State the principle of virtual work, and explain how it may be applied to determine the unknown reactions of a system. A square \(ABCD\) formed of light rods, loosely jointed, has the side \(AB\) fixed. The middle points of \(AB, BC\) are joined by a string which is kept taut by a force \(P\) acting at the middle point of \(AD\) parallel to \(AB\). Shew that the tension of the string is equal to \(P\sqrt{2}\).
Two equal particles are connected by a string 5 feet long and lie close together at the edge of a window ledge 63 feet from the ground. One of them is pushed gently over the edge. Find the time it will take to reach the ground.
The energy of 1 lb. of powder is 75 foot-tons. Shew that the weight of charge necessary to produce an initial velocity of 1500 feet per second in a projectile weighing 600 lbs. is at least 125 lbs. (Neglect the recoil of the gun.)
A particle moves in a circle of radius \(a\) with constant angular velocity \(\omega\). Shew that the acceleration is directed towards the centre and is equal to \(a\omega^2\). An elastic string of unstretched length \(a\) is stretched by an amount \(b\) when it supports a certain mass at rest. Shew that when rotating steadily at the rate of \(n\) revolutions per second round the vertical through one end (the same mass being attached to the other end) the inclination to the vertical is \[ \cos^{-1}\left(\frac{1}{a}\left(\frac{g}{4\pi^2 n^2}-b\right)\right). \] Discuss this result when \(2\pi n > \sqrt{(g/b)}\).
A particle is projected from a point on the ground at the centre of a circular wall of radius \(a\) and height \(h\). Shew that the least velocity of projection which will enable the particle to clear the wall is \(\sqrt{g\{h+\sqrt{(a^2+h^2)}\}}\).
Prove that the motion of a particle suspended from a fixed point by an inelastic string oscillating through a small angle is approximately simple harmonic. A simple pendulum 4 feet long swings through an arc of 3 inches. Find the time of a complete oscillation and the velocity at a distance of 1 inch from the centre of the arc.
A system of coplanar forces will reduce in general to a single force or a couple. Prove this and mention the exceptional case. Forces \(P, Q, R\) act along the sides \(BC, CA, AB\) of a triangle. Find the conditions that their resultant should be parallel to \(BC\) and determine its magnitude.
A rigid plane framework of five jointed bars forming two equilateral triangles \(BAC, CDA\) is in equilibrium under the action of three forces on the joints at \(A, B, D\). Prove that the directions of these forces must be parallel or concurrent lying in the plane of the framework. If the force at \(B\) be 4 cwts. acting perpendicularly to \(CD\) and the force at \(A\) be in the direction \(BA\), find the force at \(D\) and the stresses in the bars by a graphical method.
Define the moment of a force about (1) a point, (2) a straight line. A fixed smooth axis, inclined to the horizon at an angle \(\alpha\), runs through a heavy body so that it is free to slide along and to turn round the axis. If the body be kept in equilibrium by a vertical force acting on it, find the conditions of equilibrium.
Investigate the position of the centre of gravity of a homogeneous solid hemisphere. Find the centre of gravity of a semi-circular plate of radius \(a\) and of uniform small thickness such that the density at any point varies as \(\sqrt{a^2-r^2}\) where \(r\) is the distance of the point from the centre.