| Exam Board | OCR |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2011 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simple Harmonic Motion |
| Type | Small oscillations: simple pendulum (particle on string) |
| Difficulty | Standard +0.3 This is a standard M3 pendulum question requiring derivation of the equation of motion (routine application of torque/angular acceleration), recognition of SHM conditions (small angle approximation), and application of standard SHM formulas with given initial conditions. All steps follow textbook methods with no novel problem-solving required, making it slightly easier than average. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x6.05e Radial/tangential acceleration |
| Answer | Marks | Guidance |
|---|---|---|
| \([-0.2x^2\ddot{\theta} = 0.2g\sin\theta]\) \(\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2} = -4.9\sin\theta\) \(\text{For small } \theta, \sin\theta \approx \theta \text{ and } \ddot{\theta} = -4.9\theta \text{ represents SHM}\) | M1, A1, B1 [3] | For using Newton's second law transversely. Allow sign error and/or \(\sin/\cos\) error and/or missing \(0.2, g\) or \(l\). |
| Answer | Marks | Guidance |
|---|---|---|
| \(\theta = 0.15\cos(\sqrt{4.9}t)\) oe \(t = 1.04\) at first occasion \(t = 1.80\) at second occasion | A1, A1, M1, A1 [5] | For using \(\theta = A\cos(nt)\) or \(A\sin(nt + e)\). Allow \(\sin/\cos\) confusion for using \(t_1 + t_2 = 2\pi/n\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\text{Angular speed is } (-) 0.297 \text{ rads s}^{-1}\) \(\text{Linear speed is } (-) 0.594\text{ms}^{-1}\) | M1, A1, A1ft [3] | For using \(\dot{\theta} = -An\sin(nt)\) oe. Allow sign error and/or ft from \(\theta\) in (ii). |
**Part (i):**
$[-0.2x^2\ddot{\theta} = 0.2g\sin\theta]$ $\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2} = -4.9\sin\theta$ $\text{For small } \theta, \sin\theta \approx \theta \text{ and } \ddot{\theta} = -4.9\theta \text{ represents SHM}$ | M1, A1, B1 [3] | For using Newton's second law transversely. Allow sign error and/or $\sin/\cos$ error and/or missing $0.2, g$ or $l$. |
**Part (ii):**
$\theta = 0.15\cos(\sqrt{4.9}t)$ oe $t = 1.04$ at first occasion $t = 1.80$ at second occasion | A1, A1, M1, A1 [5] | For using $\theta = A\cos(nt)$ or $A\sin(nt + e)$. Allow $\sin/\cos$ confusion for using $t_1 + t_2 = 2\pi/n$ |
**Part (iii):**
$\text{Angular speed is } (-) 0.297 \text{ rads s}^{-1}$ $\text{Linear speed is } (-) 0.594\text{ms}^{-1}$ | M1, A1, A1ft [3] | For using $\dot{\theta} = -An\sin(nt)$ oe. Allow sign error and/or ft from $\theta$ in (ii). |
**General note:** In (ii) & (iii) allow M marks if angular displacement/speed has been confused with linear.One end of a light inextensible string of length $2$ m is attached to a fixed point $O$. A particle $P$ of mass $0.2$ kg is attached to the other end of the string. $P$ is held at rest with the string taut so that $OP$ makes an angle of $0.15$ radians with the downward vertical. $P$ is released and $t$ seconds afterwards $OP$ makes an angle of $\theta$ radians with the downward vertical.
\begin{enumerate}[label=(\roman*)]
\item Show that $\frac{d^2\theta}{dt^2} = -4.9 \sin \theta$ and give a reason why the motion is approximately simple harmonic. [3]
\end{enumerate}
Using the simple harmonic approximation,
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item obtain an expression for $\theta$ in terms of $t$ and hence find the values of $t$ at the first and second occasions when $\theta = -0.1$, [5]
\item find the angular speed of $OP$ and the linear speed of $P$ when $t = 0.5$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR M3 2011 Q4 [11]}}