M2 January 2005 Q3
3.
A small package \(P\) is modelled as a particle of mass 0.6 kg. The package slides down a rough plane from a point \(S\) to a point \(T\), where \(ST = 12\) m. The plane is inclined at an angle of 30\(^\circ\) to the horizontal and \(ST\) is a line of greatest slope of the plane, as shown in Figure 3. The speed of \(P\) at \(S\) is 10 m s\(^{-1}\) and the speed of \(P\) at \(T\) is 9 m s\(^{-1}\). Calculate
(a) the total loss of energy of \(P\) in moving from \(S\) to \(T\), (4)
(b) the coefficient of friction between \(P\) and the plane. (5)
| Scheme | Marks |
|---|---|
| KE lost is \(\tfrac{1}{2} \times 0.6 \times (10^2 - 9^2)\ \ (= 5.7\ \text{J})\) | B1 |
| PE lost is \(0.6 \times 9.8 \times 12\sin 30^\circ\ (= 35.28\ \text{J})\) | B1 |
| Total loss in energy is 41.0 (J) accept 41 | M1 A1 |
| (4) |
| Scheme | Marks |
|---|---|
| \(R = 0.6 \times 9.8 \times \cos 30^\circ\ (\approx 5.09)\) | B1 |
| WE \(40.98 = \mu \times 0.6 \times 9.8 \times \cos 30^\circ \times 12\) ft their (a) | M1 A1ft |
| \(\mu \approx 0.67\) or 0.671 | M1 A1 |
| (5) | |
| (9 marks) |
Alternative for (b)
| \(a = \dfrac{9^2 - 10^2}{2 \times 12}\left(= (-)\dfrac{19}{24}\right)\) awrt 0.79 | B1 |
| N2L \(mg\sin 30^\circ - \mu mg\cos 30^\circ = m\left(-\tfrac{19}{24}\right)\) ft their \(a\) | M1 A1ft |
| \(\mu \approx 0.67\) or 0.671 | M1 A1 (5) |