Four particles \(A\), \(B\), \(C\), \(D\), each of mass 1, are connected by light rods \(AB\), \(BC\), \(CD\), \(DA\) to form a square. \(A\) is attached to a fixed point, and the system hangs from it, with a thread \(AC\) maintaining the square in shape. The thread is then cut. Find the acceleration with which \(C\) begins to descend.
A train of mass \(m\) is driven by electric motors which exert a force. The force depends linearly on the velocity, decreasing to zero at a speed \(V\), and its resistance to its motion is \(m(2v/3V)^2\) at any speed \(v\). Find how far it has travelled from rest when it attains a given speed \(V\).
A smooth wedge of mass \(M\) stands on a smooth horizontal table. A particle of mass \(m\) is placed on the wedge, at a height \(h\) above the table, and slides down. A particle reaches the greatest slope, which is inclined at an angle \(\beta\) to the horizontal and passes through the vertical plane as the mass centre of the wedge. Find how far the wedge has moved when the particle reaches the bottom of the slope. Find also the time taken.
Two small rings \(P\) and \(Q\) can slide on a fixed horizontal wire \(OPQ\). The ring \(P\), of mass \(m\), is connected with \(P\) by a light spring of natural length \(l\), which exerts a force \(m\omega^2(l-i)\) when its length is \(i\). \(P\) is now attached by a rod of length \(a\) to a fixed vertical wire through \(O\). \(R\) is made to oscillate about \(O\) so that its displacement is \(s = h\sin\omega t\). Find a formula describing the possible motions of \(Q\), assuming that \((h/a)^2\) can be neglected. Comment, without calculations, on any exceptional case.
A smooth tube of length \(2a\) is constrained to rotate in a horizontal plane about its centre \(O\) with constant angular velocity \(\omega\). A particle in the tube is projected with velocity \(u\) from \(O\). Find its speed immediately after it leaves the tube. Find also the force acting on it while still in the tube, at a time \(t\) after it has left \(O\). Verify that the kinetic energy gained by the particle is equal to the work done in keeping the angular velocity of the tube constant.
A fixed hollow sphere of radius \(a\) has a small hole bored through its highest point, resting on the inside of the sphere; there is no friction. Find, for any value of \(b/a\), how many positions of equilibrium there are with the rod in a given vertical plane and which of them are stable.
Find the moment of inertia of a uniform cube of side \(2a\) about one edge. The cube is released from rest on a smooth plane at an angle \(\theta\) to the horizontal. After sliding down a distance \(b\), it meets a small inelastic ridge \(A\). Find the angular velocity with which it begins to turn about \(A\). Show that it will not get over the obstacle unless $$3b\sin\theta \geq 16\sqrt{2} \cdot a\sin^2\left(\frac{1}{4}\pi - \frac{1}{2}\theta\right).$$
A long thin uniform plank of weight \(W\) lies symmetrically along the corner at the bottom of a smooth wall. Its breadth makes an angle \(\alpha\) with the ground. A horizontal force \(F\) acting along the line of contact of the plank with the ground is applied to one end of the plank. If \(\mu\) is the coefficient of friction between the plank and the ground, find the least value of \(F\) that will cause the plank to move.
An ancient catapult consists of a uniform lever arm \(ABC\) of mass \(3M\) through \(\frac{1}{4}\pi\) and pivoted at \(B\). \(AB\) is above \(BC\) which is horizontal and of length \(l\). The projectile of mass \(m\) fills the cupped end at \(A\) which is a heavy weight of mass \(M\) is banked to a height and dropped onto \(C\). Assume that the weight remains in contact with \(C\) and that the projectile leaves as soon as it feels the impulse. If \(AB\) is of length \(\frac{l}{4}\), show that the value $$x = \left(\frac{M + M'}{m + M'}\right)^{\frac{1}{4}}$$ will give the greatest range. What is that range?
A flexible trans-Atlantic cable of density \(\rho\) and radius \(r\) hangs over a cliff in the ocean floor which forms part of the edge of the continental shelf. The density of sea water is \(\rho'\), the height of the cliff \(h\) and the length of cable hanging free between the cliff-top and where the cable again touches the horizontal surface of the mud is \(s\). Find the greatest tension in the cable.