M4 January 2006 Q2
2. A small smooth sphere \(S\) of mass \(m\) is attached to one end of a light inextensible string of length \(2a\). The other end of the string is attached to a fixed point \(A\) which is at a distance \(a\sqrt{3}\) from a smooth vertical wall. The sphere \(S\) hangs at rest in equilibrium. It is then projected horizontally towards the wall with a speed \(\sqrt{\left(\dfrac{37ga}{5}\right)}\).
(a) Show that \(S\) strikes the wall with speed \(\sqrt{\left(\dfrac{27ga}{5}\right)}\). (4)
Given that the loss in kinetic energy due to the impact with the wall is \(\dfrac{3mga}{5}\),
(b) find the coefficient of restitution between \(S\) and the wall. (7)
| Scheme | Marks |
|---|---|
| Energy: \(\ \dfrac{1}{2}m\left(\dfrac{37ga}{5} - v^2\right) = mg.2a(1 - \cos\theta)\) | M1 A1 |
| Using \(\theta = \dfrac{\pi}{3}\) & solve: \(\rightarrow v = \sqrt{\dfrac{27ga}{5}}\qquad (*)\) | M1 A1 |
| (4) |
| Scheme | Marks |
|---|---|
| Impact: \(\ u_1 = ev\sin 30\) | M1 A1 |
| KE loss \(= \dfrac{1}{2}m\left(v^2\sin^2 30 - e^2v^2\sin^2 30\right)\) | |
| \(\left[+\dfrac{1}{2}mv^2\cos^2 30 - \dfrac{1}{2}mu_2^{\,2}\right] = \dfrac{3mga}{5}\) | M1 A1 |
| [Using \(u_2 = v\cos 30\) if necessary & ] reducing to equation in \((m, g, a)\ e\) alone | |
| \(\dfrac{3mga}{5} = \dfrac{1}{2}m.\dfrac{27ga}{5}.\dfrac{1}{4}\left(1 - e^2\right)\) | A1 |
| Solve for \(e\): \(\rightarrow e = \dfrac{1}{3}\) | M1 A1 |
| (7) | |
| (11 marks) |