An ammeter has a coil and needle which rotate on a pivot. The moving system has moment of inertia \(M\), and when the deflection is \(\theta\), a spring supplies a restoring couple \(-A\theta\). When a current \(I\) flows the coil experiences a deflecting couple \(kI\). There is also a resisting couple \(-\mu\dot{\theta}\) due to friction and eddy currents whenever the coil is moving. Here \(M\), \(k\), \(\lambda\) and \(\mu\) are all constants. Show that the current and deflection are related by $$M\ddot{\theta} + \mu\dot{\theta} + A\theta = kI.$$ Find the complementary function for this equation, distinguishing between the cases where \(\mu^2\) is greater than, less than, and equal to \(4MA\). Explain how to solve the problem in which \(\theta\), \(\dot{\theta}\) and \(I\) are initially zero, and a steady current \(I\) is switched on at a time \(t = 0\). Why is the choice \(\mu^2 = 4MA\) the most convenient in practice?
A uniform rod \(AB\), of length \(a\) and mass \(m\), is pivoted about \(A\). It is released from rest with \(B\) vertically above \(A\), and given a very slight displacement so that it falls under gravity. Find the horizontal and vertical components of the reaction at the pivot when the rod makes an angle \(\theta\) with the upwards vertical.
A rocket is travelling horizontally. Its initial mass is \(M\) and it expels a mass \(m\) of gas per unit time horizontally with a velocity \(a\) relative to the rocket, where \(m\) and \(a\) are constants. If the rocket experiences a resistive force which is a constant multiple \(k\) of its velocity \(v\), show that if \(v = 0\) when \(t = 0\) $$\left(\frac{M-mt}{M}\right)^k = \left(\frac{ma-kv}{ma}\right)^m.$$ Find a similar relation for the case where the resistive force is proportional to the square of the velocity of the rocket.
A plane \(P\) passing through a point \(O\) is inclined at \(30^\circ\) to the horizontal. A ball, whose coefficient of restitution with \(P\) is \(\frac{1}{3}\), is projected from \(O\) in a vertical plane through the line of greatest slope of \(P\) with speed \(V\), at an angle of \(60^\circ\) to the horizontal and \(30^\circ\) to the line of greatest slope. Find the maximum height above \(O\) (measured vertically) that it attains between the first and second bounces.
A plane lamina in the shape of a quadrant of the unit circle has a variable density proportional to \(r^{-1}\sin(\frac{1}{2}\pi r)\) where \(r\) is the distance from the centre of the circle. Calculate its moment of inertia about an axis through its centre of gravity perpendicular to the plane of the lamina.
Two numbers \(X\) and \(Y\) between 1 and 100 (inclusive) are selected at random, all possible pairs \((X, Y)\) having equal probabilities. Let \(Z\) denote the maximum of \(X\) and \(Y\). What is the probability that \(Z \leqslant 50\)? By use of the formulae $$\sum_{r=1}^n r = \frac{1}{2}n(n+1),$$ $$\sum_{r=1}^n r^2 = \frac{1}{6}n(n+1)(2n+1),$$ or otherwise, show that the mean of \(Z\) is just over 67. Find a median of \(Z\). [A median of \(Z\) is any number \(\xi\) such that \(P\{Z \leqslant \xi\} \geqslant \frac{1}{2}\) and \(P\{Z \geqslant \xi\} \geqslant \frac{1}{2}\)]
Let \(X\) be a random variable uniformly (rectangularly) distributed over the interval \(0 < x < 1\). Derive the probability density functions of the following random variables \((a)\) \(Y = X^2 - 1\), \((b)\) \(Z = \sin\pi X\). Find the mean and standard deviation of \(Y\) and \(Z\).
A manufacturer is asked to supply steel tubing in lengths of 10 feet. Several samples are obtained from him and the mean lengths in feet of four samples each of 16 tubes found to be as follows: $$10 \cdot 16; \quad 10 \cdot 38; \quad 10 \cdot 31; \quad 10 \cdot 07.$$ What type of distribution would you expect mean lengths such as these to have and why? Samples are also obtained from another source, and in this case the mean lengths in feet of five samples of 16 tubes are found to be as follows: $$10 \cdot 15; \quad 10 \cdot 36; \quad 10 \cdot 11; \quad 10 \cdot 11; \quad 10 \cdot 07.$$ Assuming that both manufacturers produce tubing whose length has a standard deviation of \(0 \cdot 48\) feet, is there any evidence that either manufacturer's tubing has a mean length greater than \(10 \cdot 1\) feet? Is there any evidence that tubes supplied by the two manufacturers differ in mean length? [Let $$\Phi(X) = \int_{-\infty}^x \phi(x)dx,$$ where \(\phi(x) = (2\pi)^{-1}\exp(-\frac{1}{2}x^2)\). Then \(\Phi(-2 \cdot 58) = 0 \cdot 005\), \(\Phi(-2 \cdot 33) = 0 \cdot 01\), \(\Phi(-1 \cdot 96) = 0 \cdot 025\), \(\Phi(-1 \cdot 64) = 0 \cdot 05\).]
The sides of a triangle are \(p\), \(q\), \(r\); the angles opposite them are (in circular measure) \(P\), \(Q\), \(R\). Prove that $$\frac{\pi}{3} \leq \frac{pP + qQ + rR}{p + q + r} \leq \frac{\pi}{2}.$$ When, if at all, can equality occur?
Show that if \(a\), \(b\), \(c\) are integers it is always possible to find integers \(A\), \(B\), \(C\) such that $$(a + b2^i + c2^j)(A + B2^i + C2^j) = a^2 + 2b^2 + 4c^2 - 6abc.$$ Prove that the right side of this can be zero only if the integers \(a\), \(b\), \(c\) are all zero, and deduce that if now \(a\), \(b\), \(c\) are rational numbers such that \(a + b2^i + c2^j = 0\), then \(a = b = c = 0\).