M3 June 2008 Q2
2. A particle \(P\) moves with simple harmonic motion and comes to rest at two points \(A\) and \(B\) which are 0.24 m apart on a horizontal line. The time for \(P\) to travel from \(A\) to \(B\) is 1.5 s. The midpoint of \(AB\) is \(O\). At time \(t = 0\), \(P\) is moving through \(O\), towards \(A\), with speed \(u\) m s\(^{-1}\).
(a) Find the value of \(u\). (4)
(b) Find the distance of \(P\) from \(B\) when \(t = 2\) s. (5)
(c) Find the speed of \(P\) when \(t = 2\) s. (2)
| Scheme | Marks |
|---|---|
| \(T = 3 = \dfrac{2\pi}{\omega} \quad \therefore \omega = \dfrac{2\pi}{3}\) | M1A1 |
| \(u^2 = \omega^2\left(a^2 - x^2\right)\); \(a = 0.12\), \(u^2 = a^2\omega^2,\ u = 0.12 \times \omega\) | M1 |
| \(= 0.251\) ms\(^{-1}\) (0.25 m s\(^{-1}\)) | A1 |
| (4) |
| Scheme | Marks |
|---|---|
| Time from \(O \rightarrow A \rightarrow O = 1.5\)s \(\therefore t = 0.5\) | B1 |
| \(x = a\sin\omega t \quad \Rightarrow OP = 0.12\sin\left(\dfrac{\pi}{3}\right)\) | M1A1 |
| Distance from \(B\) is \(0.12 - OP = 0.12 - 0.104\ldots = 0.016\)m | M1A1 |
| (5) |
| Scheme | Marks |
|---|---|
| \(v^2 = \omega^2\left(a^2 - x^2\right)\) | M1 |
| \(v = \dfrac{2\pi}{3}\sqrt{0.12^2 - 0.104\ldots^2} = \dfrac{2\pi}{3} \times 0.0598 = 0.13\) ms\(^{-1}\) | A1 |
| (2) | |
| (11 marks) |