Problems

A curve, made of smooth wire, passing through a point \(O\) and lying in a vertical plane is to be constructed in such a manner that a smooth bead projected along the wire from \(O\) at speed \(V\) comes to rest in a time \(T(V)\), where \(T\) is a given function of \(V\). Show how an equation for the curve can be found in general, given that the solution to Abel's integral equation for \(g\), \[\int_0^x\frac{g(y)dy}{(x-y)^{\frac{1}{2}}} = f(x)\] where \(f\) is a known function, is \[g(x) = \frac{1}{\pi} \frac{d}{dx} \int_0^x \frac{f(y)dy}{(x-y)^{\frac{1}{2}}}.\] Hence show that, if \(T(V) = \text{constant}\), the curve, a tautochrone, is an inverted cycloid.

A particle moves in the \((r, \theta)\) plane under the influence of a force field \[f_r = -\mu/r^2, f_{\theta} = 0.\] Show that there exist possible motions with \(r = a\), \(\dot{\theta} = \omega\) provided \(a\), \(\omega\) are constants satisfying a certain relation. Nearly circular motion in the same field can be described by \[r = a+\delta(t)\] \[\dot{\theta} = \omega+\epsilon(t).\] By expanding the equations of motion about \(r = a\) and \(\dot{\theta} = \omega\), neglecting squares and products of \(\delta\), \(\epsilon\) and their derivatives \(\dot{\delta}\), \(\dot{\epsilon}\) show that \[\ddot{\delta}+\omega^2\delta = 0.\] Given that \(|\delta|/a\), \(|\dot{\delta}|/a\omega\) and \(|\epsilon|/\omega\) are all less than some small number \(k\) at \(t = 0\), show that \(|\delta| < 12ka\) in the subsequent motion.

A long thin pencil is held vertically with one end resting on a rough horizontal plane whose coefficient of static friction is \(\mu\). The pencil is released and starts to topple forward making an angle \(\theta(t)\) to the vertical. Show that there is a critical value of \(\mu\), say \(\mu_1\), such that

1. [(i)] if \(\mu < \mu_1\) the pencil base slips backwards before \(\cos\theta = 9/11\);
2. [(ii)] if \(\mu > \mu_1\) the pencil base slips forwards at some value of \(\theta\) lying in the range \(\frac{2}{3} > \cos\theta > \frac{1}{3}\).

Find the value of \(\mu_1\).

Particles in a certain system can only have certain given energies \(E_1\), \(E_2\) or \(E_3\). If \(n_i\) particles have energy \(E_i\) (\(i = 1, 2, 3\)) write down conditions that

1. [(i)] the total number of particles is \(N\)
2. [(ii)] the total energy is \(U\).

Hence express two of the \(n_i\) as linear functions of the third. The 'entropy' \(S\) of the system is defined as the stationary value of the function \[W = \log(n_1! n_2! n_3!)\] subject to the above two conditions. Assuming that the \(n_i\) are so large that they may be regarded as continuously varying, show that \(W\) is stationary when \[\log n_i = \alpha+E_i/T\] where \(T\) is the root of the equation \[N\sum E_ie^{E_i/T} = U\sum e^{E_i/T},\] and where \[Ne^{-\alpha} = \sum e^{E_i/T}.\] If we define \(Z(T) = \sum e^{E_i/T}\), show that \[U = -NT^2\frac{d}{dT}\log Z\] and \[S = N\left(\log \frac{N}{Z}-1\right)+\frac{U}{T}.\] If \(U\) varies while \(N\) is fixed, show that \(\frac{dU}{dS} = T\). \(\left[\frac{d}{dn}\log n! \approx \log n \text{ for large } n, \log n! \approx n(\log n-1).\right]\)