Problems

A bicycle is so geared that when the cranks turn through a radian the machine advances a distance \(k\). The couple on the pedals being \(N\) and the power applied \(P\), the rider works so that \(P = AN\left(1 - \dfrac{N}{N_0}\right)\), where \(A\) and \(N_0\) are constants. Internal friction and road resistance together are equivalent to sliding friction with coefficient \(\mu\), and air resistance is neglected. Find what acceleration the rider can produce in climbing a hill of slope \(\alpha\) with velocity \(v\), and shew that his maximum velocity is \[ Ak\left\{1 - \frac{mkg}{N_0}(\sin\alpha + \mu\cos\alpha)\right\}, \] where \(m\) is the mass of the rider and machine together. With \(k\) equal to 4\(\frac{1}{4}\) feet and \(\mu\) negligible, the rider can attain a speed of 22 ft./sec. on the flat, and 2\(\cdot\)2 ft./sec. up a hill with \(\sin\alpha = \frac{1}{50}\). Shew that by adjusting his gear suitably he can climb a hill with \(\sin\alpha = \frac{1}{15}\) with a velocity of 3\(\cdot\)05 ft./sec.

Explain the principle of the conservation of energy. A bead slides on a smooth parabolic wire in a horizontal plane, and is connected to a hanging particle of the same mass as the bead by a light inelastic string passing through a small hole at the focus. The bead is released from rest at a distance \(\frac{1}{3}a\) from the focus (where \(4a\) denotes as usual the latus rectum of the parabola.) Shew that the greatest velocity of the hanging particle in the subsequent motion is \((\frac{1}{3}ga)^{\frac{1}{2}}\).

Prove that if \(A, P, Q\) are polynomials in \(x\), and \(A\) is of lower degree than \(PQ\), then \(A/PQ\) can be expressed in the form \(M/P + N/Q\), where \(M, N\) are respectively of lower degrees than \(P, Q\). Expand \(\dfrac{\sin\phi}{1-2x\cos\phi+x^2}\) in ascending powers of \(x\) and prove that the remainder after \(n\) terms is \[ x^n\left(\frac{\sin(n+1)\phi - x\sin n\phi}{1-2x\cos\phi+x^2}\right). \]

Prove that \(\sin 7\theta / \sin\theta = c^3+c^2-2c-1\), where \(c=2\cos 2\theta\). Show that the side of a regular heptagon is approximately equal to the altitude of an equilateral triangle whose side is equal to the radius of the circle circumscribing the heptagon, and that the error is less than one-fifth per cent.

Tangents from a fixed point \(O\) to any conic of a confocal system touch it at \(P, Q\). Show that the circumcircle of \(OPQ\) passes through another fixed point. Hence show that the normals at \(P\) and \(Q\) intersect on a fixed straight line.

Show that the evolute of an equiangular spiral, whose radius vector makes a constant angle \(\alpha\) with the tangent at each point, is an equiangular spiral with the same angle \(\alpha\). If the spiral and its evolute coincide, show that \(\alpha\) must satisfy the equation \[ \tan\alpha \log(\tan\alpha) = (2n-\frac{1}{2})\pi, \] where \(n\) is an integer.

Evaluate \(\displaystyle\int\frac{x^2dx}{1+x^4}\), expressing the result in real form. Prove that \(\displaystyle\int_0^{\frac{\pi}{4}} \sqrt{\tan\theta}.d\theta = \frac{\pi}{2\sqrt{2}} + \frac{1}{\sqrt{2}}\log(\sqrt{2}-1)\).

A circular cylinder of radius \(a\) and weight \(W\) having its centre of gravity at a distance \(c\) from its axis rests in stable equilibrium on a horizontal plane. A uniform plank of thickness \(2b\) and weight \(w\) is placed on the cylinder so as to rest in a horizontal position with its length perpendicular to the axis of the cylinder. Prove that the system is in stable equilibrium for small rolling displacements if \[ a-b > wa^2/Wc. \]

A particle starts from rest at a distance \(a\) from a centre of attractive force varying as the direct distance and subject to a resistance per unit mass equal to \(k\) times the velocity. Prove that before coming finally to rest at the centre it travels a distance \(a \coth(\frac{1}{4}kT)\), where \(T\) is the period of the damped oscillation.

Two light elastic strings are fastened to a particle of mass \(m\) and their other ends to fixed points so that the strings are taut. The modulus of each is \(\lambda\), the tension \(T\) and the lengths \(a,b\). Show that the period of an oscillation along the line of the strings is \[ 2\pi\{mab/(T+\lambda)(a+b)\}^{\frac{1}{2}}. \]