Problems

State the principle of virtual work. A smooth circular cylinder of radius \(a\) is fixed with its axis horizontal. Two equal cylinders of radius \(b\) and weight \(w\) rest inside the first cylinder with their axes horizontal, and another smooth cylinder of radius \(c\) and the same weight \(w\) rests upon the last two, with its axis parallel to theirs, separating them without touching the fixed cylinder. Prove that, if the planes through the axes of each of the equal cylinders and the axis of the fixed cylinder make an angle \(\theta\) with the vertical, then \[ 8(a-b)^2 \cos^2\theta = 9\{(a-b)^2 - (b+c)^2\}. \] Prove also that this position is one of unstable equilibrium.

Shew that if a focus be taken as pole and the axis as initial line, then the equation to a conic may be written in the form \[ \frac{l}{r} = 1 + e \cos\theta. \] Derive the equation to the chord joining the points on the conic whose vectorial angles are \(\alpha+\beta\) and \(\alpha-\beta\) and deduce the equation to the tangent at the point whose vectorial angle is \(\alpha\). Hence or otherwise, prove that chords of a conic which subtend a constant angle at a focus touch a fixed conic.

\(ABC\) is an equilateral triangle inscribed in a circle of radius \(a\); \(P\) is any point on a concentric circle of radius \(r\). Shew that \(PA^2+PB^2+PC^2\) is constant, and that \(PA \cdot PB \cdot PC\) lies between \(r^3 \sim a^3\) and \(r^3+a^3\).

A circular area is rotated through 180\(^\circ\) about a coplanar axis which does not intersect the circumference of the circle. Prove that the centre of gravity of the volume generated is at a perpendicular distance \[ \frac{2}{\pi}\left(h + \frac{a^2}{4h}\right) \] from the axis, where \(a\) is the radius of the circle, and \(h\) the perpendicular distance of its centre from the axis.

Two unequal masses \(m_1\) and \(m_2\) are fixed to the ends of a light helical spring of natural length \(l\) and elastic modulus \(\lambda\). The system is placed on a smooth horizontal table and the spring is compressed through a distance \(d\) and the system is then released at both ends simultaneously. Investigate the subsequent motion of the system. It may be assumed that the axis of the spring remains straight during the motion and that the masses remain on the axis.

Prove that any conic which passes through the four common points of two rectangular hyperbolas is itself a rectangular hyperbola. Reciprocate this theorem with respect to an arbitrary point in the plane; and hence, or otherwise, shew that if \(A, B, C, D\); and \(E, F\) are the pairs of opposite vertices of a complete quadrilateral, then the three circles having \(AB, CD\), and \(EF\) as diameters belong to one and the same coaxal system.

Prove that, if \(0 < \alpha < \pi\), then \[ \int_0^{\frac{1}{2}\pi} \frac{d\theta}{1+\cos\alpha \cos\theta} = \frac{\alpha}{\sin\alpha}. \] What is the value of the integral when \(\pi < \alpha < 2\pi\)? If \[ I_p = \int (x^2+a)^p dx, \] shew that \[ (2p+1)I_p - 2pa I_{p-1} = x(x^2+a)^p, \] and, hence or otherwise, evaluate \[ \int_0^\infty \frac{dx}{(x^2+1)^4}. \]

A gun of mass \(M\) which fires a shot of mass \(m\) is able to recoil freely on a horizontal plane. If just after the explosion the total kinetic energy of the gun and the shot is in all cases the same, prove that in order that the range on a horizontal plane may be a maximum the elevation of the gun must be \[ \frac{1}{2}\cos^{-1}\left(\frac{m}{m+2M}\right). \]

If the relation between the acceleration and velocity of a body, moving in a straight line, be represented by any given curve, show how the curves relating velocity with time and velocity with displacement may be derived, provided that the acceleration never becomes zero during the interval considered. A car of mass 1 ton and moving at a speed of 60 miles per hour reaches the bottom of a hill which rises uniformly 1 foot in 8 feet (measured along the ground surface). During the ascent of the hill the power delivered to the wheels of the car is maintained at a constant value of 15 horse power. Find how long it will be before the speed of the car drops to 30 miles per hour. All resistance to motion other than the force of gravity is to be neglected.

Shew that the inverse of a circle \(C\) with respect to a circle \(\Gamma\) is a circle \(C'\), and that if \(C\) cuts \(\Gamma\) at right angles, then \(C'=C\). \(C_1\) and \(C_2\) are two circles which cut at right angles; \(C_3\) is a circle touching \(C_1\) and passing through its centre; \(C_4\) is the inverse of \(C_3\) in \(C_2\), and \(C_5\) the inverse of \(C_4\) in \(C_1\). Shew that \(C_5\) is a circle touching \(C_1\) and passing through the centre of \(C_2\).