A railway truck is at rest on an incline of slope \(\alpha\) with the lower pair of wheels locked. Show that the coefficient of friction \(\mu\) between the wheels and the rails must not be less than \((a+b)/(h+b\cot\alpha)\), where \(h\) is the distance of the centre of gravity of the truck from the plane of the rails, and \(a,b\) are the distances of the centre of gravity from the lower and upper axles measured parallel to the incline.
Explain how the potential energy of a system determines the equilibrium positions of a system and their stability. \(AB\) is the horizontal diameter of a circular wire whose plane is vertical. A bead of mass \(M\) at the lowest point \(C\) can slide on the wire and is attached to two strings which pass through small fixed rings at \(A, B\). To the other ends of the strings are attached equal masses \(m\) which hang freely. Find the potential energy of the system when it is displaced so that the radius to the bead makes an angle \(\theta\) with the vertical. Show that the equilibrium with \(M\) at \(C\) is stable if \(m < M\sqrt{2}\).
Two smooth elastic balls collide with given velocities in given directions; find the transference of momentum. If the balls approach along parallel lines with equal but opposite momenta, show that after oblique impact they move along parallel lines which are further apart than the first pair of lines.
A particle is projected under gravity with velocity \(\sqrt{2ga}\) from a point at a height \(h\) above a level plain. Show that the angle of elevation \(\theta\) for maximum range on the plain is given by \(\tan^2\theta = \displaystyle\frac{a}{a+h}\), and that the maximum range is \(2\sqrt{a(a+h)}\).
Find the horse-power required to lift 1000 gallons of water per minute from a canal 20 feet below and project it from a nozzle whose cross-section is 2 square inches, given that a cubic foot of water weighs \(62\frac{1}{2}\) lbs. and that 1 gallon of water weighs 10 lbs.
Find the acceleration of a particle moving in a circular path. Find the least angle at which a track should be banked at a curve of 500 yards radius, if a car travelling at 60 miles an hour is not to side slip, the coefficient of friction being 0.2. [Tables are provided.]
Show that motion in a straight line under a restoring force proportional to the displacement is the projection on the line of a uniform circular motion. Two light elastic strings are fastened to a particle of mass \(m\) and their other ends fastened to fixed points so that the strings are stretched. The modulus of each is \(\lambda\), their tension \(T\), and in equilibrium their stretched lengths are \(a, b\). If the particle is slightly displaced along the line of the strings, show that the period of a small oscillation is \[ 2\pi \sqrt{\frac{mab}{(T+\lambda)(a+b)}}. \]
Show that the locus of the poles of a given line with respect to a system of coaxal circles is a hyperbola whose asymptotes are respectively parallel to the radical axis and perpendicular to the given line.
Prove that \[ \frac{\sin(x-a_1)\sin(x-a_2)\dots\sin(x-a_n)}{\sin(x-\alpha_1)\sin(x-\alpha_2)\dots\sin(x-\alpha_n)} = \cos(\sum a - \sum\alpha) + \sum_{r=1}^n A_r \cot(x-\alpha_r), \] where \(A_r\) is obtained by substituting \(\alpha_r\) for \(x\) in every factor on the left-hand side except the factor \(\sin(x-\alpha_r)\).
Prove that \[ (a_1b_1+a_2b_2+\dots+a_nb_n)^2 < (a_1^2+a_2^2+\dots+a_n^2)(b_1^2+b_2^2+\dots+b_n^2), \] all the letters denoting real numbers. Prove that if \[ \frac{a_1^2+a_2^2+\dots+a_n^2}{n} \to 0 \] as \(n \to \infty\), then \[ \frac{a_1+a_2+\dots+a_n}{n} \to 0. \]