A pedestal is constructed of three uniform right circular cylinders placed with their axes vertical and in the same line. The weights of the cylinders are in the ratios 2:1:3, and their radii are in the ratios 12:11:9, where the cylinders are taken in order from the topmost downwards. If no mortar is used in the pedestal, find the greatest weight of a statue which may be placed safely anywhere on the top of the pedestal, in terms of the weight of the middle cylinder. If the three cylinders are cemented together, while the base is still not fixed to the ground, show that the greatest weight of a statue which may now be placed safely on the top of the pedestal is \(1 + k\) times its value when the cylinders were not cemented.
A light rigid wire is bent into the shape of a rectangle \(ABCD\), with \(AB = a\), \(BC = b\). Particles of weights \(w\), \(5w\), \(w\), \(2w\) are attached to the vertices, \(A\), \(B\), \(C\), \(D\) respectively, and the wire is then suspended freely from \(A\). What is the inclination of \(AB\) to the vertical when the system composed of wire and particles is in equilibrium? A rough horizontal plane is held so as to touch the wire at \(C\), and another particle of weight \(w\) is attached to \(C\). If \(\mu\) is the coefficient of friction between the wire at \(C\) and the plane, what further force must be applied vertically upwards to the plane so that sliding will just commence at \(C\)?
A hemispherical shell, with a rough inner surface, is held fixed with its rim horizontal. A uniform narrow ladder of weight \(W\) and length \(2l\) is placed with its ends \(A\), \(B\) touching the inner surface so as to lie in a vertical plane through the centre \(O\) of the hemisphere and be inclined at an angle \(\alpha\) to the horizontal. The coefficient of friction between the ladder and the inner surface of the shell is \(\tan \lambda\) at both \(A\) and \(B\). The angle \(BAO\) is \(\beta\). A man of weight \(W'\) stands on the lower end \(A\) of the ladder. What are the conditions to be imposed upon the ratio \(W'/W\), in terms of the angles \(\alpha\), \(\beta\) and \(\lambda\), so that the ladder will not slip? If these conditions are satisfied, how far may the man walk along the ladder before it will slip?
A rope of length \(L\) and weight \(w\) per unit length hangs in a vertical plane over two small rough pegs, which are parallel in the same horizontal plane and a distance \(2a\) apart. Particles of weight \(W\) are attached at the ends of the rope so that equal amounts of rope hang vertically from the pegs. If \(wLe^{\mu \pi/2} W\) is very small and all powers of this quantity above the first are neglected, show that the inclination of the rope to the horizontal between the pegs takes the value \(awe^{\mu \pi/2}/W\) at the pegs, where \(\mu\) is the coefficient of friction between the rope and the pegs, this friction assumed limiting.
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.