An aeroplane flying with uniform velocity, not vertically, drops a bomb aimed accurately to hit a fixed gun. The gun fires a shell, aimed accurately to hit the aeroplane. [Aeroplane gun, bomb, and shell are all taken as ideal points, and air resistance is ignored.] You are warned that detailed calculation should be unnecessary.]
Six equal uniform rods, each of weight \(w\), are freely jointed at their ends to form a regular hexagon \(ABCDEF\), the shape of which is maintained by two light rods \(BF\), \(CE\). The hexagon is suspended from the corner \(A\). Find the thrusts in the rods \(BF\), \(CE\).
Two equal uniform planks \(AB\), \(B'A'\), of length \(2l\), rest symmetrically across a rough circular cylinder of radius \(a\), the ends \(B\) and \(B'\) making free contact with each other vertically above the axis of the cylinder which is horizontal. The long edges of the planks are perpendicular to the axis of the cylinder, and are inclined at an angle \(\alpha\) to the horizontal. Show that equilibrium is possible if and only if \[l\cos\alpha(\cos\alpha + \mu\sin\alpha) > a\tan\alpha > l,\] where \(\mu\) is the coefficient of friction between plank and cylinder. Hence, or otherwise, show that there are such positions of equilibrium if \(\mu > l/a\), but not if \(\mu < l/a\).
A rocket without fuel has mass \(M\), and initially carries fuel of mass \(m\). When it is fired the mass of the fuel is ejected at a constant rate \(k\) with a speed \(u\) relative to the rocket. If the rocket is propelled vertically upwards, and forces other than gravity are neglected, find both its speed and the distance it has travelled by the time all the fuel is ejected.
A particle \(P\) of mass \(m\) moves in a hyperbolic orbit under the influence of a radial repulsion \(k/r^2\) from a fixed focus \(O\), where \(r = OP\). The particle starts at a great distance from \(O\) with a speed \(v\) along a line to which the perpendicular from \(O\) has length \(b\). If \(u\) is the speed of the particle when it is closest to \(O\), show from the equations of energy and angular momentum respectively that \[\left(\frac{u}{v}\right)^2 = 1 - \frac{2k\sin\alpha}{mv^2b(1+\cos\alpha)}, \quad \frac{u}{v} = \frac{\sin\alpha}{1+\cos\alpha},\] where \(\alpha\) is the acute angle between the initial direction of motion of the particle and the axis of symmetry of the orbit. Deduce that when the particle has receded a great distance from \(O\) its direction of motion has turned through the angle \[2\tan^{-1}\left(\frac{k}{mv^2b}\right).\]
The rectilinear motion of a particle is governed by the equation \[\frac{d}{dt}\left(\frac{mv}{\sqrt{1-v^2/c^2}}\right) = F,\] where \(m\), \(c\), \(F\) are positive constants, \(v\) is the speed of the particle, and \(t\) is the time. Give expressions in terms of \(t\) for \(v\) and for \(x\) (the distance travelled by the particle) under the condition \(v = x = 0\) at \(t = 0\). Show that \[\frac{mc^2}{\sqrt{1-v^2/c^2}} - Fx = mc^2,\] and that if \(v^2/c^2\) is small \[\frac{1}{2}mv^2 = Fx\] approximately.
A light spring \(ABCD\), of natural length \(3a\) and modulus \(\lambda\), lies on a smooth horizontal table to the surface of which its ends \(A\), \(D\) are fixed at points distant \(3a\) apart. Particles of mass \(m\) are rigidly attached to the points of trisection \(B\), \(C\) of the unstretched spring, and subsequently the particle at \(B\) is moved a distance \(\frac{1}{2}a\) towards that at \(C\), which is held stationary. The system is then released from rest. Determine the subsequent motion.
A uniform circular disc of mass \(m\) and radius \(a\) has a particle of mass \(m\) attached at a point on it distant \(\frac{1}{2}a\) from its centre \(C\). Initially the disc is in equilibrium in a vertical plane, with its rim resting on a rough horizontal surface and the particle vertically below \(C\). An impulse \(J\), directed towards \(C\), is then applied at a point on the rim level with \(C\). Assuming that the disc maintains contact with the surface, and rolls without slipping, show that it makes complete revolutions if \[J > m\sqrt{(7ag/2)}.\] Find the maximum displacement of \(C\) from its initial position in the case \(J < m\sqrt{(7ag/2)}\).
A ribbon of small thickness \(\xi\) is wound on a spool of radius \(a\) which rotates with angular velocity \[\frac{\omega}{1+kx},\] where \(x\) is the length of the ribbon on the spool, and \(\omega\), \(k\) are positive constants. How long is it before the depth of ribbon on the spool is \(a\)?
A uniform solid consists of a hemisphere of radius \(a\) to the base of which is fixed, symmetrically, a circular cylinder of radius \(a\) and length \(\frac{1}{4}a\). The solid rests in equilibrium with its axis vertical, on a rough sphere of radius \(b\). Show that, if the cylindrical part of the solid is uppermost, the equilibrium is stable if, and only if, \[b > \frac{4}{5}a.\]