A particle is placed inside a fixed smooth hollow sphere of internal radius \(a\). It is projected horizontally from the lowest point with speed \(u\). Show that it will leave the surface in the subsequent motion provided that \[2ag < u^2 < 5ag.\] Show that the time that elapses between the particle leaving the surface and subsequently striking it again is greatest when \[u^2 = ag(2 + \sqrt{3}).\]
A person drags a mass over a level, rough floor by pulling on a rope of length \(l\). Friction is so great that the inertia of the mass may be neglected. Show that the time-dependent position \((x, y)\) of the mass is related to the time-dependent but given position \((x_0, y_0)\) of the person by \[\frac{dy}{dx} = \frac{y_0-y}{x_0-x}.\] Hence show that if the person's locus is specified as \(y_0 = f(x_0)\), the mass's locus is determined by \[l\frac{dy}{ds}+y = f\left(l\frac{dx}{ds}+x\right),\] where \(s\) is the arc-length along the mass's locus. (i) If the person walks along the line \(y_0 = 0\), show that the mass moves along the curve \[x = l \textrm{sech}^{-1}(y/l)-\sqrt{l^2-y^2}+\text{const}.\] (ii) If the person walks along the circle \(x_0^2+y_0^2 = a^2\), \(a > l\), show that the mass ultimately moves along the circle \(x^2+y^2 = a^2-l^2\).
Derive the equation for a simple pendulum \[\ddot{\theta} = -\omega^2 \sin \theta,\] giving a value for \(\omega^2\) in terms of relevant physical quantities. Show that for small \(\alpha\) there is an approximate solution \[\theta_1(t) = \alpha \sin \omega t.\tag{1}\] By expanding \(\sin \theta\) for small \(\theta\), and using the approximation (1) in the cubic term, obtain the higher-order approximation \[\theta_2(t) = \alpha \sin \omega t - \frac{\alpha^3}{16}\omega t \cos \omega t + \frac{\alpha^3}{192}\sin 3\omega t, \tag{2}\] for suitable starting conditions at \(t = 0\). For how long an interval \(t\) would you expect this approximation to be reasonable? For a sufficiently small number of oscillations after \(t = 0\) show that \[\theta_3(t) = \alpha \sin \Omega t + \frac{\alpha^3}{192}\sin 3\Omega t, \tag{3}\] where \[\Omega = \omega(1-\alpha^2/16).\] Of the above two approximations (2) and (3), which do you prefer, and why?
An operator \(T_a\) on a vector \(\mathbf{b}\) is defined by \[T_a\mathbf{b} = \mathbf{a} \wedge \mathbf{b}.\] Show that \(T_a^3\mathbf{b} = -a^2T_a\mathbf{b}\). If \(S_a\) is defined by \[S_a\mathbf{b} = (1+T_a/1!+T_a^2/2!+...)\mathbf{b},\] show that \[S_a\mathbf{b} = \frac{1}{a^2}[(\mathbf{a}\cdot\mathbf{b})\mathbf{a}+\mathbf{a}\wedge\mathbf{b}\sin a-\mathbf{a}\wedge(\mathbf{a}\wedge\mathbf{b})\cos a],\] and that \(|S_a\mathbf{b}|^2 = b^2\).