M4 June 2009 Q4
4.
A light inextensible string of length \(2a\) has one end attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass \(m\). A second light inextensible string of length \(L\), where \(L \gt \frac{12a}{5}\), has one of its ends attached to \(P\) and passes over a small smooth peg fixed at a point \(B\). The line \(AB\) is horizontal and \(AB = 2a\). The other end of the second string is attached to a particle of mass \(\frac{7}{20}m\), which hangs vertically below \(B\), as shown in Figure 2.
(a) Show that the potential energy of the system, when the angle \(PAB = 2\theta\), is\[\tfrac{1}{5}mga(7\sin\theta - 10\sin 2\theta) + \text{constant}.\] (4)
(b) Show that there is only one value of \(\cos\theta\) for which the system is in equilibrium and find this value. (8)
(c) Determine the stability of the position of equilibrium. (4)
| Scheme | Marks |
|---|---|
| \(V = -mg2a\sin 2\theta - \tfrac{7}{20}mg(L - 4a\sin\theta)\) | M1 B1 A1 |
| \(= \tfrac{1}{5}mga(7\sin\theta - 10\sin 2\theta) - \tfrac{7}{20}mgL\) | A1 |
| (4) |
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}V}{\mathrm{d}\theta} = \tfrac{1}{5}mga(7\cos\theta - 20\cos 2\theta)\) | M1 A1 |
| \(\tfrac{1}{5}mga(7\cos\theta - 20\cos 2\theta) = 0\) | DM1 |
| \(7\cos\theta - 20(2\cos^2\theta - 1) = 0\) | DM1 |
| \(40\cos^2\theta - 7\cos\theta - 20 = 0\) | A1 |
| \((5\cos\theta - 4)(8\cos\theta + 5) = 0\) | |
| \(\cos\theta = \tfrac{4}{5}\) or \(\left(\cos\theta = -\tfrac{5}{8} \Rightarrow 2\theta \gt 180^\circ\right)\) | DM1 A1 DM1 |
| (8) |
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}^2V}{\mathrm{d}\theta^2} = \tfrac{1}{5}mga(-7\sin\theta + 40\sin 2\theta)\) | |
| \(= \tfrac{1}{5}mga(-7\sin\theta + 80\sin\theta\cos\theta)\) | M1 A1 |
| When \(\cos\theta = \tfrac{4}{5}\), | |
| \(\dfrac{\mathrm{d}^2V}{\mathrm{d}\theta^2} = \tfrac{1}{5}mga\left(\tfrac{-21}{5} + 80 \times \tfrac{3}{5} \times \tfrac{4}{5}\right) = \tfrac{171}{25}mga\) | M1 |
| \(\gt 0\) therefore stable | A1 cso |
| (4) | |
| (16 marks) |