| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simple Harmonic Motion |
| Type | Find amplitude from speed conditions |
| Difficulty | Standard +0.3 This is a standard SHM problem requiring application of the standard equations v² = ω²(a² - x²) and a = -ω²x. Part (a) involves straightforward substitution to find ω, part (b) uses the velocity equation to find amplitude, and part (c) applies v_max = ωa. While it requires knowing multiple SHM formulas and careful algebraic manipulation, it follows a predictable pattern with no novel insight required, making it slightly easier than average. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\ddot{x} = -\omega^2 x \therefore 0.12 = | -\omega^2 \times 0.03 | \) giving \(\omega = 2\) |
| period \(= \frac{2\pi}{\omega} = \pi\) seconds | M1 A1 | |
| (b) \(v^2 = \omega^2(a^2 - x^2) \therefore 0.08^2 = 2^2(a^2 - 0.03^2)\) | M1 A1 | |
| giving \(a = 0.05 \text{ m or } 5 \text{ cm}\) | A1 | |
| (c) \(v_{\max} = \omega a = 2 \times 0.05 = 0.1 \text{ ms}^{-1} \text{ or } 10 \text{ cms}^{-1}\) | M1 A1 | (9) |
(a) $\ddot{x} = -\omega^2 x \therefore 0.12 = |-\omega^2 \times 0.03|$ giving $\omega = 2$ | M1 A1 |
period $= \frac{2\pi}{\omega} = \pi$ seconds | M1 A1 |
(b) $v^2 = \omega^2(a^2 - x^2) \therefore 0.08^2 = 2^2(a^2 - 0.03^2)$ | M1 A1 |
giving $a = 0.05 \text{ m or } 5 \text{ cm}$ | A1 |
(c) $v_{\max} = \omega a = 2 \times 0.05 = 0.1 \text{ ms}^{-1} \text{ or } 10 \text{ cms}^{-1}$ | M1 A1 | (9)4. A particle moves with simple harmonic motion along a straight line.
When the particle is 3 cm from its centre of motion it has a speed of $8 \mathrm {~cm} \mathrm {~s} ^ { - 1 }$ and an acceleration of magnitude $12 \mathrm {~cm} \mathrm {~s} ^ { - 2 }$.
\begin{enumerate}[label=(\alph*)]
\item Show that the period of the motion is $\pi$ seconds.
\item Find the amplitude of the motion.
\item Hence, find the greatest speed of the particle.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 Q4 [9]}}