A uniform chain passes over a small smooth peg fixed at a height \(h\) above the edge of a table. From one side of the peg a length \(h+x\) of the chain hangs clear of the table. From the other side of the peg a length \(h\) hangs vertically, and the rest of the chain is heaped on the table below the peg. Assuming that the links are jerked into motion one by one, find a differential equation for \(x\) as a function of the time.
Forces proportional to the sides of a convex polygon are applied (a) along the sides in the same sense round the polygon, (b) at the middle points of the sides and perpendicular to them, all being directed inwards. Show that the forces in case (a) reduce to a couple proportional to the area of the polygon, and in case (b) are in equilibrium.
Two thin rods \(AB, BC\) are fixed together at \(B\), the angle \(ABC\) being \(105^\circ\). The rods are in a vertical plane with \(B\) below \(A\) and \(C\), the rod \(AB\) being inclined to the horizontal at \(45^\circ\). A uniform thin rod \(XY\) of mass \(M\) is in equilibrium with its ends \(X, Y\) attached to \(AB\) and \(BC\) respectively by light smooth rings. Determine whether the equilibrium is stable or unstable.
Show that the work done in stretching an elastic string \(AB\), of natural length \(l\) and modulus \(\lambda\), from tension \(T_1\) to tension \(T_2\) is \[ \frac{l}{2\lambda}(T_2^2 - T_1^2). \] A weight \(w\) is attached at \(B\) and weights \(w/n\) are attached at each of the points \(A_1, A_2, \dots, A_n\), where in the unstretched position \[ AA_1 = A_1A_2 = \dots = A_{n-1}A_n = A_nB. \] Show that the potential energy of the string in the equilibrium position when it is suspended from \(A\) is \(lw^2(14n+1)/(12\lambda n)\).
A frame formed of four equal light rods, each of length \(a\), freely jointed at \(A, B, C, D\), is suspended at \(A\). A particle of weight \(w\) is suspended from \(B\) and \(D\) by two strings each of length \(l\), where \(l > a/\sqrt{2}\). The frame is kept in the form of a square by a string along the diagonal \(AC\). Apply the method of virtual work to find the tension in \(AC\). In particular show that when \(l=\sqrt{5}a\), \(T=\frac{2}{3}w\).
A point \(A\) is vertically above \(B\), and \(AB=l\). The ends of a string \(ACB\) of length \(2l\) are fixed at \(A, B\). A heavy bead \(C\), which can slide freely on the string, describes a horizontal circle with angular velocity \(\omega\) about \(AB\). The plane in which \(C\) moves is at depth \(y\) below \(A\). Show that \[ y = \frac{4}{3}l + \frac{1}{3}g\omega^{-2}. \]
A particle is projected at time \(t=0\) in a fixed vertical plane from a given point \(S\) with given velocity \(\sqrt{(2ga)}\), of which the upward vertical component is \(v\). Show that at time \(t=2a/v\) the particle is on a fixed parabola independent of \(v\), that its path touches this parabola, and that its direction of motion is then perpendicular to its direction of projection.
Two uniform smooth spheres of equal mass experience an elastic collision (coefficient of restitution equal to unity). Initially one of the spheres is at rest but free to move in any direction. Show that after the collision the directions of motion of the two spheres are at right angles to each other. Discuss the special case of a head-on elastic collision. Show further that, if the mass of the sphere initially in motion is the less in the ratio \(1-\epsilon : 1\) (\(\epsilon\) small), and if \(\theta\) is the deflection in its motion, then, to the first order in \(\epsilon\), the angle between the directions of motion after collision exceeds a right angle by \[ \frac{1}{2}\epsilon\tan\theta. \]
A particle is projected from a point \(P\) in an attractive field of force \(\mu/r^5\), where \(r\) is the distance from the fixed centre of attraction \(O\). Show that when the velocity of projection is \((\mu)^{1/2}/OP^2\) the orbit is a circle passing through \(O\).
A uniform rod \(AB\) of mass \(2M\) and length \(2a\) is smoothly hinged at its end \(B\) to a point on the rim of a uniform circular disk of mass \(M\) and radius \(r\). The rod and disk are laid on a smooth horizontal table so that the direction of \(AB\) passes through the centre of the disk. A horizontal impulse \(P\) is applied at \(A\) at right angles to \(AB\). Show that the kinetic energy produced is \(\frac{9}{10}\frac{P^2}{M}\) and that the impulsive reaction at the hinge is \(P/5\).