A uniform circular ring whose centre is \(O\) is rotating in its own plane with angular velocity \(\omega\) about a fixed point \(A\) on the ring. The point \(A\) is suddenly released and a second point \(B\) on the ring fixed, where \(\angle AOB = \theta\). Find the new angular velocity about \(B\).
A uniform thin rod of length \(2a\) is supported by two small rough pegs at different levels. The upper peg lies above and the lower peg below the rod. The pegs are at a distance \(c( < a)\) apart, and the line joining them makes an angle \(\alpha\) with the horizontal. The coefficient of friction at the upper peg is \(\mu_1\) and at the lower peg \(\mu_2\). Find the greatest value of \(\alpha\) at which equilibrium can be maintained.
Six equal uniform bars, each of weight \(W\), are freely jointed together so as to form a regular hexagon \(ABCDEF\), which hangs from the point \(A\) and is kept in shape by strings \(AC, AD, AE\). Find the tensions in these strings.
Define the ``bending moment'' at a point of a beam, and explain its physical meaning. A curved rod of length \(l\) in the form of an arc of a circle of radius \(R\) has its ends connected by a string in tension. Prove that the bending moment at any point is proportional to \(\sin\frac{\alpha}{2}\sin\frac{\beta}{2}\), where \(\alpha\) and \(\beta\) are the angular distances of the point in question from the ends of the rod. Given that the rod breaks when the bending moment exceeds the value \(M_0\), find the tension in the string when the rod is about to break.
Examine the stability of a plank of thickness \(2a\) which rests horizontally across the top of a fixed horizontal rough cylinder of radius \(a\). What is the effect of giving the plank a slight bow so that it is concave towards the cylinder?
Show that the tensions at two points of a coplanar light string wrapped around a rough cylinder are related by \[ T_2=T_1 e^{\mu\theta} \] when the string is about to slip, where \(\mu\) is the coefficient of friction and \(\theta\) is the angle between the tangents at the two points. Two weights \(P, Q\) hang in limiting equilibrium from a light string which passes over a rough circular cylinder in a plane perpendicular to the axis. If \(P\) be on the point of descending, what is the maximum weight that may be added to \(Q\) without causing it to descend?
A particle \(P\) is projected from a point \(O\) with velocity \(V\). Show that, when the line \(OP\) makes an angle \(\phi\) with the upward vertical through \(O\), the distance \(OP\) cannot exceed \(V^2/\{g(1+\cos\phi)\}\). A gun, of constant muzzle velocity, is sited at a point \(O\) of a plane hillside, which makes an angle \(\alpha\) with the horizontal. The gun can fire in any direction and at any elevation; show that the region of the hillside within range has the shape of an ellipse with focus at \(O\) and eccentricity \(\sin\alpha\).
Three uniform spheres, \(A, B, C\), of masses \(2m, m, 2m\) respectively, lie in a straight line on a horizontal table, with \(B\) between \(A\) and \(C\). Initially \(B\) and \(C\) are at rest, and \(A\) is projected along the line of centres towards \(B\); the coefficient of restitution for any pair of the spheres is \(e\). Show that, if \(e>0\), there will be at least three collisions, and that there will be only three provided that \(e \ge \frac{1}{3}\).
A compound pendulum consists of a plane lamina which can swing about a horizontal axis perpendicular to the plane of the lamina. The axis can be made to pass through either of two points \(A_1, A_2\) in the lamina. The distances of the centre of gravity of the lamina from \(A_1\) and \(A_2\) are \(h_1\) and \(h_2\). The period of small oscillations when swinging about \(A_1\) is \(T_1\), and when swinging about \(A_2\) is \(T_2\). Prove that \[ \frac{8\pi^2}{g} = \frac{T_1^2+T_2^2}{h_1+h_2} + \frac{T_1^2-T_2^2}{h_1-h_2}, \] if \(h_1 \ne h_2\).
A uniform circular disc, of radius \(a\) and mass \(2m\), is freely jointed at a point \(A\) of its circumference to a uniform rod \(AB\) of length \(2a\) and mass \(m\). The system is at rest on a horizontal table, the points \(B, A\) and \(O\) (the centre of the disc) lying in a straight line. An impulse \(J\) is applied at the point \(B\) along a horizontal line perpendicular to \(AB\). Show that the impulsive reaction at \(A\) is parallel to the impulse at \(B\) and is of magnitude \(4J/11\). Find the initial values of the angular velocity and the velocity of the centre of mass of both the rod and the disc, and verify that the system acquires kinetic energy \(\frac{1}{2}JV\), where \(V\) is the initial velocity of the point \(B\).