State the principle of virtual work, and illustrate its use by solving the following problem: Three equal uniform rods \(AB, BC, CA\), each of weight \(W\), are smoothly jointed together at \(A,B\) and \(C\) and the system hangs freely from \(A\). The mid-points \(D, E\) of \(AB, AC\) respectively are joined by a light string whose tension is \(2W\). Find the horizontal and the vertical components of the reaction at \(B\).
Two uniform planks each of length \(l\) and weight \(W\) are freely hinged to the ground at two points distant \(kl\) apart. A third plank of length \(l\) and weight \(W'\) rests horizontally at its points of trisection on the other two planks when these are equally inclined upwards and towards each other. The coefficient of friction between the planks is \(\mu\). Find an equation giving the greatest possible value of \(k\) and show that this value increases with \(W'\).
A uniform heavy straight tube is supported by an endless light string which passes through the tube (in which it fits loosely) and over a fixed smooth peg. The inside of the tube is rough except that the ends of the tube are perfectly smooth. Find the positions of equilibrium. Also find the positions of equilibrium if the string is cut and the two ends are fastened to the peg.
Prove that a system of coplanar forces is in general equivalent to two forces one of which is given in magnitude, direction and line of action. Forces of magnitude 1, 2, 3, 4 act along the sides \(\vec{AB}, \vec{BC}, \vec{CD}, \vec{DA}\) of a square. The system is equivalent to two forces one of which is of magnitude 3 and acts along \(\vec{BD}\). Find the magnitude and direction of the second force and where its line of action cuts \(AB\) and \(AD\).
A shot from a gun is observed to fall a distance \(d\) short of its target, which is well within range and at the same level as the gun, where \(d\) is small compared to the distance of the target. Show that the elevation of the gun must be increased by \(gd/(2V^2\cos 2\beta)\), approximately, where \(V\) is the muzzle velocity and \(\beta\) is the elevation at which the gun was first fired.
A catapult is formed by holding a particle of mass \(m\) against the mid-point of a light elastic string of natural length \(2l\) and modulus \(\lambda\), whose ends are fixed at a distance \(2l\) apart, and then pulling back horizontally a distance \(\frac34l\). The whole system lies in contact with a smooth horizontal table. Show that when the particle is released it attains a final velocity of \[ \sqrt{(\lambda l/8m)}. \] What difference, if any, does it make if the catapult is made from two elastic strings of length \(l\) and modulus \(\lambda\) joined end to end by a non-elastic connection whose length and mass may be neglected?
A particle of mass \(m\) is placed at the top of the inclined face of a smooth wedge of mass \(M\), height \(h\) and angle \(\alpha\), which rests on a smooth horizontal plane, and is then let go. The particle slides down the face of the wedge and is caught by a small hole at the bottom of the wedge and remains there. Find the final position and state of the system.
A train of mass \(M\) travels along a horizontal track; the resistance to motion is \(kv^2\), where \(v\) is the velocity of the train. Show that, if the engine is assumed to work with constant power \(P\) and to start from rest, then the velocity of the train never exceeds \(\sqrt[3]{(P/k)}\), and that when the velocity is half this amount the distance gone is \((M/3k)\log(8/7)\).
A man of mass \(M\) carrying a hammer of mass \(m\) stands on the circumference of a light circular horizontal platform of radius \(a\) which is free to rotate about its centre. The man swings the hammer in a horizontal circle of radius \(b\) with himself as centre, where \(b< a\). The radius to the man makes an angle \(\phi\) with a fixed direction and the line from the man to the hammer makes an angle \(\theta\) with the same fixed direction. Show that \[ (M+m)a\dot{\phi}^2 + m[b^2\dot{\theta} + ab(\dot{\theta}+\dot{\phi})\cos(\theta-\phi)] = 0. \]
A circle \(A\) of radius \(a\) (\(a>b\)) rotates with angular velocity \(\omega\) about its centre \(O\) which is fixed. A circle \(C\) of radius \(\frac{1}{2}(a-b)\) touches the circle \(A\) and a fixed circle \(B\) of radius \(b\) and centre \(O\). If no slipping occurs between \(A\) and \(C\) or between \(B\) and \(C\), show that the angular velocity of \(C\) is \(\displaystyle\frac{a\omega}{a-b}\) and that the angular velocity of the line of centres is \(\displaystyle\frac{a\omega}{a+b}\).