A shell of mass \(2m\) is fired vertically upwards with velocity \(v\) from a point on a level stretch of ground. When it reaches the top of its trajectory it is split into two equal fragments by an explosion, which supplies kinetic energy amounting to \(mv^2\) to the system but leaves its momentum unchanged. Show that the greatest possible distance between the points where the two fragments hit the ground is \(2v^2/g\) if \(u \leq v\), and \((u^2 + v^2)/g\) if \(u > v\).
A rocket, whose initial mass is \((M + m)\), contains a mass \(m\) of propellant fuel. This is ejected at a constant velocity \(V\) relative to the rocket at a rate of \(\mu\) per sec. What are the conditions that the rocket (a) rises immediately; (b) rises at all? Assuming that it rises immediately show that its maximum upward velocity is $$V \log \left( 1 + \frac{m}{M} \right) - gm/\mu.$$ What is the maximum height attained? [Variation of gravity with height may be neglected.]
Two scale pans each of mass \(M\) hang in equilibrium at opposite ends of a string passing over a pulley. A freely falling particle of mass \(m\) strikes one pan and its velocity at the moment of impact is \(V\). Show that it comes to rest in the pan after a time $$\frac{2eV}{(1-e)g},$$ where \(e\) is the coefficient of restitution between the particle and the pan. What is the velocity of the pan at this instant?
A uniform circular cylinder of radius \(a\) is slightly displaced from rest along the highest generator of a fixed horizontal circular cylinder of radius \(b\). Assuming that the contact between the cylinders is always sufficiently rough to prevent slipping, find the point at which the moving cylinder leaves the fixed cylinder.
Show that, in general, the resultant of a number of parallel forces of fixed magnitude acting at fixed points of a rigid body passes through a fixed point of the body whatever the common direction of the forces. State the conditions of the exceptional case, and show that in this case by adjusting the point of action of any one of the forces the system can be made one that is in equilibrium for all directions of the forces. Explain the relevance of the general case to the usual identification of centre of mass and centre of gravity.
A uniform heavy rod of length \(2b\) has its ends attached to small light rings which slide on a smooth rigid wire in the shape of a parabola of latus rectum \(4a\) held fixed in a vertical plane with its vertex uppermost. Prove that the horizontal position of the rod is one of stable equilibrium if \(b > 2a\). Show further that in this case there are two oblique positions of equilibrium, one on either side.
Explain what is meant by \emph{moment of inertia}. Show that for a plane lamina the moment of inertia is least for a set of parallel axes in its plane when the axis contains the centre of mass. Considering axes in the plane of the lamina passing through a fixed point \(P\), if the moment of inertia for three such axes has the same value, prove that the value is equal for all such axes. Show that a uniform triangular lamina of mass \(m\) has the same moment of inertia for any coplanar axis as the system of three masses each of \(\frac{1}{3}m\) at the mid-points of the sides.
A particle is attached to one end of a light perfectly flexible string of length \(a\) whose other end \(O\) is fixed. When hanging at rest the particle is given a horizontal velocity \(u\). Find conditions to ensure that \(O\) will be the lowest point at which, in the subsequent motion the string remains taut, and show that if these conditions are not satisfied the particle will pass through \(O\) if \(u^2 = (2 + \sqrt{3})ga\).
A particle is released from rest and slides under gravity down a rough rigid wire in the shape of a loop of a cycloid held fixed in a vertical plane with its line of cusps horizontal and uppermost. If the particle starts from a cusp and comes to rest at the lowest point, prove that the coefficient of friction \(\mu\) must satisfy the equation \(\mu^2 = e^{-\mu\pi}\). [The usual parametric equations for the cycloid may be taken in the form: $$x = a(\theta + \sin\theta), \quad y = a(1 - \cos\theta).]$$
A particle moving under gravity in a medium offering resistance proportional to the speed suffers an explosion in which it splits into two parts of equal masses, the speed being relative one to the other not necessarily in the same direction as the combined velocity of the undivided particle immediately before the explosion. Prove that at any subsequent instant the distance separating the two fragments is given by $$d = \frac{v}{k}(1 - e^{-kt}),$$ where \(v\) is the initial relative velocity and \(k\) is a constant.