M1 June 2013 Q8
8.
Two particles \(A\) and \(B\) have masses \(2m\) and \(3m\) respectively. The particles are attached to the ends of a light inextensible string. Particle \(A\) is held at rest on a smooth horizontal table. The string passes over a small smooth pulley which is fixed at the edge of the table. Particle \(B\) hangs at rest vertically below the pulley with the string taut, as shown in Figure 2.
Particle \(A\) is released from rest. Assuming that \(A\) has not reached the pulley, find
| Scheme | Marks |
|---|---|
| For \(A\), \(T = 2ma\) | B1 |
| For \(B\), \(3mg - T = 3ma\) | M1 A1 |
| \(3mg = 5ma\) | DM1 |
| \(\dfrac{3g}{5} = a\) (5.9 or 5.88 m s\(^{-2}\)) | A1 |
| (5) |
Notes
B1 for \(T = 2ma\)
First M1 for resolving vertically (up or down) for \(B\), with correct no. of terms. (allow omission of \(m\), provided 3 is there)
First A1 for a correct equation.
Second M1, dependent on first M1, for eliminating \(T\), to give an equation in \(a\) only.
Second A1 for 0.6g, 5.88 or 5.9.
N.B. ‘Whole system’ equation: \(3mg = 5ma\) earns first 4 marks but any error loses all 4.
| Scheme | Marks |
|---|---|
| \(T = 6mg/5\); \(12m\); \(11.8m\) | B1 |
| (1) |
Notes
B1 for \(\dfrac{6mg}{5}\), \(11.8m\), \(12m\)
| Scheme | Marks |
|---|---|
| \(F = \sqrt{T^2 + T^2}\) | M1 A1 ft |
| \(F = \dfrac{6mg\sqrt{2}}{5}\); \(1.7mg\) (or better); \(16.6m\); \(17m\) | A1 |
| Direction clearly marked on a diagram, with an arrow, and 45\(^\circ\) (oe) marked | B1 |
| (4) | |
| (10 marks) |
Notes
M1 \(\sqrt{(T^2 + T^2)}\) or \(\dfrac{T}{\sin 45^\circ}\) or \(\dfrac{T}{\cos 45^\circ}\) or \(2T\cos 45^\circ\) or \(2T\sin 45^\circ\) (allow if \(m\) omitted)
(M0 for \(T\sin 45^\circ\))
First A1 ft on their \(T\).
Second A1 cao for \(\dfrac{6mg\sqrt{2}}{5}\) oe, \(1.7mg\) (or better), \(16.6m\), \(17m\)
B1 for the direction clearly shown on a diagram with an arrow and 45\(^\circ\) marked.