A2 June 2025 Q7
7. A particle \(P\) of mass 0.4 kg is moving in a straight line with simple harmonic motion.
The maximum speed of \(P\) is \(5\ \text{m s}^{-1}\) and the maximum magnitude of acceleration of \(P\) is \(12.5\ \text{m s}^{-2}\)
The time taken for one complete oscillation is \(T\) seconds.
| Scheme | Marks | AO |
|---|---|---|
| \(a\omega = 5\) | B1 | 3.4 |
| \(a\omega^2 = 12.5\) | B1 | 3.4 |
| Complete method to find \(T\) | M1 | 2.1/3.1a |
| \(\omega = \dfrac{12.5}{5} \Rightarrow T = \dfrac{2\pi}{\omega} = \dfrac{4\pi}{5}\) * | A1 * | 1.1b |
| (4) |
Notes
B1: Correct formula for max speed. Accept \(a^2\omega^2 = 25\)
B1: Correct formula for max acceleration. Must be same sign on both sides in their solution.
M1: Complete method e.g. solve for \(\omega\) and use \(T = \dfrac{2\pi}{\omega}\)
A1 *: cso
| Scheme | Marks | AO |
|---|---|---|
| \(a = 2 \ \Rightarrow\ x = 2\cos\dfrac{5}{2}t\) or \(x = 2\sin\dfrac{5}{2}t\) | M1 | 3.4 |
| \(\dfrac{1}{2} = 2\cos\dfrac{5}{2}t \Rightarrow t = \dfrac{2}{5}\cos^{-1}\dfrac{1}{4}\) or \(\dfrac{1}{2} = 2\sin\dfrac{5}{2}t \Rightarrow t = \dfrac{2}{5}\sin^{-1}\dfrac{1}{4}\) | M1 | 3.1a |
| Total time: \(4\left(\dfrac{2}{5}\cos^{-1}\dfrac{1}{4}\right)\) or \(\dfrac{4\pi}{5} - 4\left(\dfrac{2}{5}\sin^{-1}\dfrac{1}{4}\right)\) | M1 | 1.1b |
| \(= 2.1\) (s) | A1 | 1.1b |
| (4) |
Notes
M1: Use their \(\omega\) to solve for \(a\) and state or imply equation in \(x\) and \(t\)
M1: Find a relevant value of \(t\)
M1: Find the required value of \(t\)
Some candidates will find two times and combine:
\(a\sin\omega t = 0.5\ [t_1 = 0.10107,\ t_2 = 1.15556] \rightarrow \text{Time} = 2(t_2 - t_1)\)
\(a\cos\omega t = 0.5\ [t_3 = 0.52724,\ t_4 = 0.72939] \rightarrow \text{Time} = 4\pi/5 - 2(t_4 - t_3)\)
A1: 2.1 (s) or better (2.108985…)
| Scheme | Marks | AO |
|---|---|---|
| \(v^2 = \dfrac{25}{4}\left(4 - \dfrac{1}{25}\right)\) | M1 | 3.4 |
| \(\text{KE} = \dfrac{1}{2}mv^2 = \dfrac{1}{2} \times 0.4 \times \dfrac{99}{4}\) | M1 | 1.2 |
| \(= \dfrac{99}{20}\ \text{(J)} = 4.95\ \text{(J)}\) | A1 | 1.1b |
| (3) | ||
| (11 marks) |
Notes
M1: Use of \(v^2 = \omega^2\left(a^2 - x^2\right)\) or equivalent to find \(v\) or \(v^2\)
M1: Correct method for kinetic energy for their \(v\)
A1: cao