Problems

Prove the formulae \(v dv/ds\) and \(v^2 d\psi/ds\) for the tangential and normal accelerations of a particle describing a plane curve. A particle moving in a plane is directly opposed by a force proportional to the square of the speed, and is acted on by a force of constant magnitude perpendicular to the direction of motion. Show that when the speed has been changed from \(v_0\) to \(v\) the angle through which the direction of motion has turned is proportional to \(v_0^{-2} - v^{-2}\).

A smooth tube in the form of the portion of the cycloid \(s=4a \sin\psi\) from \(\psi = -\frac{1}{2}\pi\) to \(\psi = \frac{1}{2}\pi\) is fixed with its plane vertical and the two cusps uppermost and at the same level. A piece of uniform chain, of mass \(\rho\) per unit length, initially occupies one-half of the tube from \(\psi=0\) to \(\psi=\frac{1}{2}\pi\) and is released from rest. Show that when the chain has moved a distance \(u\) along the tube the potential energy relative to the level of \(s=0\) is \(g\rho(16a^2-12au+3u^2)/6\). Hence, or otherwise, show that the ends of the chain are at the same level after a time \(\pi\sqrt{(a/g)}\) has elapsed.

A rigid body is capable of rotation about a fixed axis. Prove that the rate of change of moment of momentum about this axis is equal to the moment of the applied forces about this axis. Point out clearly at which stage of the proof the assumption that the body is rigid is introduced. Two gear wheels, of radii \(a_1, a_2\) and of moments of inertia \(I_1, I_2\), rotate about parallel axes. At an instant when their respective angular velocities are \(\Omega_1, \Omega_2\) in the same sense the wheels are suddenly put into mesh, with their axes held fixed. Find their new angular velocities.

Prove that

1. If \(n\) is an integer greater than 2, then \[ \left(\frac{n+1}{2}\right)^n > n! > n^{n/2}, \] \[ n! \left(2-\frac{1}{n}\right)\left(2-\frac{3}{n}\right)\dots\left(2-\frac{2n-1}{n}\right) > 1; \]
2. If \(p > m > 0\), then \[ \frac{p+m}{p-m} \ge \frac{x^2-2mx+p^2}{x^2+2mx+p^2} \ge \frac{p-m}{p+m} \] for all real \(x\).

If \(a,b,c,d\) are roots of \(x^4+px^3+qx^2+rx+s=0\):

1. find the value of \(\sum \frac{a^3}{b^2}, \sum a^3b^2c^2\);
2. find the equation whose roots are \[ b^2+c^2+d^2-a^2, \dots. \]

1. If \[ y = \tan^{-1} \frac{x\sin\alpha}{1+x\cos\alpha}, \] prove \[ \frac{d^n y}{dx^n} = \frac{(n-1)!}{\sin^n\alpha} \sin n(\alpha-y)\sin^n(\alpha-y). \] (The OCR is slightly different here, but this form seems more likely based on context. Original OCR: `sin n(y-a) sin^n(y-a)` and `sin^n a`. The image is clearer showing \(\alpha - y\)).
2. In the equation \[ f(x+h) = f(x) + hf'(x) + \frac{h^2}{2}f''(x+\theta h), \] find the first three terms of the expansion of \(\theta\) in ascending powers of \(h\), and calculate them for \(f(x) = \sin x\).

Evaluate the integrals \[ \int \frac{dx}{(1+x)(4+6x+4x^2+x^3)}, \quad \int \frac{\sin^2 x \, dx}{\cos^3 x}, \quad \int_0^{\pi/2} \log(\sin x \cos x) \, dx. \]

1. Show that if \(a_1+a_2+\dots\) be a divergent series of positive terms, then the series \[ \frac{a_1}{a_1+1} + \frac{a_2}{(a_1+1)(a_2+1)} + \frac{a_3}{(a_1+1)(a_2+1)(a_3+1)} + \dots \] converges to \(+1\).
2. Find the sum of the series \[ 1 + \cos\theta\tan\theta + \frac{1}{2!}\cos 2\theta \tan^2\theta + \frac{1}{3!}\cos 3\theta\tan^3\theta + \dots. \]

Normals are drawn to an ellipse at the ends of two conjugate diameters. Find all maxima and minima distances of their common point from the centre.

Tangents at right angles are drawn to the four-cusped hypocycloid \(x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}\). Show that the bisectors of the angles between them either pass through the origin or envelop the curve \[ (x+y)^{\frac{2}{3}} + (x-y)^{\frac{2}{3}} = (2a)^{\frac{2}{3}}. \]