M2 June 2008 Q3
3.
A package of mass 3.5 kg is sliding down a ramp. The package is modelled as a particle and the ramp as a rough plane inclined at an angle of 20\(^\circ\) to the horizontal. The package slides down a line of greatest slope of the plane from a point \(A\) to a point \(B\), where \(AB = 14\) m. At \(A\) the package has speed 12 m s\(^{-1}\) and at \(B\) the package has speed 8 m s\(^{-1}\), as shown in Figure 1. Find
(a) the total energy lost by the package in travelling from \(A\) to \(B\), (5)
(b) the coefficient of friction between the package and the ramp. (5)
| Scheme | Marks |
|---|---|
| \(\Delta\text{KE} = \dfrac{1}{2} \times 3.5(12^2 - 8^2)\ \ (= 140)\) or KE at A, B correct separately | B1 |
| \(\Delta\text{PE} = 3.5 \times 9.8 \times 14\sin 20^\circ\ \ (\approx 164.238)\) or PE at A, B correct separately | M1 A1 |
| \(\Delta E = \Delta\text{KE} + \Delta\text{PE} \approx 304,\ \ 300\) | DM1 A1 |
| (5) |
| Scheme | Marks |
|---|---|
| Using Work-Energy | |
| \(F_r = \mu \times 3.5g\cos 20^\circ\) | M1 A1 |
| \(304.238\ldots = F_r \times 14\) ft their (a), \(F_r\) | M1 A1 ft |
| \(304.238\ldots = \mu 3.5g\cos 20^\circ \times 14\) | |
| \(\mu \approx 0.674,\ 0.67\) | A1 |
| (5) | |
| (10 marks) |
Alternative using N2L
| \(F_r = \mu \times 3.5g\cos 20^\circ\) | M1 A1 |
| \(v^2 = u^2 + 2as \ \Rightarrow\ 8^2 = 12^2 - 2a \times 14\) \(\left(a = \dfrac{20}{7}\right)\ (2.857\ldots)\) | |
| N2L \(\ \text{R}\nwarrow\): {their \(F_r\)} \(-\ mg\sin 20^\circ = ma\) ft their \(F_r\). | M1 A1ft |
| Leading to \(\mu \approx 0.674\) or 0.67 | A1 |
(5)