263. Describe the experiments from which it appears that, in the direct impact of elastic bodies of the same kind, the force of restitution bears a constant ratio to the force of compression. Prove that the direction and velocity of the motion of the common centre of gravity are not altered by the impact of two bodies. 264. State and explain D'Alembert's principle; and apply it to determine the motion of two weights, when one draws up the other by a wheel and axle, neglecting the inertia of the machine. 265. Of an hyperbola whose major axis is horizontal, determine the diameters down which a heavy body will descend in a given time; and that down which it will descend in the shortest time. 266. A hemisphere rests on a horizontal plane with a string fastened to its edge, which, passing over a pulley, supports a weight; determine the position of equilibrium, and, if the string be cut, the motion of the hemisphere. 267. Having given that the principal axes of a body are determined by a cubic of the form 2a'b'c'; prove that it has for a limiting equation, the quadratic to which it is reduced by making any two of the quantities d', b', c', vanish; and thence, that all its roots are real. (s—a)(s—b)(s—c) — a'2(s—-a) — b′2 (s—b) — c'2 (s—c) = 268. From a fixed point S' the straight line Sp is drawn continually representing the velocity of a body moving freely in one plane, and continually parallel to the tangent at the corresponding point of the body's path; prove that the force which acts upon the body will be continually represented by the velocity of the point p along the curve which is its locus. If the motion of the body be that of a projectile in vacuo, prove that the locus of p is a straight line. 269. If two of the principal axes, drawn through any point of a body, lie in a plane which passes through the centre of gravity, prove that every point of the body, situated in that plane, will also have two of its principal axes lying in the same place. If one of these two principal axes be always parallel to a given straight line, determine the locus of the corresponding points of the body. 270. A perfectly elastic solid of revolution, turning about its axis at a given rate, impinges on a hard smooth plane. If before impact the centre of gravity moves perpendicular to the plane with a velocity v, determine the motion of rotation after impact, and prove that the centre of gravity will move in the where p is the per p2 k2 same direction with a velocity v · - pendicular from the centre of gravity on the normal at the point of impact, and k is the radius of gyration round an axis through the centre of gravity perpendicular to the axis of the solid. 271. Determine the apparent path of a projectile to a person, who advances uniformly in a straight line towards the point of projection. 272. For what axes of suspension is the time of a small oscillation of a solid body an absolute minimum? Take the case of an ellipsoid. 273. A heavy body, symmetrical with respect to a vertical plane passing through its centre of gravity, revolves about a horizontal axis perpendicular to that plane; determine the pressure on the axis. Apply the result to the case of a cylinder whose axis is bisected by the axis of rotation. 274. Find the differential equation of the first order to the path of a projectile in a medium where the resistance ∞ (vel)2; determine the velocity at any point, and shew that it is least after the body has passed the highest point. 275. Investigate equations for finding the angular velocities of a rigid body acted upon by any forces, about three principal axes intersecting in its centre of gravity which is supposed to be fixed. When the body is acted upon by no forces, what is meant by the invariable plane? 276. Employ the equations of motion to find the path of a particle, upon a smooth horizontal plane, fastened by a thread to a point whose motion is uniform and rectilinear in that plane. 1835 277. Find the range of a projectile on a horizontal plane, not passing through the point of projection, and the direction of projection when the range is the greatest.. 278. Define mass and weight; and state the measures of moving and accelerating force. Describe some simple experiments by which it appears, that when pressure communicates motion directly, the momentum communicated in a given time is proportional to the pressure. 279. A body is projected vertically downwards with a given velocity, find the time of describing a given space; and explain the two roots of the quadratic equation. 280. A body is projected from the top of a tower with a given velocity and in a given direction; find the range on the horizontal plane passing through the foot of the tower, and the time of flight. 281. In the direct impact of two perfectly elastic bodies, the sum of each body multiplied by the square of its velocity is the same before and after impact. Shew that the same is true when the bodies impinge obliquely. 282. A heavy body oscillates in a circular arc, find the tension of the string in any position of the body. 283. If a rigid body move in any manner whatever, its vis viva at any instant is equal to the vis viva of the whole mass collected at its centre of gravity, together with the vis viva round its centre of gravity. 284 A billiard ball A in motion is struck by an equal one B moving with the same velocity, and in a direction making an angle of a with that in which A is moving, in such a manner that the line joining their centers at the time of impact is in the direction of B's motion; find the velocities of the bodies after impact, and shew that that of A will be a maximum, when α e e being the elasticity of the bodies. + e tan = e2 285. Three equal rods AW, BW, CW without weight support a weight W; their lower ends rest on a smooth horizontal plane and are connected by strings in such a manner that the rods and strings form the edges of a regular tetrahedron; if the string BC be cut, and the point A be prevented from sliding, compare the velocities of the ends of the rod BW, and shew that they are equal, when the string AB has revolved through 15°. 286. A cone of given form, and supported at G its centre of gravity, has a motion communicated to it, round an axis through G, perpendicular to the line joining G with a point in the circumference of the base, and in a plane passing through this point and the axis of the cone. Determine the position of the invariable plane; and explain clearly the motion of the cone's vertex. 287. A circular arc without weight, whose length is subtending an angle a at the centre, is placed upon a plane and acted upon at every point by a constant repulsive force ƒ tending f from one extremity of the arc; shew that it will rest with its chord e parallel to the plane if friction ÷ pressure > cota. π cosec If the position of equilibrium be slightly disturbed and friction sufficient to prevent all sliding, the time of a small oscillation 288. A sphere will roll from rest between two given curves in the shortest time possible down the involute of a cycloid. The curve will have its axis vertical, and cusp at the point from which the sphere begins to fall. It will cut the lower curve at right angles, and the two curves in points at which the tangents are parallel. 289. An uniform slender rod acted upon by gravity g is placed between two planes (one horizontal and the other vertical) having at a point in their common intersection an attractive force which at the centre of gravity of the rod 1 (dist.)2 always = §. 3π equilibrium at an inclination of to the horizon. 8 If the rod be originally placed in any given position, determine the motion. Shew that it will rest in a position of unstable the 290. A shot is fired at random with a given velocity towards a tower, whose horizontal distance from the cannon greatest range, and whose altitude subtends an angle = tan-1 at the point of projection; supposing the cannon is capable of being elevated from a horizontal position through an angle of 80°, shew that the odds in favour of the balls striking the tower are 3: 1. 291. Explain the application and use of the pendulum in the common clock. Describe the dead beat escapement, and shew in what its superiority over the recoil escapement con sists. 292. Form a given mass into a cone, such that the moments of inertia round all its principal axes may be equal. 293. Determine the moment of inertia of a cylinder round its axis, and the time of its rolling down a given inclined plane. 294. A particle oscillates in a small circular arc acted on by gravity and by a small disturbing force; determine the effect of the disturbing force on the time and arc of vibration. 295. Explain the mechanical effect of a locomotive engine. 296. When a body is acted on by a central force, the velocity at any point of the brachystochronous path between two given curves varies as the perpendicular on the tangent from the centre of force. Shew also that the path cuts the curves at right angles. 297. The time of an oscillation being π 1 find the alg teration in the time corresponding to given small and contemporaneous alterations in and g. 298. A body acted on by gravity is projected vertically upwards with a given velocity in a fluid in which the resistance varies as the square of the velocity; determine the accelerating force and the whole time of the ascent. 299. A semi-circular area acted on by gravity revolves round a horizontal axis which coincides with its diameter; find the pressure on the axis in any position. |