A point moves in the plane and its position in polar co-ordinates \((r(t), \theta(t))\) is given by \[\frac{d^2r}{dt^2} - r\left(\frac{d\theta}{dt}\right)^2 = -f(r),\] \[r^2\frac{d\theta}{dt} = h,\] where \(h\) is a constant and \(f\) a given function. Show that if \(u = 1/r\), these equations can be written in the form \[\frac{d^2u}{d\theta^2} + u = \frac{1}{h^2u^2}f\left(\frac{1}{u}\right). \tag{*}\] Solve \((*)\) in the cases (i) \(f(r) = 1/r^2\), (ii) \(f(r) = 1/r^3\).
A home-made roulette wheel is divided into 16 sections which are coloured red and black alternately and labelled with the numbers between 1 and 16. The red sections are numbered consecutively with the odd numbers in a clockwise direction, and the black sections are numbered in the opposite direction with the even numbers starting with the number 2 between the numbers 15 and 1. Unfortunately, the wheel is not true, and the probability that the ball lands in the quarter between the numbers 15 and 16 is twice that of each of the adjacent quarters and four times that of the opposite quarter. The probability is uniform within each quarter. Successive rolls are independent. What is the probability that the sum of two consecutive rolls is 28? Given that the sum of two consecutive rolls is 28, what is the probability that the ball landed in a black section both times? Find particular values of \(n\) such that, given that the sum of two successive rolls is \(n\), the probability that the ball has landed in black sections both times is (i) 0, (ii) \(\frac{1}{2}\), (iii) 1.
Balls are drawn successively at random without replacement from a box containing \(R\) red balls and \(B\) blue ones. Find the probability that the number of balls to be drawn in order to obtain \(r\) red ones (\(r \leq R\)) should be \(n\).
Let \(X_1, X_2, \ldots, X_n\) be independent random variables each uniformly distributed on the interval \((0,1)\). Find for \(0 < u < v < 1\) the probability of the event that the smallest of them is between 0 and \(u\) and the largest is between \(u\) and \(v\).
The following is from an advertisement for `X' beer. We've tried our famous `X' Taste Test on twenty beer experts, pouring three glasses, one from the tap, one from the can, and one from the bottle. And then we've asked which is which. Result? No one identified the three correctly. Why? Because all three glasses have the same famous `X' Taste. What confidence can you have in the reasoning in this advertisement?
A large horizontal disc has a toy gun mounted on it in such a way that the barrel of the gun lies in a vertical plane through the centre of the disc and the muzzle of the gun is in the plane of the disc and at a distance \(a\) from its centre. The gun is directed upwards at an angle \(\alpha\) to the horizontal and towards the axis of the disc. The disc is set rotating with angular velocity \(\omega\) about its axis and the gun fires a projectile with velocity \(V\) relative to the gun. Allowing for gravity but ignoring air-resistance, find the value of \(V\) which minimises the distance between the axis of the disc and the point at which the projectile strikes the disc.
A particle which is moving freely under gravity has a perfectly elastic collision with a vertical wall. Show that the path followed after the collision is the mirror image of the path that would have been followed if the wall were absent. A cubical room has a horizontal floor. A particle is projected from a point of the floor at an angle of elevation \(\alpha\), so as to move in a vertical plane parallel to one of the walls. It bounces successively off a wall, the ceiling, and the opposite wall, and then strikes the floor at the point of projection. All the bounces are perfectly elastic. Show that \(1 < \tan\alpha < 2\).
A hollow cylinder of internal radius \(3a\) is fixed with its axis horizontal. There rests inside it in stable equilibrium a uniform solid cylinder of radius \(a\) and mass \(M\). The axes of the cylinders are parallel and no slipping can occur between them. A particle of mass \(m\), with \(m > M\), is now attached to the top of the inner cylinder. Show that this equilibrium position is no longer stable. If the equilibrium is slightly disturbed, show that the particle touches the outer cylinder in the subsequent motion only if \(m \geq 2M\).
An aeroplane flies at a constant air speed \(v\) around the boundary of a circular airfield. When there is no wind it takes a time \(T\) to complete one circuit of the airfield. Show that when there is a steady wind blowing, whose speed \(u\) is small compared with \(v\), the increase in the time required for one circuit is approximately \(3Tu^2/4v^2\).
A uniform cylinder of radius \(a\) and mass \(M\) rests on horizontal ground with its axis horizontal. A uniform rod of length \(2l\) and mass \(m\) rests against the cylinder and has one end attached to the ground by a smooth hinge. The rod makes an angle \(2\alpha\) with the horizontal such that \(a\cot\alpha < 2l\), and it lies in the vertical plane through the centre of the cylinder which is perpendicular to its axis. The coefficients of friction between the cylinder and the rod, and between the cylinder and the ground, both have value \(\mu\). Show that the system is in equilibrium provided that \(\mu > \tan\alpha\). A force \(P\) is now applied at the centre of the cylinder along a line parallel to the rod and directed away from the hinge. Find the smallest value of \(P\) for which the cylinder will move, on the assumption that slipping occurs first between the cylinder and the rod.