A uniform rectangular rough plank of weight \(W\) and thickness \(2b\) rests in equilibrium across the top of a fixed horizontal circular cylinder of radius \(a\). The length of the plank is perpendicular to the axis of the cylinder. Find an expression for the increase in potential energy if the plank is turned without slipping through an angle \(\theta\), and deduce that the horizontal position is stable provided that \(a > b\). Discuss the case \(a = b\).
A uniform plane lamina of mass \(M\) has the form of a semicircular area of centre \(C\) and diametral base \(OO'\) of length \(2a\). The radius perpendicular to this is \(CD\). Find the moment of inertia of the lamina about the chord \(OD\).
A uniform heavy perfectly flexible chain hangs under gravity with its ends attached to two small light smooth rings that can slide on a horizontal rod. The rod is made to rotate with constant angular velocity \(\omega\) about a vertical axis that intersects it, and the chain is at relative rest in a vertical plane with the axis of rotation as an axis of symmetry. Rectangular coordinates \(x\), \(y\) are measured horizontally from the axis of rotation and vertically upwards from the lowest point of the chain. Find the tension at a general point \((x, y)\) of the chain given that \(T_0\) is the tension at the lowest point and \(\rho\) is the line-density of the chain. Show that the total length of the chain is \[2y_1 - \frac{\omega^2 x_1^2}{g} + \frac{2T_0}{\rho g},\] where \((x_1, y_1)\) are the coordinates of either of the rings.
A particle of unit mass moves under an attractive force of magnitude \(\mu/r^2\), where \(r\) is its distance from the centre of force. If the particle starts at rest at distance \(R\), show that it will be at distance \(r\) after a time \[\{(R^2r - Rr^2)^{1/2} + R^{1/2} \cos^{-1}(r/R)^{1/2}\}/(2\mu)^{1/2}.\] On the assumption that the Earth's orbit is circular, find how many days it would take for the Earth, if suddenly stopped in its orbital motion, to fall into the sun.
A heavy particle of mass \(2M\) is attached at one end of a light, inextensible string passes over a small smooth peg and carries at its other end a bead of mass \(M\) that can slide freely on a smooth fixed vertical rod at perpendicular distance \(a\) from the peg, so that there is a unique position of equilibrium, and that the period of small oscillations about it is \(\pi(6Mr^2/3y)^{1/2}\).
A railway engine with its tender contains a quantity of fuel that is being consumed at a constant rate of \(m\) units of mass per unit time in doing a constant amount of work equal to \(P\) per unit time, while the combustion involves the intake of surrounding air at the constant rate of \(s\) units of mass per unit time. The engine is running on a level track, and the resistance is \(kv\), where \(v\) is the speed and \(k\) is a constant. If the total mass is initially \(M_0\) and the speed is initially \(v_0\), show that at time \(t\) \[\frac{P - (k + s)v^2}{P - (k + s)v_0^2} = \left(1 - \frac{mt}{M_0}\right)^{\frac{2(k+s)}{m}}.\]
A heavy uniform rod \(AB\) is suspended in equilibrium under gravity by two equal inextensible light strings \(AC\), \(BC\) attached to its ends and to a fixed point \(C\). One of the strings is now suddenly cut. Find, in terms of the angle \(A\), the factor by which the tension in the other string is instantaneously reduced.
Two equal toothed wheels of mass \(M\) and radius \(a\), which may be regarded as uniform circular discs, are rotating in the same sense with angular velocities \(\omega_1\) and \(\omega_2\) about parallel spindles of negligible inertia at distance apart slightly greater than \(2a\). The spindles are moved towards each other so that the teeth suddenly engage. Find the amount of energy lost. If the wheels are not toothed but slightly rough, so that a finite time is taken before they come to roll on each other without slipping, what is the loss of energy?
An arm \(OQ\) of length \(a\) revolves in the plane \(OXY\) with constant angular velocity \(\omega\) and a rod \(PQ\) of length \(b (> a)\) is freely pivoted to it at \(Q\), while \(P\) is constrained to move along \(OX\). Show that, when \(POQ = \theta\) and \(OP = x\), then \[\frac{dx}{d\theta} = \frac{a\omega \sin \theta}{a \cos \theta - x}.\] Interpret this result in relation to the instantaneous centre of the rod. Find an expression for the acceleration of \(P\), and show that if \(a/b\) is small it is approximately \[\omega^2 a \left(\cos \theta + \frac{a}{b} \cos 2\theta \right).\]
A water-cistern has the form of a right circular cylinder of radius \(a\) and height \(h\). It is open at the top and is made of uniform thin metal. Find the ratio of \(a\) to \(h\) if the volume of the cistern is to be a maximum for a given amount of metal. What would be the value of the ratio if the cistern were closed at both ends?