A uniform thin rod of length \(2a\) and weight \(W\) is freely hinged at one end to a fixed support. The other end is joined by a light elastic string of modulus \(\lambda\) and unstretched length \(l\) \((l < 4a)\) to a fixed hook located vertically above and at distance \(2a\) from the hinge. Find the range of values of \(\lambda\) for which there exists a position of equilibrium with the rod inclined to the vertical. Show also that if such a position of equilibrium exists then it is stable.
A uniform chain of weight \(w\) per unit length forms a closed loop and hangs at rest over a smooth cylindrical peg having a section of arbitrary convex shape. Prove that if \(T\) is the tension in the chain at height \(y\) above a fixed level, then \(T - wy\) is the same for all points of the chain whether or not they are in contact with the peg.
A particle of unit mass is projected with speed \(v\) at an inclination \(\theta\) above the horizontal in a medium whose resistance is \(k\) times the velocity. Find the time \(T\) that elapses before the motion of the particle is inclined below the horizontal at the same angle \(\theta\). Is the particle above or below the point of projection after this time \(T\) has lapsed?
Two masses \(M\), \(m\) are connected by a string that passes through a hole in a smooth horizontal table, the mass \(m\) hanging vertically. Show that, so long as the string remains taut, \(M\) describes a curve whose differential equation is \begin{align} \left(1 + \frac{m}{M}\right)\frac{d^2u}{d\theta^2} + u = \frac{mg}{Mh^2u^2}, \end{align} where \(h\) is a constant and \(u = r^{-1}\). Find an expression for the tension in the string in terms of \(M\), \(m\), \(g\), \(h\) and \(u\).
A particle falls from a position of limiting equilibrium near the top of a nearly smooth glass sphere. Show that it will leave the sphere at a point whose radius is inclined to the vertical at an angle \begin{align} \alpha + \mu\left(2 - \frac{4}{3\sin\alpha}\right), \end{align} approximately, where \(\cos\alpha = \frac{2}{3}\) and \(\mu\) is the coefficient of friction.
\(AB\) and \(CD\) are two equal uniform rods connected by a string \(BC\). The system is on a smooth table so that \(AB\), \(BC\) and \(CD\) form three sides of a square. The rod \(AB\) is struck a blow in the direction \(AD\), show that the initial velocity of \(A\) is \(\frac{3}{4}\) times that of \(D\).
A horizontal plane lamina is free to rotate in its own plane about an axis intersects the lamina in the point \(O\). The moment of inertia of the lamina about this axis is \(I\). An insect of mass \(m\) is at the point \(A\), where \(OA = a\). The whole system is initially at rest when the insect begins to walk along a straight line in the lamina perpendicular to \(OA\). Show that the maximum angle through which the lamina can turn is \begin{align} \frac{a^2}{a^2 + k^2} \cdot \frac{\pi}{2}. \end{align}
The period of small oscillations of a compound pendulum is \(T\). It is hanging from a pivot and suddenly set in motion with angular velocity \(\omega_0\). Show that it makes complete revolutions in a vertical plane if \(\omega_0 T > 4\pi\).
A particle of mass \(m\) is travelling uniformly in a straight line with energy \(E\) when it breaks up into two particles of mass \(\frac{1}{2}m\). One of the final particles is observed to be moving at right angles to the original direction. If the release of energy in the process is \(Q\), find the direction and speed of the other particle. Show that the phenomenon is possible only if \(Q > E\).
A machine gun of mass \(M\) stands on a horizontal plane and contains a shot of mass \(M'\). The shot is fired horizontally at the rate of mass \(m\) per unit time with velocity \(v\) relative to the gun. If the coefficient of friction between the gun and the plane is \(\mu\), and sliding begins at once, show that the velocity of the gun after all the shot is fired is \begin{align} u\log\left(1 + \frac{M'}{M}\right) - \frac{\mu gM'}{m}. \end{align}