A light helical spring stands in a vertical position on a table: a mass is placed on the top of the spring and compresses it 2 inches. The mass is then pushed down a further distance of 3 inches and released. Find the velocity with which the mass is shot off the spring and the time that elapses before this happens, measured from the instant of release.
A disc is rotated about its axis, which is vertical, from rest with uniform angular acceleration \(\alpha\). A particle rests on it at distance \(a\) from the centre of the disc. The coefficient of friction between disc and particle is \(\mu\). After a time the particle slips; when does this happen and in what direction over the disc does slipping begin?
Prove that every recurring simple continued fraction of the form \[ a_1 + \frac{1}{a_2 +} \dots \frac{1}{a_n +} \frac{1}{b_1 +} \dots \frac{1}{b_k +} \frac{1}{b_1 +} \dots, \] in which the \(a\)'s and \(b\)'s are positive integers, is the root of a quadratic equation \[ Lx^2 + Mx + N = 0, \] in which \(L, M, N\) are integers, and show that if there are no non-recurring quotients, \(M\) cannot vanish. Evaluate \[ 1 + \frac{1}{2+} \frac{1}{1+} \frac{1}{1+} \frac{1}{2+} \frac{1}{1+} \frac{1}{1+} \dots, \] in which the recurring sequence of quotients is 1, 2, 1.
Establish the principle of virtual work; and give an account of its application to determine the conditions of equilibrium of a system of rigid bodies.
A semicircular track is made on a hillside, which is inclined at 20° to the horizontal, so that the diameter is a line of greatest slope. Draw a graph showing how the inclination of the track to the horizontal varies, and hence find the average inclination for small equidistant intervals round it.
An electric train weighing 150 tons is running down a gradient of 1 in 100 at a speed of 15 feet per sec. and with an acceleration of 0.5 feet per sec. per sec. The frictional resistance to motion may be taken as 30 lbs. wt. per ton of the train. At the instant under consideration the supply point, at which the voltage between the two current rails is kept constant at 500 volts, is 2000 yards distant. If each current rail has a resistance of 0.025 ohm per 1000 yards and the motors of the train have an efficiency of 80\%, find the current taken from the rails and the voltage between them at the train.
A 50-ton locomotive starts from rest with a 10-ton truck, the coupling chain being initially slack. Immediately before the coupling becomes taut steam is cut off and the engine is moving at 1.5 ft. per sec. Find the maximum tension in the coupling chain (supposed inextensible) if each coupling hook is attached to its vehicle by a spring which extends \(\frac{1}{8}\) inch per ton of pull.
Prove the identity \[ \cos \frac{\pi}{11} + \cos \frac{3\pi}{11} + \cos \frac{5\pi}{11} + \cos \frac{7\pi}{11} + \cos \frac{9\pi}{11} = \frac{1}{2}. \] Sum the series (\(n\) terms) \[ \sin\theta + 2 \sin 2\theta + 3 \sin 3\theta + \dots + n \sin n\theta. \]
State Newton's Laws of Motion, and shew how some of the fundamental theorems of Statics are involved in them. In particular, deduce the principles of the independence of forces, and of the transmissibility of force in a rigid body. Establish the relations which obtain between the change of motion of a body moving in a straight line and (1) the time average, (2) the space average, of the force acting on it. If \(v\) is the velocity of the body, and \(s\) its distance from a fixed point in the line of motion, shew that the time taken to travel any distance can be deduced from the graph in which \(1/v\) is taken as ordinate and \(s\) as abscissa.
Find the equations of the tangent and normal at the point \((am^2, 2am)\) on the parabola \(y^2 = 4ax\); find also the coordinates of the point where the normal meets the parabola again.