A rod of mass \(M\) is free to rotate in a vertical plane about a fixed point \(O\). The moment of inertia of the rod about \(O\) is \(I\), and the distance of the centre of gravity from \(O\) is \(h\). When the rod makes an angle \(\alpha\) with the downward vertical, its angular velocity is \(\omega\). Determine the horizontal and vertical components of the reaction on the hinge when the rod makes an angle \(\theta\) with the downward vertical. Shew that these two components cannot vanish simultaneously in a position in which the rod is not vertical, unless \(I=Mh^2\) and the angular velocity \(\omega\) satisfies the inequalities \[ \frac{2g\cos\alpha}{h} < \omega^2 < \frac{g(2\cos\alpha+3)}{h}. \] If the two components vanish simultaneously when the rod is vertical, prove that \[ \omega^2 = \frac{g}{h}\left[\frac{4Mh^2\cos^2\frac{\alpha}{2}}{I} + 1\right]. \]
A square framework formed of four equal uniform rods each of weight \(W\) is hung up by one corner. The rods are freely jointed at each corner, and a weight \(W\) is suspended from each of the three lower corners. The shape of the square is preserved by means of a light rod along the horizontal diagonal. Prove that the thrust in this rod is \(4W\).
A rod is in equilibrium resting over the rim of a smooth hemispherical bowl fixed with its rim horizontal. One end of the rod rests on the curved surface. Show that the inclination, \(\theta\), of the rod to the horizontal is given by \(4r\cos 2\theta = l\cos\theta\), where \(l\) is the length of the rod and \(r\) the radius of the bowl. Show also that \(l\) must satisfy \[ \frac{2\sqrt{6}}{3}r < l < 4r. \]
A sash window of breadth \(a\), height \(b\), and weight \(W\) hangs in its frame with one of its cords broken. The remaining cord passes over its pulley and is attached to a counterpoise of weight \(W/2\). The window is slightly loose in its frame so that it makes contact only at two opposite corners. Show that the least value for the coefficient of friction between the window and the frame consistent with equilibrium is given by \(\mu=b/a\).
A perfectly rough uniform plank of thickness \(t\) rests horizontally on the top of a fixed circular cylinder of radius \(a\) whose axis is horizontal and perpendicular to the long edges of the plank. Show that equilibrium is stable if \(t<2a\). Investigate the case \(t=2a\). If \(t<2a\), find an equation for the greatest angular displacement which can be made consistent with the plank returning towards the horizontal position if released.
A uniform horizontal beam which is to carry a uniformly distributed load is supported at one end and at one other point. Assuming that the beam will break if the bending moment at any point exceeds a certain value, show that, in order that the greatest possible load may be carried by the beam, the second support must be placed at a point which divides the beam in the ratio \(1:(\sqrt{2}-1)\).
An engine and train of total mass \(M\) move on horizontal rails, the pull of the engine being constant. The total resistance to the motion is equal to \(fv/v_0\) per unit mass, where \(v\) is the speed of the train, \(v_0\) is its maximum speed, and \(f\) is a constant. The train is moving with its maximum speed when the last coach, whose mass is \(m\), is slipped. Show that after a time \(v_0/f\) the rest of the train has moved a distance \[ \frac{v_0^2}{f}\left\{\ln\left(\frac{M}{M-m}\right) - \frac{m}{M}\right\}, \] where \(e\) is the base of Napierian logarithms.
A long light inextensible string passes over a light frictionless pulley and carries a bucket of mass \(M\) at one end and a counterpoise of mass \(M\) at the other. The bucket and counterpoise are in equilibrium when an elastic particle of mass \(m\) is dropped into the bucket so that it hits the horizontal bottom of the bucket with a velocity \(u\). Show that the particle ceases to bounce after a time \[ T = \frac{2eu}{g(1-e)}, \] where \(e\) is the coefficient of restitution between the particle and the bucket. By considering the fact that the total momentum of the system in the direction of the string is unaltered by impulsive tensions in the string, or otherwise, prove that the velocity of the bucket after time \(T\) is \[ \frac{1+e}{1-e}\frac{mu}{2M+m}. \]
A particle is attached by a light inextensible string of length \(a\) to a fixed point. The particle hangs in equilibrium and is then given a horizontal velocity \(\sqrt{(7ga/2)}\). Show that during the subsequent motion the maximum height of the particle above its initial position is \(27a/16\).
A particle of mass \(m\) hangs from a fixed point by an elastic string of natural length \(l\) and modulus of elasticity \(2mg\), and a second particle of the same mass hangs from the first by a similar string. The whole system lies in a vertical plane which rotates with uniform angular velocity \(\omega\) about the vertical through the point of suspension, the strings making constant angles \(\alpha\) and \(\beta\) with the vertical. Show that \begin{align*} \tan\alpha &= \frac{l\omega^2}{4g}\{4\tan\alpha+4\sin\alpha+\tan\beta+2\sin\beta\}, \\ \tan\beta &= \frac{l\omega^2}{2g}\{2\tan\alpha+2\sin\alpha+\tan\beta+2\sin\beta\}. \end{align*}