M1 January 2012 Q8
8.
A particle \(P\) of mass 4 kg is moving up a fixed rough plane at a constant speed of 16 m s\(^{-1}\) under the action of a force of magnitude 36 N. The plane is inclined at 30\(^\circ\) to the horizontal. The force acts in the vertical plane containing the line of greatest slope of the plane through \(P\), and acts at 30\(^\circ\) to the inclined plane, as shown in Figure 2. The coefficient of friction between \(P\) and the plane is \(\mu\). Find
(a) the magnitude of the normal reaction between \(P\) and the plane, (4)
(b) the value of \(\mu\). (5)
The force of magnitude 36 N is removed.
(c) Find the distance that \(P\) travels between the instant when the force is removed and the instant when it comes to rest. (5)
| Scheme | Marks |
|---|---|
| \(R + 36\sin 30^\circ = 4g\cos 30^\circ\) | M1 A1 |
| \(R \approx 15.9,\ 16\) | M1 A1 |
| (4) |
| Scheme | Marks |
|---|---|
| Use of \(F_r = \mu R\) | B1 |
| \(36\cos 30^\circ = F + 4g\sin 30^\circ\) | M1 A1 |
| \(\mu = \dfrac{36\cos 30^\circ - 4g\sin 30^\circ}{R} \approx 0.726\) 0.73 | M1 A1 |
| (5) |
| Scheme | Marks |
|---|---|
| After force is removed | |
| \(R = 4g\cos 30^\circ\) | B1 |
| \(-\mu 4g\cos 30^\circ - 4g\sin 30^\circ = 4a\) | M1 A1 |
| \(a = (-)11.06\ldots\) | |
| \(v^2 = u^2 + 2as \ \Rightarrow\ 0^2 = 16^2 - 2 \times 11.06\ldots \times s\) | M1 |
| \(s = \dfrac{16^2}{2 \times 11.06\ldots} \approx 11.6\ \ (\text{m})\) 12 | A1 |
| (5) | |
| (14 marks) |