Problems

Prove that, if a body is in equilibrium under three forces, the lines of action of the three forces are coplanar and either meet in a point or are parallel. The mass per unit length of a ladder increases uniformly from the top to the bottom of the ladder, and is twice as great at the bottom as at the top. The ladder stands on a horizontal plane and rests against a vertical wall. The angle of friction at both ends of the ladder is \(\epsilon\). If the ladder is just about to slip, prove that its inclination \(\theta\) to the horizon is given by \[ \tan\theta = \frac{5}{6}\cot\epsilon - \frac{2}{3}\tan\epsilon. \]

State the principle of virtual work. A weightless tripod, consisting of three legs of equal length \(l\), smoothly jointed at the apex, stands on a smooth horizontal plane. A weight \(W\) hangs from the apex. The tripod is prevented from collapsing by three inextensible strings, each of length \(l/2\), joining the mid-points of the legs. Shew that the tension in each string is \(\displaystyle\frac{\sqrt{2}}{3\sqrt{3}}W\).

Define mechanical advantage and efficiency. Shew that the mechanical advantage in the pulley system shown is twice the efficiency, the radii of the pulleys being \(a\) and \(a/2\), and \(A\) being a fixed point directly below the axis of the upper pulley. The lower pulley runs smoothly on its bearings, whereas the rotation of the upper pulley is opposed by a frictional couple proportional to the pressure of the pulley on its bearings. If \(E\) is the efficiency, shew that the least force \(P\) which will just prevent the weight from slipping down is \(\frac{1}{2}EW\). (Neglect the weight of the upper pulley and assume that the rope does not slip.)

\(PA_1A_2...A_{2n}Q\) is the chain of a suspension bridge. Each of the vertical bars \(A_1B_1, A_2B_2,...A_{2n}B_{2n}\) bears an equal portion of the weight of the roadway. The distances \(B_0B_1, B_1B_2, ... B_{2n}B_{2n+1}\) are all equal. The weights of the chain and bars may be neglected in comparison with the weight of the roadway. By means of a force diagram, or otherwise, shew that the points \(P, A_1, A_2, ... A_{2n}, Q\) lie on a parabola whose axis is vertical. If \(W\) is the total weight of roadway supported by the bars \(A_1B_1, ... A_{2n}B_{2n}\), \(d\) the depth of \(A_nA_{n+1}\) below \(PQ\), and \(l\) the total span of the bridge, shew that the tension in the chain at \(P\) or \(Q\) is \[ \frac{W}{2}\sqrt{1+\frac{(n+1)^2l^2}{4(2n+1)^2d^2}}. \]

The end \(P\) of a straight rod \(PQ\) describes with uniform angular velocity a circle whose centre is \(O\), while the other end \(Q\) moves on a fixed line through \(O\) in the plane of the circle. The end \(Q'\) of an equal straight rod \(PQ'\) moves on the same fixed line through \(O\). Prove that the velocities of \(Q\) and \(Q'\) are in the ratio \(QO:OQ'\).

A battleship is steaming ahead with velocity \(V\). A gun is mounted on the battleship so as to point straight backwards, and is set at an angle of elevation \(\alpha\). If \(v\) is the velocity of projection (relative to the gun), shew that the range is \(\displaystyle\frac{2v}{g}\sin\alpha(v\cos\alpha-V)\); also shew that the angle of elevation for maximum range is \(\cos^{-1}\{(V+\sqrt{V^2+8v^2})/4v\}\).

A particle moves under a force directed towards a fixed point \(O\). Shew that its path lies in a plane and that \(pv\) is constant, where \(v\) is the velocity of the particle at any instant and \(p\) the length of the perpendicular from \(O\) to the tangent to the path. A particle is repelled from a centre of force \(O\) with a force \(\mu r\) per unit mass, where \(r\) is the distance of the particle from \(O\). Shew that, if the particle is projected from a point \(P\) in any direction with velocity \(OP\sqrt{\mu}\), its path is a rectangular hyperbola with \(O\) as centre.

A railway truck is at rest at the foot of an incline of 1 in 70. A second railway truck of equal weight starts from rest at a point 1000 feet up the incline, and runs down under gravity. The trucks collide at the foot of the incline, the coefficient of restitution being \(\frac{1}{3}\). Find how far each truck travels along the level, the frictional resistances for each truck being 16 lbs. wt. per ton, both on the incline and on the level. Where the incline meets the level, the rails are slightly curved, each in a vertical plane, so that there is no vertical impact, and at the instant of collision both trucks are on the level. (Assume that \(g=32\) ft. sec.\(^{-2}\))

State Hooke's law. A mass \(m\) hangs from a fixed point by means of a light spring, which obeys Hooke's law. Shew that, if the mass be given a small vertical displacement, the ensuing motion of the mass is simple harmonic. If \(n\) is the number of oscillations per second in this simple harmonic motion, and if \(l\) is the length of the spring when the system is in equilibrium, find the natural length of the spring, and shew that, when the spring is extended to double its natural length, the tension is \(m(4\pi^2n^2l-g)\).

A rope hangs over a pulley, whose moment of inertia is \(I\), and which is perfectly smooth on its bearings, but perfectly rough to the rope. Two monkeys of equal mass \(m\) hang one on each end of the rope. The monkeys can climb with constant speeds \(u_1\) and \(u_2\) relative to the rope (\(u_1 > u_2\)). Shew that in a race through a height \(h\) the monkey of speed \(u_1\) can give the other monkey any start up to \[ hI(u_1-u_2)/\{(I+ma^2)u_1+ma^2u_2\}, \] where \(a\) is the radius of the pulley. (The system is at rest before the monkeys start climbing.)