Define angular velocity. A particle P is projected from a point O freely under gravity. Prove that the angular velocity of the line OP is \(gtON/OP^2\) where P is the position of the particle at time \(t\) after projection and N is the foot of the perpendicular from P on the horizontal plane through O.
State the principle of the conservation of linear momentum. Two particles each of mass \(m\) are attached to a string ABC at A and B. The angle ABC is \(2\pi/3\) and an impulse P is applied at C along BC. Find the velocities with which A and B begin to move. What would the difference be if the particle B were a smooth ring free to slide on the string?
Define simple harmonic motion. Find the potential energy of a particle possessed of such a motion, and also the time of a complete oscillation. A ring P of mass \(m\) slides on a smooth circular wire whose centre is O, and is acted on by an attractive force \(m\mu CP\) towards C, where C is a point within the circle. Initially the ring is at rest at A where O, C, A are collinear and C between O and A, if the ring is slightly disturbed and performs small oscillations about A, prove that the time of a complete oscillation is \(2\pi\sqrt{OA/\mu OC}\).
A uniform wire ABC is bent at B to form two sides of a triangle ABC, and is then hung up by the end A. Show that BC will be horizontal if \[ \sin C = \sqrt{2} \cdot \sin\frac{B}{2}. \]
Two uniform spheres of equal weight but unequal radii a, b are connected by a cord of length \(l\), attached to a point on each surface. They rest in contact, the string hanging over a smooth peg. Show that the two portions of the string make equal angles \[ \sin^{-1}\frac{a+b}{a+b+l} \] with the vertical.
A thin uniform rod passes over one peg and under another, the coefficient of friction between each peg and the rod being \(\mu\). The distance between the pegs is \(a\), and the straight line joining them makes an angle \(\beta\) with the horizontal. Show that equilibrium is not possible unless the length of the rod is greater than \[ \frac{a}{\mu}(\mu+\tan\beta). \]
A moving staircase has a speed of 90 feet per minute, and the vertical rise is 44 feet. 150 people, of average weight 120 lb. and all ascending, use the staircase per minute. Neglecting all other considerations, prove that if the people stand still 24 horse-power are required to maintain the motion of the staircase; and find the horse-power required if the people walk up, relatively to the staircase, at 70 feet per minute.
A scale-pan weighing 1 lb. is attached to a light spiral spring and causes it to extend 2 inches. A 2 lb. weight is then placed in the pan and suddenly released. Find how far the pan will fall, the tension of the spring when the pan is at its lowest point, and the period of oscillation.
A body makes complete revolutions about a fixed horizontal axis, about which its radius of gyration is \(k\), and the centre of gravity of the body is at a distance \(c\) from the axis. If the greatest and least angular velocities are \(p\) per cent. greater and \(p\) per cent. less than a quantity \(\omega\), prove that \[ \omega = \sqrt{\frac{200gc}{k^2}}. \]
If a shot travelling with velocity \(v\) is subject to a retardation \(kv^3\) on account of air resistance, prove that (neglecting gravity) \[ 2kt = \frac{1}{v^2} - \frac{1}{u^2} \] and \[ kx = \frac{1}{v} - \frac{1}{u}, \] where \(u\) is the initial velocity, \(v\) the velocity after \(t\) seconds, and \(x\) the space described in that time.