M3 January 2012 Q2
2. A particle \(P\) is moving in a straight line with simple harmonic motion. The centre of the oscillation is the fixed point \(C\), the amplitude of the oscillation is 0.5 m and the time to complete one oscillation is \(\tfrac{2\pi}{3}\) seconds. The point \(A\) is on the path of \(P\) and 0.2 m from \(C\).
Find
(a) the magnitude and direction of the acceleration of \(P\) when it passes through \(A\), (3)
(b) the speed of \(P\) when it passes through \(A\), (2)
(c) the time \(P\) takes to move directly from \(C\) to \(A\). (3)
| Scheme | Marks |
|---|---|
| \(T = \dfrac{2\pi}{\omega} = \dfrac{2\pi}{3},\quad \omega = 3\) | B1 |
| \(|a| = \omega^2 x = 9 \times 0.2 = 1.8\) ms\(^{-2}\) towards \(C\) | M1 A1 |
| (3) |
| Scheme | Marks |
|---|---|
| \(v^2 = \omega^2\left(a^2 - x^2\right) = 9(0.25 - 0.04) = 1.89\) | M1 |
| \(v = 1.37\) ms\(^{-1}\) | A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(x = 0.5\sin 3t = 0.2\) | M1 A1ft |
| \(t = \dfrac{1}{3}\sin^{-1} 0.4 \approx 0.137\) s | A1 |
| (3) | |
| (8 marks) |