Problems

A horizontal conveyor belt moves with a constant velocity \(u\). At time \(t = 0\), a parcel of mass \(m\) is dropped gently onto the belt. If the coefficient of friction between the parcel and the belt is \(\mu\), find

1. the time that elapses before the parcel is at rest relative to the belt;
2. the distance the parcel slides relative to the belt;
3. the energy dissipated during this sliding;
4. the impulse of the frictional force.

Two particles of masses \(m\) and \(2m\) are suspended over a movable pulley of mass \(m\) by a light string of length \(l\). The movable pulley is itself connected to a particle of mass \(4m\) by a light string of length \(L\) which passes over a fixed pulley. Find the acceleration of the particle of mass \(4m\). [You may neglect the moments of inertia of the pulleys.]

A uniform spherical dust cloud of mass \(M\) expands or contracts in such a way as to remain both uniform and spherical. The gravitational force on a particle of mass \(m\) at a distance \(r\) from the origin is radial and given by \[F = -\frac{4\pi}{3}G\rho mr,\] \(\rho\) being the density of the cloud and \(G\) the gravitational constant. By considering a particle on the surface of the cloud at distance \(R\) from the centre of the cloud, or otherwise, show that \[\frac{1}{2}\dot{R}^2 - \frac{GM}{R} = -\frac{GM}{R_M},\] \(R_M\) being a constant. Verify that for \(R_M > 0\) this equation has a solution of the form \begin{align*} R &= a\sin^2\chi\\ t &= b(\chi-\sin\chi\cos\chi), \end{align*} where \(a\) and \(b\) are constants. Evaluate \(a\) and \(b\) in terms of \(G\), \(M\) and \(R_M\). Show that this solution describes a cloud that expands from infinite density at \(t = 0\) and which collapses back to infinite density at time \[t_\infty = \pi\sqrt{\frac{R_M^3}{2GM}}.\]

A spherical raindrop has mass \(m\), radius \(r\) and downward speed \(v\) as it falls through a cloud of water vapour, which is moving upwards at speed \(U\). The raindrop grows by the condensation of water vapour on its surface, so that the increase of mass per unit time is proportional to the surface area. The raindrop starts from rest with radius \(r_0\) at time \(t = 0\).

1. Prove that the equation of motion of the raindrop is \[\frac{d(mv)}{dt} = mg-U\frac{dm}{dt}.\]
2. Show that the radius of the drop grows linearly with time.
3. Find the speed of the drop as a function of time by integrating the equations in (i) and (ii).
4. Show that the acceleration of the drop tends to a constant value as \(t \to \infty\), and find this value.

A uniform fine chain of length \(l\) is suspended with its lower end just touching a horizontal table. The chain is allowed to fall freely. If the mass of the chain is \(M\), find the force on the table when a length \(x\) has reached it. [You may assume that the part of the chain on the table does not interfere with the subsequent motion.]

Let \(f(x)\) be a polynomial in \(x\). Explain why, if \(z\) is an approximation to a root of \(f(x)\), then \(z-f(z)/f'(z)\) is often a closer approximation. By considering polynomials of the form \(x^r+a\), and suitable real values of \(z_0\), show that the iteration \[z_n = z_{n-1}-f(z_{n-1})/f'(z_{n-1}) \quad (n = 1, 2, \ldots)\] may exhibit any of the following three behaviours.

1. For every real value of \(z_0\) it approaches a root of \(f(x)\).
2. For no real value of \(z_0\) does it approach a root of \(f(x)\).
3. For some, but not for all, real values of \(z_0\) it approaches a root of \(f(x)\).