A particle of weight \(2W\) is attached to the end \(A\), and a particle of weight \(W\) attached to the end \(B\), of a light rod \(AB\) of length \(2a\). The rod hangs from a point \(O\) by light strings \(AO\), \(BO\), each of length \(b\). Prove that in equilibrium the inclination of the rod to the horizontal is \(\theta\), where \[ \tan \theta = \frac{a}{3\sqrt{(b^2-a^2)}}. \] Find the tension in the string \(AO\) in terms of \(a\), \(b\), and \(W\).
A uniform straight beam \(ABCDE\) of weight \(W\) rests on supports at the same level at \(B\) and \(D\), and weights \(W\) are hung from the end-points \(A\) and \(E\), where \(AB\), \(BC\), \(CD\), \(DE\) are each of length \(a\). Sketch a graph of the bending moment at each point of the beam, and find in particular the bending moment at \(B\) and at \(C\).
A heavy uniform chain of length \(l\) is attached at one end to a point \(A\) at a height \(l\) above a rough horizontal table, where \(l < l\). The point \(A\) is moved parallel to the table with constant velocity \(v\). Show that the length \(s\) of chain which comes clear of the table satisfies \[ s^2 + 2\mu ls = h^2 + 2\mu lh, \] where \(\mu\) is the coefficient of friction.
A system of forces acts in one plane on a rigid body. Prove that, if \(O\) is a fixed point in the plane, a force through a point \(P\) of the plane is equivalent to an equal force through \(O\) and a couple. Hence prove that the system is equivalent either to a single force or to a single couple. Forces \(W\), \(2W\), \(3W\), \(4W\) act along the sides \(AB\), \(BC\), \(CD\), \(DA\) of a square \(ABCD\). Prove that the system is equivalent to a single force, and find its magnitude and direction and the point in which its line of action meets the line \(AB\).
A smooth wire has the shape of a parabola whose latus rectum is of length \(l_0\) and whose axis is vertical and vertex upwards. Two beads \(A\) and \(B\), whose masses are \(m_1\) and \(m_2\), where \(0 < m_1 < m_2\), slide on the wire, and are joined by a light inelastic string of length \(l\), where \(l > 2a\), which passes through a small smooth ring at the focus of the parabola. Prove that the only positions of equilibrium for which the two beads are not on the same side of the vertex are those in which at least one of the beads is at the vertex of the parabola, and determine which positions are stable and which are unstable. How is the problem altered if \(m_1 = m_2\)?
A plane is inclined at an angle \(\alpha\) to the horizontal. Its surface is rough, but not uniformly rough, the coefficient of friction \(\mu\) being proportional to the distance \(r\) from a point \(O\) in the plane, \(\mu = kr\). A particle of mass \(m\) is placed on the plane at \(O\) and released from rest. How far does the particle travel before it comes to rest, and how long is it in motion before it comes to rest? Verify that the work wasted through friction is equal to the potential energy lost.
A car of mass \(m\) moves in a straight line on a level road. It is acted on by a constant propulsive force \(kv^2\), and the motion is opposed by a resisting force \(kv^2\) when the speed is \(v\). Prove that the steady speed at which the car can travel is \(c\), and that if it starts from rest it attains the speed \(v\) when it has travelled a distance \[ \frac{m}{2k} \log \frac{c^2}{c^2-v^2}. \] If the mass of the car is one ton, the steady speed is 60 miles per hour, and the horse-power developed by the engine at this speed is 30, find, correct to the nearest foot, the distance travelled when the car attains a speed of 30 miles per hour. [Assume \(g = 32\) ft. sec.\(^{-2}\).]
A bead of mass \(m\) slides on a smooth wire in the form of a circle of radius \(a\) which is fixed in a vertical plane. The bead is projected from the lowest point of the circle at the instant \(t = 0\) with velocity \(2\sqrt{(ga)}\), and in the subsequent motion the radius from the centre of the circle to the bead makes an angle \(\theta\) with the downward vertical at time \(t\). Prove that \[ \sin \frac{\theta}{2} = \tanh nt, \] where \(n^2 = g/a\). If \(R\) is the reaction of the wire on the bead at any time during the motion, \(R\) being measured towards the centre of the circle, express \(R\) (i) as a function of \(\theta\), and (ii) as a function of \(t\).
The maximum range of a certain gun on a horizontal plane is \(2h\). The gun is placed at the highest point of a hill in the form of a hemisphere of radius \(a\), where \(a > h\). Prove that the area of the part of the surface of the hill which is commanded by the gun is \[ \pi a \{a - \sqrt{(a-4h)}\}^2. \] Examine the limit to which this expression tends as \(a\) tends to infinity.
One point \(O\) of a rigid lamina of mass \(M\) is fixed, and the lamina is free to swing about \(O\), without friction, in a vertical plane. If the lamina executes small oscillations about the position of stable equilibrium, prove that the length of the equivalent simple pendulum is \[ h + \frac{l^2}{h}, \] where \(h\) is the distance of the centre of gravity \(G\) from \(O\), and \(Ml^2\) is the moment of inertia of the lamina about \(G\). If the lamina has uniform surface density, and has the form of an annulus bounded by concentric circles of radii \(a\) and \(b\), where \(a < b\), and if the point of suspension \(O\) can have any position in the annulus, prove that the least possible value for the length of the equivalent simple pendulum is \(\sqrt{2(a^2 + b^2)}\).