A particle moves in a straight line under a force \(F\), its mass increasing by picking up matter whose previous velocity was \(u\). If the mass and velocity of the particle at time \(t\) are \(m\) and \(v\), respectively, show that \[\frac{d}{dt}(mv) - \frac{dm}{dt}u = F.\] A particle whose mass at time \(t\) is \(m_0(1 + at)\) is projected vertically upwards under gravity at time \(t = 0\) with velocity \(V\), the added mass being picked up from rest. Show that it rises to a height \[\frac{g+2aV}{4a^2}\log\left(1+\frac{2aV}{g}\right) - \frac{V}{2a}.\]
A particle is projected horizontally from a point \(A\) on a vertical wall directly towards a parallel wall, which is a distance \(d\) away. The particle strikes the ground, which is horizontal, at \(B\), a distance \(b\) from the first wall, before bouncing on to hit the parallel wall at \(C\). It then rebounds towards the first wall. Assuming all impacts are perfectly elastic, find the condition on \(b/d\) for the particle to hit the first wall again before it hits the ground a second time. If this condition is satisfied and the particle hits this wall at a height \(\frac{1}{2}h\) above the ground, where \(h\) is the height of \(A\), calculate the height of \(C\) as a fraction of \(h\).
A bead of mass \(m\) slides down a rough wire in the shape of a circle. The wire is fixed with its plane vertical and the coefficient of friction between the bead and the wire is \(\mu\). Show that the reaction \(R\) between the bead and the wire satisfies \[\frac{dR}{d\theta} - 2\mu R + 3mg\sin\theta = 0,\] where \(\theta\) is the angle the radius to the bead makes with the downward vertical. Show that this equation is satisfied by \[R = A\cos\theta + B\sin\theta + Ce^{2\mu\theta},\] where \(C\) is arbitrary and \(A\) and \(B\) are to be determined. If the bead is released from rest at the same level as the centre of the circle and comes to rest at its lowest point show that \[(1 - 2\mu^2)e^{\mu\pi} = 3\mu.\]
Four identical spheres rest in a pile on a table, three touching each other and the fourth symmetrically on top. Let \(\alpha\) be the angle between the vertical and any nonhorizontal line-of-centres (\(\sin \alpha = 1/\sqrt{3}\)). Show that the spheres will stay in place without slipping provided that (a) the coefficient of limiting friction between two spheres is greater than \(\tan(\frac{1}{2}\alpha)\), and (b) the coefficient of limiting friction between a sphere and the table is greater than \(\frac{1}{3}\tan(\frac{1}{2}\alpha)\).
A particle bounces down a staircase, one bounce on each step. The coefficient of restitution is \(e\), and the bounces are exactly repetitive. Show that the maximum height of each bounce above the step just bounced off is \(\frac{e^2}{(1 - e^2)}\) times the height of each step.
A bead of mass \(m_1\) can slide freely and without friction on a straight horizontal wire. A second bead of mass \(m_2\) hangs from the first bead by a string of constant length \(l\). Find the frequency of small oscillations about the equilibrium configuration. [You may assume that the centre of gravity of the two beads does not move horizontally.]