M1 June 2008 Q8
8.
Two particles \(P\) and \(Q\), of mass 2 kg and 3 kg respectively, are joined by a light inextensible string. Initially the particles are at rest on a rough horizontal plane with the string taut. A constant force \(\mathbf{F}\) of magnitude 30 N is applied to \(Q\) in the direction \(PQ\), as shown in Figure 4. The force is applied for 3 s and during this time \(Q\) travels a distance of 6 m. The coefficient of friction between each particle and the plane is \(\mu\). Find
(a) the acceleration of \(Q\), (2)
(b) the value of \(\mu\), (4)
(c) the tension in the string. (4)
(d) State how in your calculation you have used the information that the string is inextensible. (1)
When the particles have moved for 3 s, the force \(\mathbf{F}\) is removed.
(e) Find the time between the instant that the force is removed and the instant that \(Q\) comes to rest. (4)
| Scheme | Marks |
|---|---|
| \(s = ut + \tfrac{1}{2}at^2\ \ \Rightarrow\ \ 6 = \tfrac{1}{2}a \times 9\) | M1 |
| \(a = 1\tfrac{1}{3}\ \left(\text{ms}^{-2}\right)\) | A1 |
| (2) |
| Scheme | Marks |
|---|---|
| N2L for system \(30 - \mu 5g = 5a\) ft their \(a\), accept symbol | M1 A1ft |
| \(\mu = \dfrac{14}{3g} = \dfrac{10}{21}\) or awrt 0.48 | DM1 A1 |
| (4) |
| Scheme | Marks |
|---|---|
| N2L for \(P\) \(T - \mu 2g = 2a\) ft their \(\mu\), their \(a\), accept symbols \(T - \dfrac{14}{3g} \times 2g = 2 \times \dfrac{4}{3}\) | M1 A1 ft |
| Leading to \(T = 12\ \ (\text{N})\) awrt 12 | DM1 A1 |
| (4) |
Alternatively
| N2L for \(Q\) \(30 - T - \mu 3g = 3a\) | M1 A1 |
| Leading to \(T = 12\ \ (\text{N})\) awrt 12 | DM1 A1 |
| Scheme | Marks |
|---|---|
| The acceleration of \(P\) and \(Q\) (or the whole of the system) is the same. | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| \(v = u + at\ \ \Rightarrow\ \ v = \dfrac{4}{3} \times 3 = 4\) | B1 ft on \(a\) |
| N2L (for system or either particle) \(-5\mu g = 5a\) or equivalent \(a = -\mu g\) | M1 |
| \(v = u + at\ \ \Rightarrow\ \ 0 = 4 - \mu gt\) | DM1 |
| Leading to \(t = \dfrac{6}{7}\ \ (\text{s})\) accept 0.86, 0.857 | A1 |
| (4) | |
| (15 marks) |