M2 June 2006 Q7
7. A particle \(P\) has mass 4 kg. It is projected from a point \(A\) up a line of greatest slope of a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan\alpha = \tfrac{3}{4}\). The coefficient of friction between \(P\) and the plane is \(\tfrac{2}{7}\). The particle comes to rest instantaneously at the point \(B\) on the plane, where \(AB = 2.5\) m. It then moves back down the plane to \(A\).
(a) Find the work done by friction as \(P\) moves from \(A\) to \(B\). (4)
(b) Using the work-energy principle, find the speed with which \(P\) is projected from \(A\). (4)
(c) Find the speed of \(P\) when it returns to \(A\). (4)
| Scheme | Marks |
|---|---|
| \(R = 4g\cos\alpha = 16g/5\ \ \Rightarrow\ \ F = 2/7 \times 16g/5\) | M1 A1 |
| Work done \(= F \times 2.5 = \underline{22.4\ \text{J}}\) or 22 J indep | M1 A1 |
| (4) |
| Scheme | Marks |
|---|---|
| \(\tfrac{1}{2} \times 4 \times u^2 = 22.4 + 4g \times 2.5 \times 3/5\) | M1 A2,1,0 f.t. |
| \(\Rightarrow u \approx \underline{6.37\ \text{m s}^{-1}}\) or 6.4 ms\(^{-1}\) | A1cao |
| (4) |
| Scheme | Marks |
|---|---|
| \(\tfrac{1}{2} \times 4 \times v^2 = \tfrac{1}{2} \times 4 \times u^2 - 44.8\) | M1 A2,1,0 f.t. |
| [OR \(\tfrac{1}{2} \times 4 \times v^2 = 0 + 4g \times 2.5 \times 3/5 - 22.4\)] | |
| \(\Rightarrow v \approx \underline{4.27\ \text{m s}^{-1}}\) or 4.3 ms\(^{-1}\) | A1 |
| (4) | |
| (12 marks) |