time of a body's descent in a straight line towards a given centre of attraction vary inversely as the square of the distance fallen through, determine the law of the attraction. (20) Assuming the velocity of a body falling to a centre of attraction to be as where a is the initial and Va the variable distance from the centre, find the law of the attraction. (21) Find the time of falling to the centre when the attraction (dist.). (22) Shew that the time of descent, to a centre of attraction oc (dist.), through the first half of the initial distance, is to that through the last half as π + 2 : π — 2. (23) A particle descends to a centre of attraction, intensity (dist.). Find n so that the velocity acquired from infinity to distance a, shall be equal to that acquired from distance a to distance a, from the centre. (24) A particle is placed at the extremity of the axis of a thin attracting cylinder of infinite length and of radius a, shew that its velocity after describing a space is proportional to (25) A particle falls to an infinite homogeneous solid bounded by parallel plane faces, find the time of descent. (26) Every point of a fine uniform ring repels with an intensity (dist.)-2, find the time of a small oscillation in its plane, about the centre. (27) Shew that a body cannot move so that the velocity shall vary as the distance from the beginning of the motion. And if the velocity vary as the cube root of that distance, determine the acceleration, and the time of describing a given distance. (28) Shew that the time of quickest descent down a focal chord of a parabola whose axis is vertical is 131 where is the latus rectum. (29) An ellipse is suspended with its major axis vertical, find the diameter down which a particle will fall in the least time, and the limiting value of the excentricity that this may not be the axis major itself. (30) Particles slide down chords from a point 0 to a curved surface, under the attraction of a plane whose attraction is as the distance, and they reach the surface in the same time; shew that the surface is generated by the revolution (about a line whose length is a through O perpendicular to the plane) of the curve whose polar equation about O is (31) If the particles commence their motion at the surface, and reach O after a given time, the equation of the generating curve is p cos = a {sec (k cos 0) — 1}. (32) Prove that the times of falling through a given distance AC towards a centre S, under the action of two attractions, one of which varies as the distance, and the other is constant and equal to the original value of the first, are as the arc (whose versed sine is AC) to the chord, in a circle whose radius is AS. (33) The earth being supposed a thin uniform spherical shell, in the surface of which a circular aperture of given radius is made, if a particle be dropped from the centre of the aperture, determine its velocity at any point of the descent. (34) If a particle fall down a radius of a circle under the action of an attraction x (D) in the centre, and ascend the opposite radius under the action of a repulsion of equal intensity at equal distances from the centre, shew that it will acquire a velocity which is a geometric mean between the radius and the intensity at the circumference. (35) If a particle fall to a centre of attraction of intensity x (D); determine the constant attraction which would produce the effect in the same time, and compare the final velocities. (36) Find the equation of the curve down each of whose tangents a particle will slide to the horizontal axis in a given time. (37) A sphere is composed of an infinite number of free particles, equally distributed, which gravitate to each other without interfering; supposing the particles to have no initial velocity, prove that the mean density about a given particle will vary inversely as the cube of its distance from the centre. (38) Prove that if PQ be a chord of quickest descent from one curve in a vertical plane to another, the tangents at P and Qare parallel and PQ bisects the angles between the normals and the vertical. (39) A rough horizontal plane has the coefficients of friction at any point proportional to the distance from a fixed point S to which an attraction tends whose intensity is (dist.), prove that if a particle be placed at a distance a tan a from S it will arrive at S in time a being the distance at which the particle must be placed so as to be on the point of moving. (40) If a particle P move from rest under the action of an attraction tending to a point S measured by the acceleration n'SP, determine the time from rest to rest; and shew that, if a small constant retardation fact through a portion of the path extending equally on each side of S the time will be unaltered, and the diminution of the amplitude of one oscillation will be 2 cos nt, 7 being the time when the disturbance 2f begins. n2 (41) A fine thread having two masses each equal to P suspended at its extremities is hung over two smooth pegs in the same horizontal line; a mass is then attached to the middle point of the portion of the string between the pegs, and allowed to descend under gravity; shew that the velocity of Q at any depth x below the horizontal line is (42) An elastic string has its ends fastened to the ends of a rod of equal length. The middle point of the string is fastened, and at that point is placed a centre of repulsion, which repels every particle of the rod with an intensity μ (dist.) The rod is then moved parallel to itself through a distance equal to half its length. If in this position the elasticity of the string is such that the rod is in equilibrium, shew that if slightly displaced perpendicular to its length, the time of a small oscillation ✓ (43) A particle moves in a straight line under an attraction to a centre in the straight line μx+2μ' and starts from rest at a distance a from the centre; shew that after a time t the distance from the centre will be CHAPTER IV. PARABOLIC MOTION. 106. In this chapter we intend to treat principally of the motion of a free particle which is subject to the action of forces whose resultant is parallel to a given fixed line. The simplest case of course will be when that resultant is constant. The problem then becomes the determination of the motion of a projectile in vacuo and unresisted, since the attraction of the earth may be considered within moderate limits as constant and parallel to a fixed line. This we will now consider. 107. A free particle moves under the action of a vertical attraction whose intensity is constant; to determine the form of the path, and the circumstances of its description. Taking the axis of a horizontal and in the vertical plane and sense of projection, and that of y vertically upwards, it is evident that the particle will continue to move in the plane of xy, as it is projected in it, and is subject to no force which would tend to withdraw it from that plane. The equations of motion then are if g be the kinetic measure of the attraction per unit of mass. Suppose that the point from which the particle is projected is taken as origin, that the velocity of projection is V, and that the direction of projection makes an angle a with the axis of x. The first and second integrals of the above equations will then be x = V cos a. t, y = V sin a. t - gt... ..(2). |