Explain how necessary and sufficient conditions for the equilibrium of a coplanar system of forces can be expressed in terms of (i) Force Components, (ii) Moments. \par A smooth sphere of radius \(a\) and weight \(W\) is suspended from a fixed point \(O\) by a light inextensible string of length \(l\) attached to a point on its surface. Weights \(W_1\) and \(W_2\) are attached to smooth inextensible strings suspended from \(O\) and hanging over the surface of the sphere on opposite sides in the same vertical plane containing the first string. \par Determine the inclination of the first string to the vertical.
Explain what is meant by a couple acting on a body and define the moment of a couple. From your definitions establish the equivalence of coplanar couples of equal moment. \par A rough uniform rod of weight \(W\) lies on a horizontal table. The coefficient of friction is \(\mu\). \par Find the least value of a horizontal force applied at one end of the rod in a direction perpendicular to it which will disturb equilibrium.
A uniform heavy rod rests in equilibrium with its ends supported by rings which can slide on a rough circular wire held fixed in a vertical plane. The rod subtends an angle \(2\alpha\) at the centre. The angle of friction is \(\lambda\). Show that the greatest possible inclination of the rod to the horizontal is given by \(\cot\theta = \cot 2\alpha + \cos 2\alpha \csc 2\lambda\), provided the right-hand side is positive. Explain what conclusion is to be drawn if the right-hand side is either zero or negative.
Three rigid uniform rods \(AB, BC, CD\) are of unequal length and their weights are \(W, W'\) and \(W\) respectively. They are smoothly jointed at \(B\) and \(C\) and hang from smooth pivots \(A\) and \(D\) not necessarily on the same horizontal level. Shew that in the position of equilibrium the mid-point of \(BC\) is in its lowest possible position.
Two particles each of mass \(m\) are connected by a light inextensible string passing through a hole in a smooth horizontal table. One particle moves on the table; the other hangs below from the hole. When the lower particle is instantaneously at rest the other particle has speed \(v\) perpendicular to the string. Assuming that the string is of sufficient length, shew that in the subsequent motion the lower particle moves between two levels, one of which is the initial one.
Shew that the loss of energy due to impact of two smooth uniform spheres moving in the same straight line is proportional to the energy of their motion before impact relative to their joint centre of mass. \par Shew also that the maximum deviation which can be produced in the direction of motion of a smooth sphere by any collision with an equal stationary sphere is \[ \tan^{-1} \left\{ (1+e) (8-8e)^{-\frac{1}{2}} \right\}, \] where \(e\) is the coefficient of restitution.
A particle is projected vertically upwards with speed \(u\). Assuming that the particle encounters resistance per unit mass of \(kV^2\), where \(V\) is its speed, find the height to which it will rise and shew that its speed on passing the initial position is \[ u \left( 1 + \frac{ku^2}{g} \right)^{-\frac{1}{2}}. \]
A smooth tube is constrained to rotate with constant angular velocity in a horizontal plane about a point of itself. A particle is attached to the end of a light elastic string of natural length \(a\), the other end of which is attached to the tube at the centre of rotation. It is found that the particle can rest in relative equilibrium at a distance \(2a\) from the centre. Shew that if the particle is released from relative rest at a distance \(a\) from the centre, the greatest distance it attains subsequently is \(3a\), assuming that the tube is of sufficient length.
Two uniform smooth rods each of length \(2a\) and mass \(M\) are smoothly jointed together and move on a smooth horizontal table. Initially they are in the same straight line and have a velocity \(u\) in a direction perpendicular to their length. \par A point at a distance \(a\) from the end of one rod is seized and held fixed with the rod free to rotate about it. Find the subsequent instantaneous motion and determine the loss of kinetic energy.
A rough circular wire is held fixed in a vertical plane. A bead on the wire is released from rest at the point where the radius is horizontal. Shew that the particle will not reach the lowest point of the wire unless \(1-2\mu^2 > 3\mu e^{-\mu\pi}\), where \(\mu\) is the coefficient of friction.