The axis of a right circular cylinder of radius \(a\) passes through the centre of a sphere of radius greater than \(a\). The length of a generator cut off by the sphere is \(2b\). Prove that the volume of the ring lying within the sphere but outside the cylinder is the same as that of a sphere of radius \(b\). Find the surface area of the same ring.
Explain what is meant by the statement that two systems of forces acting on a rigid body are equivalent, and show that any system of forces whose lines of action all lie in a plane is equivalent either to a single force or to a couple. Four distinct points \(A\), \(B\), \(C\), \(D\) lie in a plane, no three of them being collinear. Forces whose magnitudes are proportional to the lengths of the sides \(AB\), \(BC\), \(CD\), \(DA\) of the quadrilateral \(ABCD\) act along the lines \(BA\), \(BC\), \(DC\), \(DA\), respectively. Show that the system is in equilibrium if \(ABCD\) is a parallelogram, but otherwise is equivalent to a single non-zero force acting in the line joining the mid-points of the lines \(AC\), \(BD\).
A rope attached to a ship is wound a number of times round a bollard on a quay. Obtain from first principles an equation which explains why a small pull on the free end of the rope is sufficient to sustain a large pull by the ship on the other end. Calculate to 2 significant figures the value of the coefficient of friction between the rope and the bollard, if the smallest force sufficient to sustain a pull of 20 tons weight by the ship is 14 lb. wt. when the rope is wound 3 times round the bollard. [The weight of the rope is to be neglected.]
A thin straight heavy beam passes through a number of fixed rings in a horizontal line and rests in equilibrium. Derive differential equations connecting the shearing force and bending moment in the beam and the weight of the beam per unit length (which is not necessarily constant along the beam). What happens to the shearing force and bending moment at one of the rings? Show that, if the beam is supported by two rings only and the centre of gravity of the beam does not lie between them, then the greatest numerical value of the bending moment is attained at the ring nearer to the centre of gravity. Show also that this need not be so if the centre of gravity lies between the rings.
A thin uniform circular disc of radius \(r\) and mass \(6m\) is attached along a diameter to a thin uniform straight rod of length \(6r\) and mass \(m\). The compound pendulum so formed is suspended from one end and can oscillate about a smooth horizontal axis whose direction is normal to the plane of the disc. It is found that the length of the equivalent simple pendulum equals the distance of the centre of the disc from the axis. Find this length. A clock governed by this pendulum (making small oscillations) loses 1 min. every 24 hr. Show that this loss can be corrected by moving the disc along the rod a distance approximately 0.0076\(r\).
A particle of mass \(m\) is attached to two light springs each of natural length \(2l\). The other ends of the springs are fixed at points \(A\), \(B\) distance \(6l\) apart on a rough horizontal table. The springs are such that the tension in either when it is extended to twice its natural length is \(Mg\), and the coefficient of friction between the particle and the table is \(\mu\). Show that, if \(m\mu < 3M\), the particle can rest in equilibrium at any point of the line \(AB\) whose distance from its mid-point does not exceed \(d = (\mu m/M)l\). Show also that, if it is released from rest at a point on \(AB\) distant \(3d\) from \(C\), where \(1 < 3d < 6l\), the particle will eventually rest in equilibrium after oscillating for a time \(\pi\mu/\omega\), where \(\omega^2 = \mu g/d\) and the integer \(n\) is defined by \(2n - 1 < \lambda \leq 2n + 1\). [The springs are assumed to satisfy Hooke's law.]
A bead of mass \(m\) is free to move on a smooth circular wire of radius \(r\) which is fixed in a vertical plane. A light, perfectly elastic string of natural length \(r\) has one end attached to the bead and the other fixed to the highest point \(P\) of the wire. The tension in the string when the bead is at the lowest point \(Q\) of the wire is \(T_0\). Show that there are positions of equilibrium other than \(P\) and \(Q\) if and only if \(T_0 > 2mg\), and that in this case equilibrium at \(Q\) is unstable. In the particular case \(T_0 = 4mg\) the bead is slightly disturbed from rest at \(Q\). Show that in the subsequent motion the string becomes slack, and find the reaction of the wire on the bead at the instant when this occurs.
A shell of mass \(2m\) is fired vertically upwards with velocity \(v\) from a point on a level stretch of ground. When it reaches the top of its trajectory it is split into two equal fragments by an explosion, which supplies kinetic energy amounting to \(mv^2\) to the system but leaves its momentum unchanged. Show that the greatest possible distance between the points where the two fragments hit the ground is \(2v^2/g\) if \(u \leq v\), and \((u^2 + v^2)/g\) if \(u > v\).
A rocket, whose initial mass is \((M + m)\), contains a mass \(m\) of propellant fuel. This is ejected at a constant velocity \(V\) relative to the rocket at a rate of \(\mu\) per sec. What are the conditions that the rocket (a) rises immediately; (b) rises at all? Assuming that it rises immediately show that its maximum upward velocity is $$V \log \left( 1 + \frac{m}{M} \right) - gm/\mu.$$ What is the maximum height attained? [Variation of gravity with height may be neglected.]
Two scale pans each of mass \(M\) hang in equilibrium at opposite ends of a string passing over a pulley. A freely falling particle of mass \(m\) strikes one pan and its velocity at the moment of impact is \(V\). Show that it comes to rest in the pan after a time $$\frac{2eV}{(1-e)g},$$ where \(e\) is the coefficient of restitution between the particle and the pan. What is the velocity of the pan at this instant?