Problems

The random variables \(X_1, X_2, \ldots, X_n\) are independent and have identical probability distributions. The function \(\phi\) of \(n\) arguments is such that \(\phi(X_1, X_2, \ldots, X_n)\) has expectation \(\mu\) and variance \(\sigma^2\). Furthermore, \(\phi\) is not symmetric, so that there is at least one pair of suffixes \((i, j)\) such that with positive probability \[\phi(X_1, \ldots, X_i, \ldots, X_j, \ldots, X_n) \neq \phi(X_1, \ldots, X_j, \ldots, X_i, \ldots, X_n).\] The symmetrisation \(\psi\) of \(\phi\) is defined by \[\psi(X_1, \ldots, X_n) = \frac{1}{n!}\sum\phi(X_{i_1}, \ldots, X_{i_n})\] where the summation is over all \(n!\) permutations \((i_1, i_2, \ldots, i_n)\) of \((1, 2, \ldots, n)\). Prove that \(\psi(X_1, X_2, \ldots, X_n)\) has expectation \(\mu\) but variance less than \(\sigma^2\). [A simpler version, using exactly the same strategy of proof, has \(n = 2\).]

A massless hoop, of radius \(a\), stands vertically on a rough plane. A weight is attached to the rim of the hoop so that the radius to the weight makes an angle \(\theta_0\) (\(0 \leq \theta_0 < \pi\)) to the upward vertical. In the subsequent motion the hoop remains vertical and rolling occurs without slipping until the vertical reaction at the point of contact with the plane is zero. Show that this occurs when \(\theta = \theta_1\) where \(\frac{1}{2}\pi \leq \theta_1 < \pi\). At the moment when the vertical reaction is zero, the plane is removed. Show that the velocity of the weight when it reaches the former level of the plane is \[2\sqrt{2ag}\cos\left(\frac{\theta_1}{2}\right).\]

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 \(\frac{dU}{dS} = T\). \(\left[\frac{d}{dn}\log n! \approx \log n \text{ for large } n, \log n! \approx n(\log n-1).\right]\)