Problems

The bank of a river whose surface lies in the \((x, y)\)-plane is given by \(y = 0\). The surface current is in the \(x\)-direction and is given by \(ky\). A man who swims steadily at speed \(V\) starts from the point \((0, y_0)\) wishing to reach the point \((0, 0)\). Assuming that \(V > ky_0\), calculate the time it takes him to reach his destination

1. if he arranges to swim so that his path is a straight line;
2. if he swims towards the bank until he reaches it and then swims along the bank;
3. if he always points himself towards his destination.

Show that each time can be written in the form \(T = k^{-1}F(ky_0 V^{-1})\). By means of a series expansion show that, for \(ky_0V^{-1} \ll 1\), method (c) is fastest. \([\sin^{-1}u = u + \frac{1}{6}u^3 + \frac{3}{40}u^5 + \ldots]\)

A block of mass \(M\) rests on a rough horizontal table, and is attached to one end of an unstretched spring of length \(l\) and modulus \(\lambda\). The other end is suddenly put into motion with uniform velocity \(V\) away from the block. The limiting coefficient of static friction \(\mu_s\) is larger than the coefficient of dynamic friction \(\mu_d\). Show that the motion of the block repeats itself every \[2\left\{\left(\frac{(\mu_s-\mu_d)g}{\alpha^2V} + \frac{1}{\alpha}\left[\pi-\tan^{-1}\frac{(\mu_s-\mu_d)g}{2V}\right]\right)\right\}\] units of time, where \(\alpha^2 = \lambda/Ml\). (It may be assumed that the tension in the spring is always positive.)

A uniform sphere of radius \(a\) and mass \(M\) moves under gravity in a vertical plane on the inside of a circular cylinder of radius \(2a\) and mass \(M_1\) which is pivoted about its own fixed horizontal axis. The centre of the sphere moves in a plane perpendicular to this axis. The centre of gravity of the cylinder (which is not of uniform density) is a distance \(a\) from its axis and its radius of gyration about its axis is \(2a\). Let \(\phi\) be the angle by which the cylinder departs from its equilibrium position and \(\theta\) the angle made with the vertical by a line drawn through the centre of the sphere perpendicular to the axis of the cylinder. In terms of \(\theta\) and \(\phi\), what are the equations of motion when the sphere and cylinder are (a) perfectly smooth; (b) perfectly rough? Show that the motion in (a) must, for small disturbances about equilibrium, be periodic with period either \(2\pi\sqrt{(a/g)}\) or \(4\pi\sqrt{(a/g)}\), interpreting the result physically. Explain, by reference to the equations of motion, how periodic motions can arise in case (b).