\(A\), \(B\) and \(C\) are three smooth horizontal parallel pegs, \(A\) and \(C\) being a distance \(a\) from \(B\) in a horizontal plane. Two uniform rods, \(DB\) and \(EF\), of the same weight per unit length and of length \(3a\) and \(a\), respectively, are smoothly jointed at \(E\) and are laid perpendicularly across the pegs \(A\), \(B\), \(C\). Show that, if the rods are to remain horizontal, \(D\) must be \(\frac{5a}{6}\) beyond an end peg.
A uniform rod of length \(l_0\) and mass \(m\) is hinged at one end to the point \(A\) and is free to rotate in a vertical plane through the point \(B\) which is at a distance \(l_0\) horizontally from \(A\). The other end of the rod is attached to the point \(B\) by an elastic string of unstretched length \(l_0\). The tension in the string when stretched to a length \(l(l > l_0)\) is given by \(mg R(l-l_0)/l_0\). Derive an equation in terms of \(\theta\), the angle between the rod and the line \(AB\), for the positions of equilibrium of the system with the rod lying below \(AB\), and obtain a criterion for their stability. If \(f'(\omega) > 0\) for all \(\omega > 0\), show that only one such position exists, and discuss its stability.
A solid circular cylinder of radius \(a\) rolls on the inside of a fixed hollow circular cylinder of radius \(b(> a)\). Calculate the greatest magnitude of the acceleration experienced by the centre of gravity of the moving cylinder during a motion in which the highest point of contact reached subtends an angle \(\alpha(\cos \alpha < 3/4)\) with the downward vertical at the centre of the fixed cylinder.
Three springs of unit length and modulus \(M\) are joined together end to end and restricted to lie on a horizontal line. Two masses \(m\) are fixed to the junctions and the outer ends are held fixed. By taking the coordinates \(x_1\) and \(x_2\) to represent the displacements of the two masses from their respective positions of equilibrium, show that two simple harmonic motions are possible, in which either \(x_1 + x_2\) or \(x_1 - x_2\) is zero. What is the ratio of their periods? If the masses are released from rest at arbitrary values of \(x_1\) and \(x_2\), show that in general at no later time are both particles at rest.
A uniform rod of length \(a\) and mass \(m\) is rotating freely on a smooth horizontal table with angular velocity \(\Omega\) about one end whose position is held fixed. If the rod strikes a particle of mass \(m\) at rest at a distance \(b\) from the fixed end, and the coefficient of restitution is \(\frac{1}{2}\), show that the final angular velocity of the rod is $$\frac{1}{2}\left(\frac{2a^2 - 3b^2}{a^2 + 3b^2}\right)\Omega,$$ and that the loss of energy in the collision is $$\frac{3m}{8}\frac{a^2b^2}{a^2 + 3b^2}\Omega^2.$$
The stars of a globular cluster may be taken to move independently under the influence of smooth mean gravitational field of the whole cluster. A star moves on a straight-line `orbit' through the centre of the cluster under the influence of an attractive force \(F = -\psi'(r)\) per unit mass towards the centre. Here \(\psi(r)\) is a known function of distance \(r\) from the centre of the cluster. Find as an integral the time that it takes for a star that starts at rest at \(r = R_0\) to reach \(r = R < R_0\) for the first time. $$\psi = \frac{GM}{b+(r^2+b^2)^{1/2}}$$ If where \(GM\) and \(b\) are constants, use the variable \(\chi(r) = \{(r/b)^2 + 1\}^{1/2}\) to show that the period of the motion is $$P = \left(\frac{8b^2(1+\chi_0)}{GM}\right)^{1/2} \int_1^{\chi_0} \frac{\chi d\chi}{[(1-\chi_0)(\chi-1)]^{1/2}},$$ where \(\chi_0 = \chi(R_0)\).
Two particles collide elastically on a smooth horizontal plane. Write down the law of the conservation of energy for the system. An observer moving with uniform velocity in the same plane evaluates the kinetic energy of the particles before and after collision using the particle velocities as seen by him. Show that the necessary and sufficient condition for all such observers to agree on the conservation of energy in the collision is that momentum should be conserved. What is the corresponding theorem if energy is lost in the collision?
Two equal cones of semi-vertical angle \(\alpha\) are mounted with their axes parallel. They are in contact along a generator for a length \(l\) between the vertices and the normal component of the force between them is \(F/l\) per unit length of the generator. If the coefficient of friction between the cones is \(\mu\) and each is freely pivoted about its axis, show that the least couple that must be applied to one cone about its axis in order to move the system is $$\frac{\mu Fl \sin \alpha}{1 + \sqrt{2}}.$$
A uniform solid hemisphere of mass \(M\) and radius \(a\) is freely pivoted at the centre and its flat surface and hangs downward under gravity. A bullet of mass \(m\), travelling horizontally, embeds itself at the lowest point of the hemisphere. Show that the final angular velocity of the hemisphere is $$V^2 > \frac{(2M + 5m)(3M + 8m)}{10m^2}ga.$$
A shell is such that when exploded at rest the maximum velocity of a piece of shrapnel is \(V\). It is given a proximity fuse set to explode just before hitting the ground. If it is fired from a gun of muzzle velocity \(V\) at a fixed angle \(\theta\) show that the maximum range of a piece of shrapnel is $$\frac{3V^2}{2g}\left(\sqrt{3}\cos\frac{2\theta}{3} - \sin\frac{2\theta}{3}\right).$$ Hence show that if \(\theta\) is allowed to vary the range of a piece of shrapnel cannot exceed $$\frac{3\sqrt{3}V^2}{2g}.$$