State Newton's Second Law of Motion and shew how it leads to the equation \(P=mf\). A pulley of mass \(m\) is connected with a mass \(4m\) by a string which hangs over a fixed smooth pulley; a string with masses \(m\) and \(2m\) at its extremities is hung over the pulley. If the system is free to move, find the acceleration of each of the masses.
Find the velocity of the centre of inertia of two particles whose masses and velocities are given. Shew that the kinetic energy of two masses is equal to that of the sum of the masses moving with the velocity of the centre of inertia together with that of each mass moving with its velocity relative to the centre of inertia.
Find the range of a projectile on an inclined plane through the point of projection. Two particles are projected with velocities \(u, u'\) and elevations \(\theta, \theta'\) from the same point at the same time and in the same vertical plane. Shew that the difference of the times of their passing through the other point common to their paths is \[ 2uu' \sin(\theta \sim \theta') / g(u\cos\theta+u'\cos\theta'). \]
State the principles by which we are enabled to calculate the changes in velocity produced by the impact of two smooth elastic spheres. A smooth elastic sphere falls vertically with velocity \(u\) on a smooth wedge which lies on a smooth table. Calculate the velocity of the wedge after the impact, the wedge and the table being supposed inelastic.
An elastic string of natural length \(a\) has one end fixed and a weight attached to the other. When it hangs vertically it is stretched a length \(c\), and when it revolves as a conical pendulum making \(n\) revolutions per second it is stretched a length \(z\). Prove that \(gz = 4\pi^2 n^2 c(a+z)\).
Find the conditions that \(ax^2+bx+c\) may be positive for all real values of \(x\). Shew that for real values of \(x\) the fraction \((2x^2-5x+2)/(x^2-4x+3)\) assumes all values from \(-\infty\) to \(+\infty\). Draw a graph of the function for all values of \(x\) from \(-\infty\) to \(+\infty\).
Prove that if \((x-b)(x-c)+(x-c)(x-a)+(x-a)(x-b)\) be a perfect square in \(x\), then \(a=b=c\). Determine \(\lambda\) so that \[ (3x+2y-1)(2x+3y-1) + \lambda(x+4y-1)(4x+y-1)=0 \] may be the product of linear factors.
Prove that an infinite series \(u_1+u_2+u_3+\dots\) is convergent or divergent according as when \(n\) tends to infinity \(u_{n+1}/u_n\) tends to a limit less than or greater than unity. State and prove a test for the case in which the limit of \(u_{n+1}/u_n\) is unity. Examine the convergency or divergency of the series whose \(n\)th terms are \(n^4/n!, (n!)^2 x^n/3n!\).
Prove that, if \(\omega\) is an imaginary cube root of unity, then \(1+\omega+\omega^2=0\). Shew how to use the cube roots of unity to find the sum of a series obtained by picking out every third term from a known series; and prove that \[ 1+\frac{x^3}{3!} + \frac{x^6}{6!} + \frac{x^9}{9!} + \dots = \frac{1}{3}\left\{e^x+2e^{-x/2}\cos\frac{\sqrt{3}}{2}x\right\}. \]
Shew, graphically or otherwise, that the cubic equation in \(\theta\), \[ \frac{x^2}{a^2-\theta} + \frac{y^2}{b^2-\theta} + \frac{z^2}{c^2-\theta} = 1, \quad a>b>c, \] has three real roots \(\lambda, \mu, \nu\) which are such that \(a^2>\lambda>b^2>\mu>c^2>\nu\). Also shew that \[ x^2 = \frac{(a^2-\lambda)(a^2-\mu)(a^2-\nu)}{(a^2-b^2)(a^2-c^2)}. \]