A car is moving along a straight horizontal road at a speed \(v\). It is desired to fire a shell which hits the car from a gun placed a distance \(p\) from the road, the trajectory of the shell being along horizontal. The gun fires the shell with muzzle velocity \(v_0\) immediately. The resistance to the motion is \(kv^2\) per unit mass when the speed of the shell is \(v\). Determine the resistance to the point that, when \(kp\) and \(v/v_0\) are small quantities of the same order of magnitude, the value of \(\alpha\) is approximately \(\frac{1}{2}\pi - (1 + \frac{1}{2}kp) u/v_0\).
A smooth wire \(AB\) of length \(a\) is originally in a vertical line, \(B\) being above \(A\). A stop is attached to the wire very near the end \(B\) and a heavy bead is threaded on to the wire just above the stop (so that the bead cannot move nearer to \(A\), but is free to leave the wire after moving a negligibly small distance away from the stop). The wire is then suddenly constrained to rotate with uniform angular velocity in a vertical plane about the end \(A\), which remains fixed. Find where the bead leaves the wire, and at what distance from \(A\) it meets the horizontal plane through \(A\).
A particle of mass \(m\) is hanging freely at one end of an elastic string whose other end is held fixed. The particle has caused the string to extend a distance \(\frac{2l}{3}\) and is then released. Find the velocity with which it reaches the point of support of the string. If the particle rebounds elastically from a horizontal plane at the point of support, describe briefly the subsequent motion.
A straight light rigid rod \(ABC\) is bent at \(B\) so that \(AB\) and \(BC\) are at right angles, with \(AB = BC\). Particles of mass \(m\) are attached at \(A\) and \(C\), and the system moves without rotation on a smooth horizontal plane, with uniform angular velocity \(u\). An impulse \(P\) is applied to \(C\) in a direction parallel to \(BA\). Find the angular velocity of the system immediately after the blow. If the system is initially moving in a direction perpendicular to \(AC\), with \(B\) foremost, show that there is no change in kinetic energy if \(P = 3mu/\sqrt{2}\).
A thin uniform plate in the shape of a square \(ABCD\) is of mass \(M\) and side \(2a\), and can rotate freely and smoothly about the side \(AB\), which is horizontal. The plate is held along \(CD\) so that its plane is horizontal, and it is sufficiently rough to prevent the slipping of a particle of mass \(m\) on its upper surface, lying on the perpendicular bisector of \(AB\) at a distance \(d\) from it. Find the condition that the particle initially remains on the plate, and if it is satisfied find the initial angular acceleration of the plate. If the particle is now fixed firmly to the plate and the system performs small oscillations about its equilibrium position, find the length of the equivalent simple pendulum.
A smooth uniform wedge of angle \(\alpha\) and mass \(M\) rests on a fixed horizontal table. A particle of mass \(m\) is placed at the mid-point of the top of the wedge and both slides down the sloping face till it meets \(S_1\) when it coalesces with the wedge. Find the motion of the system.
A particle of weight \(2W\) is attached to the end \(A\), and a particle of weight \(W\) attached to the end \(B\), of a light rod \(AB\) of length \(2a\). The rod hangs from a point \(O\) by light strings \(AO\), \(BO\), each of length \(b\). Prove that in equilibrium the inclination of the rod to the horizontal is \(\theta\), where \[ \tan \theta = \frac{a}{3\sqrt{(b^2-a^2)}}. \] Find the tension in the string \(AO\) in terms of \(a\), \(b\), and \(W\).
A uniform straight beam \(ABCDE\) of weight \(W\) rests on supports at the same level at \(B\) and \(D\), and weights \(W\) are hung from the end-points \(A\) and \(E\), where \(AB\), \(BC\), \(CD\), \(DE\) are each of length \(a\). Sketch a graph of the bending moment at each point of the beam, and find in particular the bending moment at \(B\) and at \(C\).
A heavy uniform chain of length \(l\) is attached at one end to a point \(A\) at a height \(l\) above a rough horizontal table, where \(l < l\). The point \(A\) is moved parallel to the table with constant velocity \(v\). Show that the length \(s\) of chain which comes clear of the table satisfies \[ s^2 + 2\mu ls = h^2 + 2\mu lh, \] where \(\mu\) is the coefficient of friction.
A system of forces acts in one plane on a rigid body. Prove that, if \(O\) is a fixed point in the plane, a force through a point \(P\) of the plane is equivalent to an equal force through \(O\) and a couple. Hence prove that the system is equivalent either to a single force or to a single couple. Forces \(W\), \(2W\), \(3W\), \(4W\) act along the sides \(AB\), \(BC\), \(CD\), \(DA\) of a square \(ABCD\). Prove that the system is equivalent to a single force, and find its magnitude and direction and the point in which its line of action meets the line \(AB\).