Problems

Shew that the loss of energy due to impact of two smooth uniform spheres moving in the same straight line is proportional to the energy of their motion before impact relative to their joint centre of mass. \par Shew also that the maximum deviation which can be produced in the direction of motion of a smooth sphere by any collision with an equal stationary sphere is \[ \tan^{-1} \left\{ (1+e) (8-8e)^{-\frac{1}{2}} \right\}, \] where \(e\) is the coefficient of restitution.

A particle is projected vertically upwards with speed \(u\). Assuming that the particle encounters resistance per unit mass of \(kV^2\), where \(V\) is its speed, find the height to which it will rise and shew that its speed on passing the initial position is \[ u \left( 1 + \frac{ku^2}{g} \right)^{-\frac{1}{2}}. \]

A smooth tube is constrained to rotate with constant angular velocity in a horizontal plane about a point of itself. A particle is attached to the end of a light elastic string of natural length \(a\), the other end of which is attached to the tube at the centre of rotation. It is found that the particle can rest in relative equilibrium at a distance \(2a\) from the centre. Shew that if the particle is released from relative rest at a distance \(a\) from the centre, the greatest distance it attains subsequently is \(3a\), assuming that the tube is of sufficient length.

Two uniform smooth rods each of length \(2a\) and mass \(M\) are smoothly jointed together and move on a smooth horizontal table. Initially they are in the same straight line and have a velocity \(u\) in a direction perpendicular to their length. \par A point at a distance \(a\) from the end of one rod is seized and held fixed with the rod free to rotate about it. Find the subsequent instantaneous motion and determine the loss of kinetic energy.

A rough circular wire is held fixed in a vertical plane. A bead on the wire is released from rest at the point where the radius is horizontal. Shew that the particle will not reach the lowest point of the wire unless \(1-2\mu^2 > 3\mu e^{-\mu\pi}\), where \(\mu\) is the coefficient of friction.

Solve the equations \begin{align*} \lambda x + 2y + z &= 2\lambda, \\ 2x + \lambda y + z &= -2, \\ x+y+z &= -1, \end{align*} when \(\lambda\) is not equal to 2 or to 0. \par Shew that the equations have no solution when \(\lambda = 2\), and that they have an infinite number of solutions when \(\lambda = 0\).

The roots of the equation \(x^3 + px + q = 0\) are \(\alpha, \beta, \gamma\), and \(\omega\) is a complex number such that \(\omega^3=1\). If \[ \phi = \alpha + \omega\beta + \omega^2\gamma, \quad \psi = \alpha + \omega^2\beta + \omega\gamma, \] find the values of \(\alpha, \beta, \gamma\) in terms of \(\phi\) and \(\psi\). \par Shew that \[ \phi\psi = -3p, \quad \phi^3+\psi^3 = -27q, \quad (\phi^3-\psi^3)^2 = (27q)^2 + 108p^3. \]

If all the numbers \(a_i, b_i\) and \(c_i\) are positive, and if \(m\) is a positive integer, shew that \[ \left\{ \frac{a_1^m c_1 + a_2^m c_2 + \dots + a_n^m c_n}{b_1^m c_1 + b_2^m c_2 + \dots + b_n^m c_n} \right\}^{1/m} \] lies between the greatest and the least of the numbers \[ \frac{a_1}{b_1}, \frac{a_2}{b_2}, \dots, \frac{a_n}{b_n}. \]

(i) Shew that, if \(x > 0\), then \(x^{1/n} \to 1\) as \(n \to \infty\). \par (ii) Shew that, if \(a>0\) and \(a_n \to a\), then \(\sqrt{a_n} \to \sqrt{a}\).

Define the modulus \(|z|\) of the complex number \(z\). \par Shew that \(|z_1+z_2| \leq |z_1|+|z_2|\), and give the geometrical interpretation. \par Shew that, if \(z \neq 1\), then \(\left| \frac{1}{1-z} \right| \leq \frac{1}{|1-|z||}\). \par Shew that, if \(|z-1|+|z+1| = 2a\), where \(a>1\), then \(|z| \leq a\).