State one set of conditions for the equilibrium of forces acting in a plane upon a rigid body. The ends A and C of two uniform unequal rods AB, BC which are smoothly jointed at B are free to move on a fixed smooth horizontal wire. Shew that the smaller rod, in a position of equilibrium, is vertical; and that the reaction at the wire for the larger rod is independent of the weight of the smaller.
A thin uniform rod is bent at one end to form a walking-stick with a semicircular handle. The straight portion AB is of length \(2l\), and the curved portion BC is of radius \(a\) (less than \(l\)). The straight line joining the ends B and C of the semicircular portion is perpendicular to AB. The stick hangs from a horizontal table, supported at the end C. Prove that the straight part makes an angle \(\psi\) with the vertical, where \[ \tan\psi = a(\pi a + 4l)/2(l^2 - a^2). \]
A heavy elastic string whose weight per unit length when unstretched is \(w\), and whose modulus of elasticity is \(\lambda\), is hung freely from one end A, and supports a weight \(W\) at the other. Prove that the natural length \(l\) is increased to \[ l + \frac{wl^2}{2\lambda} + \frac{Wl}{\lambda}. \] If a rigid horizontal plane is placed underneath \(W\), partly supporting it, at a distance \(h\) from A, prove that the reaction at the plane is \[ W + \lambda + \frac{wl}{2} - \frac{\lambda h}{l}, \] where \[ l + \frac{wl^2}{2\lambda} \le h \le l + \frac{wl^2}{2\lambda} + \frac{Wl}{\lambda}. \]
A heavy uniform inelastic string of length \(l\) has one end attached to a fixed peg, and passes through a smooth ring distant \(2h\) from the peg and on the same horizontal level. Shew that the length \(y\) of string which hangs vertically from the ring is given by \[ \frac{l-y}{2y} = \tanh \frac{2h}{\sqrt{\{(3y-l)(y+l)\}}}. \]
A bead of mass \(m\) is free to move on a smooth rod which is constrained to rotate about one end in a vertical plane with constant angular velocity \(\omega\). Initially the bead is at rest at the fixed end of the rod, and the rod is pointing vertically downwards. Find the differential equation satisfied by the distance \(r\) moved by the bead along the rod. Deduce the value of \(r\) at time \(t\), and shew that the reaction between the bead and the rod at that instant is \(mg(\sinh\omega t + 2\sin\omega t)\).
A light string of natural length \(2l\) and modulus of elasticity \(mg\) is attached to two points at a distance \(2\sqrt{3}l\) apart on the same horizontal level. A small stone of mass \(m\) is held against its middle point as for a catapult, and then depressed a distance \(3l\) vertically downwards. Shew that the stone rises when released. Shew also that it is projected (the stone and the string being in contact till the horizontal position is reached) to a distance \(2(3-\sqrt{3})l\) above the two fixed points.
A shell is projected vertically upwards from the ground, its kinetic energy initially being \(E\). When its velocity has been reduced by one half, it explodes into two parts of masses \(M_1\) and \(M_2\), of which \(M_1\) is the upper. The kinetic energy just after the explosion exceeds that just before by \(E\), and the explosion acts along the line in which the shell is travelling. Shew that the upper portion reaches a height \[ \frac{M_1+M_2+\sqrt{M_1 M_2}}{M_1(M_1+M_2)} \frac{E}{g} \] above the ground.
A particle of mass \(m\) is at the centre of the base of a smooth rectangular box of mass \(M\) which rests on a smooth horizontal table. The particle is projected along the base so as to meet normally the face P of the box. The coefficient of restitution of the particle and either the face P or the opposite face is \(e\). The velocity of the particle and of the box respectively, measured in the direction and sense of the initial velocity, just before the \(n\)th impact with the face P are denoted by \(u_n\) and \(V_n\) respectively; where \(v_1\) is the initial velocity \(v\) of the particle, and \(V_1\) is zero. By considering the values of the functions \(\lambda_n = u_n - V_n\) and \(\mu_n = MV_n + mu_n\), evaluate \(V_n\) and \(u_n\), and shew that they tend to a common limit, which is the value which they would have had if the particle had stuck to the box on its first impact.
A lamina of uniform density \(\rho\) is free to turn about an axis in its own plane through the centre of mass. It is acted on by a constant couple whose moment about the axis is \(L\). The resistance of the atmosphere produces at every point of the lamina a resisting normal pressure which, per unit area, is \(k\) times the velocity. Shew that the equation of motion is \[ I\ddot{\theta} = L - (kI/\rho)\dot{\theta}, \] where \(I\) is the moment of inertia about the axis of rotation. If the lamina is initially at rest, find its angular velocity at time \(t\).
A uniform circular disc of radius \(a\) and mass \(m\) rolls without slipping in a vertical plane on a horizontal table. Its centre is attached by two elastic strings each of natural length \(l\) and modulus of elasticity \(\lambda\), to two fixed points, one on either side of it, each at a height \(a\) above the table and in the same vertical plane as the disc. Assuming that each string remains taut throughout the motion, shew that the disc, if slightly disturbed from its position of equilibrium, executes a simple harmonic motion of period \(\pi\sqrt(3ml/\lambda)\).