A uniform plank is held at rest with one end on a smooth horizontal floor and with the other end against a smooth vertical wall. The plank makes an angle of \(60^{\circ}\) with the vertical wall. If the plank is released from rest, show that the top end of the plank loses contact with the wall after it has slipped down the vertical wall through a distance equal to \(\frac{1}{6}\) of the length of the plank.
A particle of unit mass moves in a plane under the influence of a force which is directed towards a fixed point \(O\) in the plane and whose magnitude is \(3a^2u^2/(4r^3)\), where \(r\) is the distance of the particle from \(O\) and where \(a\) and \(u\) are constants. If the particle is projected from a point \(A\) in the plane such that \(OA = a\), with speed \(u\) in a direction in the plane perpendicular to \(OA\), show that the particle eventually approaches infinity along a line parallel to \(AO\).
A uniform rectangular lamina of mass \(M\) moves on a smooth horizontal plane with velocity \(u\) in the direction of an axis \(AB\) of symmetry of the lamina. At time \(t = 0\) a uniform sphere, of mass \(m\) and radius \(a\), whose centre is at rest but which has angular velocity \(\Omega\) about a horizontal axis at right angles to \(AB\), is placed on the lamina at a point on \(AB\). The coefficient of friction between the sphere and the lamina is \(\mu\). The sense of \(\Omega\) is such that at \(t = 0\) the lowest point of the sphere is moving in the direction \(BA\). Show that slipping ceases after a time \(T\) given by \begin{align*} \mu gT(7M + 2m) = 2M(u + a\Omega), \end{align*} and obtain an expression for the angular velocity of the sphere at \(t = T\), assuming that the sphere has not reached the edge of the lamina before \(t = T\). State briefly the nature of the motion after \(t = T\) and before the sphere reaches the edge of the lamina.
Two particles of equal mass \(m\) are connected by a light inextensible rod and lie upon a smooth horizontal table. One of them is struck by a blow of impulse \(I\) in a direction that makes an angle \(\theta\) with the rod, creating an impulsive thrust in the rod. Show that the kinetic energy created by the blow is \(I^2(1 + \sin^2\theta)/4m\). Four particles, of equal mass, connected by light inextensible rods smoothly jointed to the particles, lie upon a smooth horizontal table in the configuration of a square \(ABCD\). An impulse is applied at \(A\) in the plane of the square. Show that the kinetic energy created is independent of the direction of the impulse.
A uniform sphere of radius \(a\) and mass \(m\) with centre \(B\) has a particle of mass \(m\) embedded in it at a point \(A\) just below its surface. It is placed upon a perfectly rough fixed sphere of radius \(a\) and centre \(O\) in such a way that \(O\), \(A\) and \(B\) are in the same vertical plane. Let \(\alpha\) be the angle between \(OB\) and the upward vertical, and let \(\beta\) be the angle between \(BA\) and the downward vertical. Use a geometrical argument to derive a condition for equilibrium in terms of \(\alpha\) and \(\beta\). Show that there is a position of equilibrium for any fixed \(\alpha\) in the range \(0 \leq \alpha \leq \pi/6\). For rolling displacements of the movable sphere upon the fixed sphere in which \(A\) remains in the same vertical plane as \(O\) and \(B\), show that \(d\beta/d\alpha = 2\) and evaluate \(dV/d\alpha\), where \(V\) is the potential energy of the system; hence show that the positions of equilibrium of the system are unstable.
A rocket is programmed to burn its propellant fuel and eject it at a variable rate but at a constant velocity \(u\) relative to the rocket. Its initial mass is \(M_0\) and its mass at time \(T\) after all its fuel has been burned is \(M_0(1-e)\), where \(e\) is a constant, \(e < 1\). The rocket is launched from rest in a vertical direction under the influence of a constant gravitational acceleration \(-g\). Show that the velocity \(w\) of the rocket at time \(T\) is given by \begin{align*} w = -gT - u\log_e(1-e) \end{align*} independently of all details of the fuel burning program other than the fact that the burning takes time \(T\). In the special case where the mass of the rocket at time \(t\) is \(M_0(1-pt)\) for \(0 \leq t \leq T\), \(p\) being constant, show that the rocket rises to a height \(H\) given by \begin{align*} H = -\frac{1}{2}gT^2 - \frac{u}{p}[pT\log_e(1-pT) - pT - \log_e(1-pT)] \end{align*} at time \(T\). Show that \(H\) is certainly positive if \(up > g\).