- elks friction at the axis of suspension is taken into account, and s' the corresponding arc when the friction is neglected, prove that 2kl e2ks f g f being the constant effect of friction, and g gravity. 179. If two bodies A and B of elasticity e impinge directly on each other with velocities a and b respectively, and if u and 1 be their respective velocities after impact, and р and a the velocities lost and gained respectively, 1 e Ite 180. Shew that the centres of oscillation and suspension are reciprocal, and explain the use of this property in finding practically the length of a pendulum. 181. A body attracting with a force varying directly as the distance moves uniformly fin a straight line; determine the motion of another body situated in the same plane and subject to its influence. 182. A body descends down the arc of a vertical catenary having its vertex at the lowest point; find the curve of ascent when the oscillations are isochronous, the two curves being so united at the lowest point as to have a common tangent. 183. A corpuscle is attracted by two straight lines at right angles to each other, the particles of which attract with forces 1 varying as : having given the position of the corpuscle and the length of one of the lines, find the length of the other when the direction in which the corpuscle begins to move is towards their common intersection. 184. A body descends in a straight line in a medium whereof the density varies as the square root of the distance from a given point, and is urged by a constant force tending to that point; find the velocity and time corresponding to a given space, supposing the resistance to vary as the density and velocity jointly. 185. Two balls connected together by an inflexible and inextensible line are constrained to move, the one on a horizontal D2 : D3 m plane, the other on an inclined plane which is at liberty to move freely on the horizontal plane ; find the motions of the balls and of the plane, supposing the motion of the rod to be in a vertical plane. 186. A right-angled triangle vibrates in its own plane about an axis passing through its vertex, find the length of the isochronous simple pendulum ; and if one of the sides be slightly diminished and the other as much increased, determine the variation of the pendulum. 187. A body falls towards a centre of force which varies as 1 1 in a medium of which the density varies as and the re D3 sistance varies as (velocity)2. Prove that at any distance r from the centre, (velocity)? 1 a2 ħ where m = force at distance 1, h = density at distance 1, and a = distance from centre at the beginning of the motion, 188. A uniform rod vibrates in a medium the resistance of which varies as the velocity; find the time of one of its small oscillations. 189. A body moving on a curve is acted on by forces X and Y parallel to the axes of the curve; find the reaction, and ply it to find the tension of a string, at the lowest point, when a body oscillates in a circle through an arc of 120°. 190. A ladder rests with one end on a smooth horizontal plane, and the other against a smooth vertical wall; find the horizontal force at its foot which will keep it at rest; and when the force is removed determine its motion. 191. A pile of weight w is driven by a hammer H impinging with a velocity v, the friction being represented by F; find the motion; and when the velocity given to the pile is small, approximate to the whole space through which it is driven, 192. A body is acted on by gravity ; find the tautochronous curve in a medium in which the resistance varies partly as the velocity and partly as the square of the velocity; and from the result prove that it is a cycloid when the resistance vanishes, or varies as the velocity. ap 1831 193. Find the moment of inertia of an ellipse revolving in its own plane about any axis. 194. If a system move in any manner whatever, prove that f Emvds is a minimum. 195. Find the time of an oscillation in a hypocycloid, the body being acted upon by a force varying as the distance from the centre of the globe. 196. Find the moment of inertia of a system about any axis passing through the origin of the coordinates, and the moment of inertia about any axis in terms of the moments about the principal axes. 197. If a body acted upon by gravity be projected in a medium the resistance of which varies as the square of the velocity, find the equation to the curve described; and when the resistance vanishes, shew that it is the equation to a parabola. 198. Define inertia, moss, weight, moving force, accelerating force. 199. A body acted on by gravity descends in a straight line; find the space described in a given time, and prove that it is equal to half the space which would be described in the same time with the last acquired velocity continued uniform. 200. A body descends on an inclined plane, find the accelerating force; and prove that at any point in the descent, the velocity is equal to that which would be acquired down the perpendicular height. 201. If the material particles m, m', of any system in motion, pass from the positions a, a', . to b, b,... during the very small time d', and c, d, be the positions they would have had, if during ot only the impressed forces had acted ; then forces proportional to and in the directions of bc, 6'c', will produce equilibrium with the pressures on the fixed points and axes of the system. 202. Also, if e, é ... be the places m, m' . . . would have had, if the impressed motions had been compounded with uniform motions during ot along the actual paths with the . velocities at a, a ; then forces proportional to and in the directions of be, bé, will keep the system at rest; and if Yo y, ... be any other positions compatible with the conditions of the system 2. meb? 2 3. mey?. 203. An elastic chord AabcB is stretched between two fixed points, A, B; the portion abc is made to assume the form of two straight lines ab, bc, the points a and c being in the straight line joining A, B, and b at a sinall distance from it: when the chord is suddenly left to itself, what motion will take place, and what will happen when the motion reaches the fixed points ? 204. Having given the direction and velocity of projection of a body acted upon by gravity, find the equation of the trajectory. 205. Find the time of oscillation in a cycloidal arc. 206 A system of material points moveable about a horizontal axis, has all its parts acted on by gravity; it is required to determine the accelerative force, and to find a point of the system which shall be accelerated exactly as much as a single point in the same position. 207. Shew that if a material particle, moving in any manner in space, be solicited by the forces X, Y, Z, in the directions of three rectangular axes, and x, y, z be the coordinates of its plac : at the time t, dax daz Z dt2 dt2' dt2 dạy Y = X = 208. If two equal bodies which attract each other with forces 1 varying as are constrained to move in two straight lines (dist.)2 at right angles to one another, shew that they will arrive together at the point of intersection of the lines, from whatever points their motions commence. And having given their distance at the beginning of the motion, find the time to the point of intersection. 209. A given hemisphere rests with its base upon a horizontal plane, and a given uniform rod, one end of which is moveable about a horizontal axis fixed in the plane, is palced against the hemisphere so as to be a tangent to a great circle of it; and the rod by its pressure puts the hemisphere in motion ; find the equation for determining the motion of the rod when the friction of the plane varies as the pressure of the hemisphere upon it. And when there is no friction, find the angular velocity of the rod when it comes to the plane. 1 210. If a body be attracted by a force which varies as (dist.)n) find the value of n when the velocity acquired from an infinite distance to a distance r from the centre is equal to the velocity that would be acquired from r to to 211. A body being acted upon by any forces, prove that the motion of the centre of gravity will be the same as if all those forces acted at that centre, and that the motion of rotation will be affected as if the centre of gravity were fixed, and the same forces applied. 212. The moment of inertia of any system, with respect to any given axis, is equal to the moment about an axis parallel to this passing through the centre of gravity, together with the moment of the whole body collected in its centre of gravity about the given axis. 213. If any number of bodies be acted upon by their mutual attractions, their centre of gravity will either be at rest, or move uniformly in a straight line, 214. A body which is symmetrical with respect to a vertical plane passing through the centre of gravity revolves about a horizontal axis, the body being acted on by gravity only; find the pressure on the axis. 215. State and prove the principle of the conservation of vis viva; and when the vis viva is a maximum, shew that the body passes through a position of stable equilibrium, and when a minimum through a position of unstable equilibrium. 216. A cylindrical body unrolls itself from a vertical string, the other end of which passes over a fixed pulley and supports a weight; it is required to determine the motion. |