My house lies between two bus stops, one of which lies 90 yards to the right and one 270 yards to the left. If I catch a bus at the left-hand stop it costs me 6 pence, and if I catch it at the right-hand stop it will cost me 7 pence and if I miss the bus I must take a taxi which will cost 20 pence. The bus comes from the the right and comes to a stop at a point 90 yards further away from my house than the bus stop. I reckon to walk 2 yards a second until I see the bus and then to run at 6 yards a second. The bus travels at 15 yards a second until it reaches the first bus stop where it waits for 3 seconds and then goes round a corner out of sight. When I leave the house there is no bus in sight, and I reckon that it does not matter which stop I go to. How frequent are the buses?
A farmer wishes to provide his cattle with three nutrients \(A, B\) and \(C\), for which the minimum requirements of 21, 9 and 12 units respectively. Two animal foods \(F_1\) and \(F_2\) are available; their content for unit cost are given in the following table.
A heavy horizontal carriageway of uniform weight \(w\) per unit length is suspended from a heavy flexible wire attached to two pillars a distance \(2d\) apart. The weight of the wire per unit length at any point is chosen to be \(k\) times the tension it has to sustain. Assuming that the carriageway acts as a continuous vertical load on the wire, and that \(kd < \pi\), show that the vertical load on each pillar is given by \(T_0\beta\tan\beta kd\) where \(T_0\) is the minimum tension in the wire and \(\beta^2 = (w+T_0k)/T_0k\).
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
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).