Three particles, \(A\), \(B\), \(C\), each of mass \(m\), lie at rest on a smooth horizontal table. The particles are joined by two light straight inelastic strings \(AB\), \(BC\), and the angle between \(AB\) (produced) and \(BC\) is \(\alpha\) (\( < \frac{1}{2}\pi\)). A horizontal impulse \(P\) is applied to the mass at \(C\) in a direction \(DC\), where \(D\) is the foot of the perpendicular from \(C\) to \(AB\) produced. Find the velocity with which the particle at \(A\) begins to move.
Two small smooth pegs are situated at a distance \(2h\) apart at the same level. A light string, which hangs symmetrically across the pegs, carries a mass \(m\) at each end and a mass \(M\) (\( < 2m\)) at the middle point. If the masses move vertically and the string remains taut and below the level of the pegs, write down the kinetic energy and the potential energy in terms of the angle \(\theta\) that the inclined portions of the string make with the vertical. By writing \(\theta = \alpha + \psi\), where \(\alpha\) is the equilibrium value of \(\theta\) and \(\psi\) and its derivatives with respect to the time are assumed to remain small, deduce from the energy equation that the period of small vertical oscillations of the system is the same as that of small oscillations of a simple pendulum of length \(h\cos\alpha\cot\alpha(\cos\alpha + \cot\alpha)\).
Two gear-wheels, of moments of inertia \(I_1\), \(I_2\) and of effective radii \(a_1\), \(a_2\) respectively, are mounted on light parallel axles and are rotating in the same sense with angular velocities \(\omega_1\), \(\omega_2\) respectively. If the axles are moved so that the teeth engage suddenly, find the new angular velocities, (i) the impulsive reactions on the axles (the smooth bearings of which are held fixed). If the wheels continue to run in mesh, find the magnitude of the constant braking couple that must be applied to the wheel \(I_2\) in order to bring the wheels to rest while the wheel \(I_1\) is making \(N\) revolutions. Find also the time taken to come to rest.
A uniform thin straight rod \(AB\), of mass \(M\) and length \(2l\), is initially at rest on a smooth horizontal table. If the end \(A\) is constrained to move from rest with constant acceleration \(f\) in a horizontal straight line at right angles to the rod, find the components of the force being exerted on the rod at \(A\) at the instant when the rod has turned an angle \(\theta\) from its initial direction. Discuss whether the rod will make complete revolutions.
A uniform elastic ring rests horizontally on a smooth sphere of radius \(a\). The natural length of the ring is \(2\pi a \sin \alpha\), and the tension needed to double its length is \(k; 2\pi\) times its weight. By consideration of potential energy, or otherwise, show that the ring rests in equilibrium at a height \(a \cos \theta\) above the centre of the sphere, where \(\theta\) is given by \[\tan \theta + k = k \sin \theta/\sin \alpha.\] Show graphically that there is a value below which \(k\) must not fall if such an equilibrium position is to exist. What is the physical meaning of this restriction?
A uniform circular cylinder of weight \(W\) rests on a rough horizontal plane with coefficient of friction \(\mu_1\). A second cylinder, of weight \(kW\), rests partly on the first, touching it along one generator, with coefficient of friction \(\mu_2\); it is also supported (along a generator) by an inclined plane that makes an angle \(2\beta\) with the horizontal and with which the coefficient of friction is \(\mu_3\). The plane through the axes of the cylinders makes an angle \(2\alpha\) with the vertical. Show that for equilibrium \[\mu_1 \geq \frac{k \tan \alpha \tan \beta}{\tan \alpha + (1 + k) \tan \beta}, \quad \mu_3 \geq \tan \alpha, \quad \mu_3 \geq \tan \beta.\]
An inelastic hammer of mass \(M\), initially moving with velocity \(V\), strikes a nail of mass \(m\) into a block of wood of mass \(M'\) that is free to recoil. The motion of the nail takes place in one horizontal line. Assuming that the resistance of the wood block to the nail can be represented by a constant force \(R\), prove that the nail penetrates into the wood a distance \[\frac{M^2 M' V^2}{2R(M + m)(M + m + M')}.\]
Taking the Earth as a sphere within which gravitational acceleration towards and varies directly as the distance from the centre, and neglecting air resistance, prove that the period of oscillation of a body sliding smoothly along a straight line connecting any two points of the surface would be equal to the orbital period of a satellite in grazing circular motion. Find this period approximately in minutes, assuming that the radius of the Earth is 4000 miles and that \(g = 32\) ft/sec\(^2\). Explain why the period of any actual satellite is greater than this.
A particle of unit mass moves along a straight line under a constant force of magnitude \(2a\) directed along the line and is subject to the resistance \(a + 2bv + cv^2\), where \(v\) is speed. Prove that if \(a\), \(b\), \(c\) are positive and the particle starts with zero velocity at time \(t = 0\), then in the subsequent motion the displacement is given by \[\frac{1}{c} \log \frac{\cosh (kt + \alpha)}{\cosh \alpha} - bt/c,\] where \(k = (b^2 + ac)^{\frac{1}{2}}\) and \(\tanh \alpha = b/k\). Describe the nature of this motion.