M2 January 2007 Q1
1. A particle of mass 0.8 kg is moving in a straight line on a rough horizontal plane. The speed of the particle is reduced from 15 m s\(^{-1}\) to 10 m s\(^{-1}\) as the particle moves 20 m. Assuming that the only resistance to motion is the friction between the particle and the plane, find
(a) the work done by friction in reducing the speed of the particle from 15 m s\(^{-1}\) to 10 m s\(^{-1}\), (2)
(b) the coefficient of friction between the particle and the plane. (4)
| Scheme | Marks |
|---|---|
| \(\dfrac{1}{2}0.8(15^2 - 10^2) = 50\ \ (\text{J})\) | M1 A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(F = \mu R = \mu 0.8g\) | M1 |
| Work-energy \(\mu 0.8g \times 20 = 50\) ft their (a) | M1 A1ft |
| \(\mu \approx 0.32\) accept 0.319 | A1 |
| (4) | |
| (6 marks) |
Alternative for (b)
| \(v^2 = u^2 + 2as\ \ \Rightarrow\ \ a = \dfrac{15^2 - 10^2}{2 \times 20} = 3.125\) | M1 |
| N2L \(F = \mu mg = ma = 3.125m\) | M1 A1ft |
| \(\mu \approx 0.32\) accept 0.319 | A1 (4) |
Alternative for (b)
| WE \(F = \dfrac{50}{20}\ \ (= 2.5)\) | M1 |
| \(F = \mu R \Rightarrow \dfrac{50}{20} = \mu 0.8g\) ft their (a) | M1 A1 ft |
| \(\mu \approx 0.32\) | A1 (4) |
The first M1 for (b) could be scored in (a):
| \(v^2 = u^2 + 2as \Rightarrow 10^2 = 15^2 - 2 \times 20 \times (-)a \Rightarrow a = (-)\dfrac{125}{40}\) | (b)M1 |
| \(F = ma \Rightarrow F = 2.5\) \(WD = F \times d \Rightarrow 2.5 \times 20 = 50J\) | (a)M1A1 |