A uniform rod \(AB\) is suspended from a point \(O\) by light inelastic strings \(OA\), \(OB\) attached to its ends. Prove that the tensions in the strings are proportional to their lengths. Examine whether the result can be extended to (i) a uniform triangular lamina \(ABC\) suspended by strings \(OA\), \(OB\), \(OC\); (ii) a uniform polygonal lamina with four or more vertices suspended by strings attached to its vertices.
A uniform elastic ring has weight \(W\), unstretched length \(2\pi r\) and modulus of elasticity \(\lambda\). It rests horizontally around a smooth cone of semi-angle \(\alpha\) of which the axis is vertical. Find the depth below the apex of the cone at which the ring will be in equilibrium.
A rocket containing a mass \(m\) gm. of propellant has a total initial mass of \((M + m)\) gm. The propellant issues from the rocket at a rate \(k\) gm. per sec. with a velocity \(V\) cm./sec. relative to the rocket. The rocket is fired vertically, and gravity may be assumed to be constant. What will be (i) the height of the rocket when the fuel is exhausted; (ii) the maximum vertical velocity of the rocket?
A submarine making 9 knots (304 yd. per min.) due north sights a target on a bearing of 80° at a range of 10,000 yd. Ten minutes later the bearing is 70° and the range is 9000 yd. Assuming that the target's speed and course are constant, the submarine immediately alters course without changing speed so as to approach the target as closely as possible. By accurate drawing or by calculation find the angle from due north of the submarine's new course, and the minimum range to the target.
Starting from Newton's laws of motion, deduce the principle of conservation of momentum for a system of particles. A locomotive of mass \(M\) is attached to a train of \(n\) trucks each of mass \(m\), and initially all the inelastic chain couplings are slack to an extent \(a\) each. The locomotive starts from rest, and the propulsive force exerted by its driving wheels has the constant value \(P\). Neglecting all frictional effects, find the velocity of the train just after the last truck has been jerked into motion, and show that the energy dissipated in the jerks is $$\frac{1}{2}Pna\left[\frac{(n+1)m^2}{M+nm}\right].$$
A portable electric drill contains a motor whose shaft carries a pinion having 15 teeth. The parallel drill spindle carries a gear wheel of 60 teeth and meshes externally with the pinion. The moment of inertia of the motor armature and pinion is \(I_1\), and that of the gear and drill spindle is \(I_2\). When the motor is switched on its torque is \(T\). What torque must be exerted on the casing of the drill to prevent it from turning as the spindle accelerates freely?
A body of mass 40 lb. moves in a straight line under the influence of an applied force which varies with time \(t\) roughly as shown in the figure. The area under the curve for the interval 0 to 10 sec. represents an impulse of 50 lb. sec. and the centroid of this area has abscissa \(t = 6\) sec. If the body has a velocity of 2 ft. per sec. when \(t = 0\), find the distance that it travels during the interval 0 to 10 sec. [A force-time graph is shown with a curved line from 0 to 10 seconds on the x-axis, with peak around 6 seconds]
The top of a light spring is fixed. A weight is attached to the bottom of the spring and causes it to assume a static deflection \(\delta\). A force \(F_0(1 - \cos\omega t)\) is then applied vertically downwards to the weight when it is at rest at \(t = 0\). Calculate the tension in the spring at any subsequent time due to this force, provided that the forcing frequency is not equal to the undamped natural frequency of the system.
A light rod \(AB\) of length \(r\) is hinged at \(A\); a second light rod \(BC\) of length \(nr\) is hinged at \(B\), the point \(C\) being so guided that \(AC\) is always horizontal, while \(ABC\) is in a vertical plane. A mass \(m\) is attached to \(B\) and a mass \(M\) to \(C\). In the equilibrium position, \(AB\) is vertical. Find the natural frequency of oscillation of the system for small displacements of \(AB\) through an angle \(\theta\) with the vertical, and show that it is independent of \(n\). (Neglect all friction.)
A rigid plank of length \(l\), breadth \(b\) and thickness \(h\) is laid across a rough log of radius \(r\) to act as a seesaw. Find the relationship between \(r\) and \(h\) for the plank to rest stably on the log if it is initially placed symmetrically across it. Could you have solved this problem by any other methods? If so, describe them briefly.