M2 January 2011 Q4
4.
A box of mass 30 kg is held at rest at point \(A\) on a rough inclined plane. The plane is inclined at 20\(^\circ\) to the horizontal. Point \(B\) is 50 m from \(A\) up a line of greatest slope of the plane, as shown in Figure 1. The box is dragged from \(A\) to \(B\) by a force acting parallel to \(AB\) and then held at rest at \(B\). The coefficient of friction between the box and the plane is \(\dfrac{1}{4}\). Friction is the only non-gravitational resistive force acting on the box. Modelling the box as a particle,
(a) find the work done in dragging the box from \(A\) to \(B\). (6)
The box is released from rest at the point \(B\) and slides down the slope. Using the work-energy principle, or otherwise,
(b) find the speed of the box as it reaches \(A\). (5)
| Scheme | Marks |
|---|---|
| Work done against friction \(= 50 \times \mu\text{R}\) | M1 |
| \(= 50 \times \frac{1}{4} \times 30\cos 20^\circ \times 9.8\) | A1 |
| Gain in GPE \(= 30 \times 9.8 \times 50\sin 20^\circ\) | M1 A1 |
| Total work done \(=\) WD against Friction \(+\) gain in GPE | DM1 |
| \(= 8480\ (\text{J}),\ 8500\ (\text{J})\) | A1 |
| (6) |
| Scheme | Marks |
|---|---|
| Loss in GPE \(=\) WD against friction \(+\) gain in KE 3 terms | M1 |
| \(30 \times 9.8 \times 50\sin 20^\circ = 50 \times \frac{1}{4} \times 30 \times 9.8 \times \cos 20^\circ + \frac{1}{2} \times 30 \times v^2\) −1 ee | A2,1,0 |
| \(\frac{1}{2}v^2 = 50 \times 9.8 \times \left(\sin 20^\circ - \frac{1}{4}\cos 20^\circ\right)\), | DM1 |
| \(v = 10.2\) m s\(^{-1}\). | A1 |
| (5) | |
| (11 marks) |