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.
A system of particles moves under external and internal forces. Prove that (a) the centroid moves as if the whole mass were concentrated there and the vector sum of the external forces acted upon it, and (b) the rate of change of angular momentum of the system relative to the centroid is equal to the sum of the moments about the centroid of the external forces. Two particles of masses \(m_1\) and \(m_2\) are connected by an inextensible string of length \(l\) and placed on a smooth horizontal plane. \(m_1\) is first held fixed and \(m_2\) whirls around it, moving with speed \(v\) on a circle. \(m_1\) is then let go. Describe the nature of the subsequent motion in detail, and find the tension in the string before and after the release of \(m_1\).
In a nuclear collision, in which linear momentum is conserved but mass is not necessarily conserved, a particle of mass \(m_1\) with kinetic energy \(T_1\) hits a particle \(m_2\) that is initially at rest. As a result, particles of masses \(m_3\) and \(m_4\) move away in directions making angles \(\theta_3\) and \(\theta_4\) with the original direction of motion of \(m_1\), and energy \(Q\) is absorbed. The masses \(m_3\) and \(m_4\) are known, and \(\theta_3\), \(T_3\) can be measured, where \(T_3\) is the kinetic energy of the particle of mass \(m_3\). Prove the formula for \(Q\), namely \[Q = T_1 \left(1 - \frac{m_1}{m_4}\right) - T_3 \left(1 + \frac{m_3}{m_4}\right) + 2\sqrt{\frac{m_1 m_3 T_1 T_3}{m_4^2}} \cos \theta_3.\]
Prove that if a parabola rolls on a fixed straight line the path of the focus is a catenary. [The formula \(\int_0^\psi \sec^3\psi \, d\psi = \frac{1}{2} \sec \psi \tan \psi + \frac{1}{2} \log (\sec \psi + \tan \psi)\) may be used without proof.]