A uniform hemisphere of mass \(M\) and radius \(a\) rests with its plane face upon a smooth horizontal table. A particle of mass \(m\) falling vertically strikes the hemisphere and rebounds horizontally. After the impact the hemisphere slides along the table. If the coefficient of restitution between the particle and the hemisphere is \(e\) and the effect of friction is neglected, find the height above the table at which the hemisphere is struck.
A particle is released from rest on the surface of a smooth fixed sphere at a point whose angular distance from the highest point is \(\alpha\). Find the point where the particle leaves the surface, and prove that the angular distance of this point from the highest point of the sphere cannot be less than about \(48^\circ\). Show that the radius of curvature of the trajectory of the particle after it leaves the surface is initially equal to the radius of the sphere.
A rigid body rotates without friction about a fixed horizontal axis; the radius of gyration about the axis is \(k\) and the distance of the centre of gravity from the axis is \(h\). Prove that the angular displacement of the body from its position of stable equilibrium varies with time in the same way as that of a simple pendulum, of length \(k^2/h\), set in motion in a corresponding way. A rod \(AB\), not necessarily uniform, is of length \(l\). When it swings about the end \(A\) in a vertical plane the length of the equivalent simple pendulum is \(a\), and when it swings about the other end \(B\) in a vertical plane the length of the equivalent simple pendulum is \(b\). Find the distance of the centre of gravity from \(A\).
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.