Problems

The lower end of a uniform rod of length \(a\) slides on a light smooth inextensible string of length \(2a\) whose ends are fixed to two points distant \(2\sqrt{(a^2-b^2)}\) apart in a horizontal line, and the upper end of the rod slides on a fixed smooth vertical rod which bisects the line joining the two fixed points. If the rod makes an angle \(\theta\) with the vertical at any time, find expressions for the kinetic and potential energy in terms of \(\theta, \dot{\theta}\). Prove that, if \(2b>a\), the time of a small oscillation about the vertical position of equilibrium is \(2\pi/\mu\), where \[ \mu^2 = \frac{3g(2b-a)}{2a^2}. \]

Discuss the theory of the motion of a wheel, whose plane is vertical, in contact with rough horizontal ground, and set in motion by

1. [(a)] a horizontal force P applied at its centre in its plane; or
2. [(b)] a couple G about its axis,

determining limits for the coefficient of friction in order that the wheel may roll without slipping. A uniform circular cylinder of radius \(a\) and radius of gyration \(k\) about its axis is set rolling and sliding on rough horizontal ground; the initial velocity of the axis is \(V\) and the initial angular velocity \(\omega\) is such that the points on the ground have the greatest velocity. Prove that, if \(k^2\omega > aV\), then after time \(\frac{k^2V+a\omega}{a^2+k^2} \frac{1}{\mu g}\) the cylinder rolls back with uniform velocity \(\frac{a}{a^2+k^2}(k^2\omega - aV)\), where \(\mu\) is the coefficient of friction.

Prove that, if an angle of a triangle and the length of the opposite side and the length of the bisector of the angle intercepted by the opposite side are given, the triangle is uniquely determined. Deduce that, if the lengths of the bisectors of two of the angles of a triangle intercepted by the opposite sides are equal, the triangle is isosceles.

TP, TQ are two tangents to a conic, touching it at P and Q. A line through Q cuts the curve in B and cuts TP in A. The tangent at B cuts TP in C. Prove that the range TACP is harmonic. State the dual theorem and draw a figure to exhibit it.

Shew that a triangular prism with parallel plane ends can be divided into three tetrahedra of equal volume; and that the volume of a tetrahedron is given by one-sixth of the product of a pair of opposite edges, the shortest distance between them and the sine of the angle between their directions.

A shuffled pack of 52 cards contains 20 honours. Express in terms of factorials the chance of securing exactly half the honours in taking half the pack.

Prove that, if \(u = f(X) + g(Y)\), where \[ X = x^2+y^2 \quad \text{and} \quad Y=xy, \] then \[ xy \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right) - (x^2+y^2)\frac{\partial^2 u}{\partial x \partial y} = 4Y\frac{df}{dX} - X\frac{dg}{dY}. \]

Prove that, if \(F(x)\) is a polynomial of degree \(r\), \(n\) an integer greater than \(r\), and \(c>b>a\), then \[ \int_a^b \frac{F(x)\,dx}{(x-c)^{n+1}} = -\frac{1}{n!} \frac{d^n}{dc^n} \left\{ F(c) \log \frac{c-a}{c-b} \right\}. \]

Show that, in general, a system of coplanar forces can be reduced to a single force acting at a specified point in the plane, together with a couple. Show that, if \(G_1, G_2, G_3\) denote the sums of the moments of a set of coplanar forces about the corners A, B, C of a triangle in the plane, then the sum \(G\) of the moments of the forces about any point O in the plane is given by the relation \[ G \cdot \triangle ABC = G_1 \cdot \triangle OBC + G_2 \cdot \triangle OCA + G_3 \cdot \triangle OAB. \]

A uniform rod OA of weight \(W\) and length \(2a\) can turn in a vertical plane about the end O. It is supported in equilibrium in a horizontal position by a light string attached to the end A, passing over a smooth peg B vertically above the middle point of OA, and carrying a weight; and the angle OAB is \(30^\circ\) in the equilibrium position of the rod. Prove that to turn the rod about O through a small angle \(\theta\) requires an amount of work \(\sqrt{3}Wa\theta^2/4\). What do you infer about the stability of the equilibrium?