M3 June 2011 Q5
5. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(l\) and modulus of elasticity \(3mg\). The other end of the string is attached to a fixed point \(O\) on a rough horizontal table. The particle lies at rest at the point \(A\) on the table, where \(OA = \dfrac{7}{6}l\). The coefficient of friction between \(P\) and the table is \(\mu\).
(a) Show that \(\mu \geqslant \dfrac{1}{2}\). (4)
The particle is now moved along the table to the point \(B\), where \(OB = \dfrac{3}{2}l\), and released from rest. Given that \(\mu = \dfrac{1}{2}\), find
(b) the speed of \(P\) at the instant when the string becomes slack, (5)
(c) the total distance moved by \(P\) before it comes to rest again. (3)
| Scheme | Marks |
|---|---|
| \(T = \dfrac{3mg}{l}\left(\dfrac{1}{6}l\right) = \dfrac{1}{2}mg\) | B1 |
| R\((\uparrow)\quad R = mg \qquad\) R\((\rightarrow)\quad F = T = \dfrac{1}{2}mg\) | M1 |
| \(F \leqslant \mu R\) \(\dfrac{1}{2}mg \leqslant \mu mg\) | M1 |
| \(\mu \geqslant \dfrac{1}{2}\) * | A1 |
| (4) |
| Scheme | Marks |
|---|---|
| E.P.E. lost \(= \dfrac{1}{2} \times \dfrac{3mg}{l}\left(\dfrac{1}{2}l\right)^2 = \dfrac{3mgl}{8}\) | B1 |
| Work done by friction \(= \dfrac{1}{2}mg\left(\dfrac{l}{2}\right)\) | B1 |
| \(\dfrac{3mgl}{8} = \dfrac{1}{2}mv^2 + \dfrac{1}{2}mg\left(\dfrac{l}{2}\right)\) | M1 A1ft |
| \(v^2 = \dfrac{gl}{4}\) \(v = \dfrac{1}{2}\sqrt{gl}\) | A1 |
| (5) |
| Scheme | Marks |
|---|---|
| \(\dfrac{3mgl}{8} = \dfrac{1}{2}mgx\) | M1 A1 ft |
| \(x = \dfrac{3l}{4}\) | A1 |
| (3) | |
| (12 marks) |