Problems

State carefully the principle of virtual work. Illustrate the applications of its converse by solving the following problems:

1. Two uniform rods \(AB, BC\), of weights \(W_1\) and \(W_2\) respectively, are freely hinged together at \(B\), and to a wall at \(A, C\), so that \(A\) is vertically above \(C\). The rod \(AB\) is horizontal and the angle \(ABC\) is equal to \(\beta\). Find the horizontal and vertical components of the actions at \(A\) and \(B\).
2. A smooth paraboloid of revolution, the latus rectum of whose generating parabola is \(4a\), is fixed with its axis vertical and its vertex uppermost. An inextensible chain, of length \(2\pi r\) and of mass \(M\), in the form of a horizontal circle rotates with angular velocity \(\omega\), in contact with the paraboloid. Shew that the tension in the chain is \(Mr(g+2a\omega^2)/4\pi a\).

A light inextensible thread is wound on a reel, which may be considered as a uniform circular cylinder of mass \(M\); to the free end of the thread is fastened a mass \(M\), which lies on a smooth horizontal table. The portion of thread on the table is taut, and is perpendicular to an edge of the table. The reel is held below this edge with the portion of the string between the reel and the table vertical, and the system released from rest. Shew that the reel descends vertically. Shew also that the tension in the thread throughout the motion is \(\frac{1}{4}Mg\).

One end of a light inextensible string \(OAB\), in which \(OA=a, AB=b\), is fixed at \(O\), and masses \(m, M\) are carried at \(A, B\) respectively. If the masses move in a vertical plane with the strings taut, write down the equations of motion of the particles in terms of the angles \(\theta, \phi\) which \(OA\) and \(AB\) make with the vertical. If \(m=M\) and the system is released from rest with \(\theta=\alpha, \phi=\alpha+30^\circ\), where \(0 < \alpha < 60^\circ\), shew that the initial tensions in \(OA\) and \(AB\) are \(\frac{8}{5}mg \cos\alpha\) and \(\frac{2\sqrt{3}}{5} mg\cos\alpha\) respectively. Find also the initial angular acceleration of \(OA\).

A particle, moving under gravity, is resisted by a frictional force which acts in the opposite direction to the velocity and is a function of the speed \((g)\) only. Shew from the equations of horizontal and vertical motion that the horizontal velocity continually diminishes, and that \[ \frac{d}{dx}\left(\frac{v}{u}\right) = -\frac{1}{2}\frac{d}{dy}\left(\frac{v^2}{u^2}\right) = -\frac{g}{u^2}, \] where \(x\) is measured horizontally and \(y\) vertically upwards, and \((u,v)\) are the components of velocity in these directions. Deduce that

1. the path is always concave downwards;
2. if the particle is projected from a point of a horizontal plane the angle of impact with the plane is greater than the angle of projection.

Shew that a plane system of forces acting on a rigid body is equivalent either to a single force or to a couple. Forces of magnitudes 1, 1, 2, 2 act on a rigid body in the sides \(AB, BC, CD, DA\) of a square \(ABCD\). Shew that the system is equivalent to a single force, and find its magnitude and line of action.

Three uniform heavy rods \(AB, BC, CA\) of lengths 3, 4, 5 feet, and weights \(3W, 4W, 5W\), are freely jointed together at their ends to form a triangular framework. The framework is suspended by a string attached to a point \(H\) of \(AC\), and \(AC\) is horizontal. Prove that the length \(AH\) is 12/5 feet. Find the horizontal and vertical components of the reaction at \(A\) on the rod \(AB\).

Four uniform rods \(AB, BC, CD, DE\), each of length \(2a\) and weight \(w\), are freely hinged together at \(B, C\) and \(D\), and the chain hangs in equilibrium from two supports at the same level to which the ends \(A\) and \(E\) are freely attached. If the rods \(AB, BC\) make angles \(\theta, \phi\) with the horizontal, prove that \(\tan\theta = 3\tan\phi\). If the horizontal component of the force exerted by a support is \(3w/2\), prove that the distance \(AE\) is about \(6.6a\).

A bead of mass \(m\) slides on a smooth circular hoop which is fixed in a vertical plane, and the bead is attached to a particle of mass \(M\) by a light inelastic string passing through a small smooth ring which is fixed at the highest point of the hoop. Shew that the potential energy of the system is \[ C-mga\left(x - \frac{M^2}{m^2} \frac{1}{x}\right), \] where \(a\) is the radius of the circle, and \(ax\) is the distance of the bead from the ring. Shew that, if \(M \le 2m\), there is a position of unstable equilibrium in which \(x=M/m\). Find the positions of equilibrium, and discuss their stability, if \(M>2m\).

Find the form of a uniform flexible inelastic string which hangs at rest under the action of gravity with its ends attached to fixed supports. The ends of a uniform string of length \(2na\) are held at the same level at a distance \(2a\) apart. Prove that the depth of the lowest point of the string below the level of the supports is \[ na - \frac{1-e^{-\theta}}{\theta}a, \] where \(\theta\) is defined by the equation \[ \sinh\theta = n\theta. \]

A particle moves with constant acceleration on a straight line. Shew that the velocity at the middle of any time-interval is equal (i) to the mean velocity in the interval, (ii) to the average of the velocities at the beginning and end of the interval. A particle moving on a straight line travels distances \(AB, BC, CD\) of lengths 44, 51, 40 feet in three successive intervals of 2, 3, 4 seconds. Shew that these observations are consistent with the hypothesis that the particle has constant retardation, and, on this hypothesis, find the distance from \(D\) to the point \(E\) where the particle comes to rest, and find the time taken in the motion from \(D\) to \(E\).