A projectile of mass \(m\) lb., moving horizontally with velocity \(v\) feet per second, strikes an inelastic nail of mass \(m'\) lb. projecting horizontally from a mass of \(M\) lb. which is free to slide on a smooth horizontal plane. Prove that the nail is driven \[ \frac{m^2M}{(M+m+m')(m+m')}\frac{6v^2}{gP} \text{ inches} \] into the block, where \(P\) lb. weight is the mean resistance of the block to penetration by the nail.
State the principle of the conservation of linear momentum. A smooth inclined plane of angle \(\alpha\) and of mass \(M\) is free to slide on a smooth horizontal plane. A particle of mass \(m\) is placed on its inclined face and slides down under gravity. Find its acceleration in space and the pressure between it and the plane.
Two equal light rods \(AB, BC\) are smoothly jointed at \(B\) and \(A\) is smoothly jointed to a fixed point. Masses \(m,m'\) are attached to \(B\) and \(C\) respectively. \(C\) is released from rest when \(AC\) is horizontal and when the angle \(ABC\) is \(2\pi/3\). Find the acceleration of \(C\) and the tension in \(AB\) immediately after the system is released.
Calculate the loss of kinetic energy when a ball of mass \(m\) moving with velocity \(u\) strikes directly a ball of mass \(m'\) moving with velocity \(u'\). Two equal balls are lying in contact on a smooth table, and a third equal ball, moving along their common tangent strikes them simultaneously. Prove that \(\frac{2}{3}(1-e^2)\) of its kinetic energy is lost by the impact, \(e\) being the coefficient of restitution for each pair of balls.
A uniform rigid rod \(AB\) weighing 12 lb. is hung from a rigid horizontal beam by three equal vertical wires, one at each end and one at the middle point. A weight of 18 lb. is attached to the rod at \(C\), where \(AC=\frac{1}{4}AB\). If the wires obey Hooke's law, find the pull in each wire.
Seven equal uniform rods \(AB, BC, CD, DE, EF, FG, GA\), are freely jointed at their extremities and rest in a vertical plane supported by small light rings at \(A\) and \(C\), which can slide on a smooth fixed horizontal rod. If \(\theta, \phi, \psi\) are the angles that \(BA, AG, GF\), make with the vertical, prove that \[ \tan\theta = 4\tan\phi = 2\tan\psi. \]
A thin uniform straight rod \(PQ\) of weight \(W\) rests partly within and partly without a uniform cylindrical jar of weight \(4W\), which stands on a horizontal table. The rod rests in contact with the smooth rim of the jar, with its end \(P\) pressing against the rough curved surface of the jar. If the rod is about to slip and the jar is about to upset simultaneously, prove that the rod makes with the vertical an angle \[ \frac{1}{2}\lambda + \frac{1}{4}\cos^{-1}\left(\frac{1}{3}\cos\lambda\right), \] where \(\lambda\) is the angle of friction.
A ball whose coefficient of restitution is \(e\) is projected with velocity \(v\) at an inclination \(\alpha\) to the horizontal from a point \(A\) on a horizontal plane. \(A\) is at a distance \(d\) from a vertical wall. The ball strikes the wall, and then after rebounding once on the horizontal plane returns to \(A\). Prove that \[ v^2e\sin2\alpha=gd. \]
Two particles, each of mass \(m\), are attached to the ends of a long fine inextensible string, which hangs over two small smooth pegs which are at the same level and \(2a\) apart. A particle of mass \(2m\) is attached to the string midway between the pegs and is then let go. Prove that during the subsequent motion, if \(\phi\) is the angle between the two non-parallel parts of the string, the velocity of the mass \(2m\) is \[ 2\sqrt{ag\frac{1-\tan\frac{\phi}{4}}{3+\cos\phi}}. \]
A uniform rectangular plate \(ABCD\) is hinged at the fixed point \(A\) and is supported in such a position that \(AB\), one of the longer sides, is horizontal, and \(AD\) is vertical. When the plate is released it swings in its own plane about the fixed hinge \(A\) and comes to rest with \(AB\) vertical. The stiffness of the hinge produces a constant retarding couple during motion. Prove that the plate stays in the new position if \[ \frac{AB}{AD} > 1+\frac{\pi}{2}. \]