| Exam Board | AQA |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dimensional Analysis |
| Type | Derive dimensions from formula |
| Difficulty | Moderate -0.8 This is a straightforward dimensional analysis exercise requiring systematic application of standard mechanics dimensions. While it involves multiple steps and careful bookkeeping of units, it's purely procedural with no conceptual difficulty or problem-solving insight needed—easier than average A-level questions. |
| Spec | 6.01a Dimensions: M, L, T notation6.01b Units vs dimensions: relationship |
| Answer | Marks | Guidance |
|---|---|---|
| \(\left[E\right] = \left[\frac{Gm_1m_2}{r}\right] = MLT^{-2}L^2M^{-1}MML^{-1} = ML^2T^{-2}\) | M1 dM1 A1 | M1: Working with the 2nd term in the expression. dM1: Correct unsimplified expression. A1: CAO |
| Answer | Marks | Guidance |
|---|---|---|
| \(\left[\frac{K^2}{2m_1r^2}\right] = ML^2T^{-2}\) leading to \(\left[K^2\right] = M^2L^4T^{-2}\) and \(\left[K\right] = ML^2T^{-1}\) | M1 A1 A1 | M1: Working with their answer to (a) and the first term of the expression. A1: Correct dimensions for \(K^2\). A1: CAO |
**(a)**
| $\left[E\right] = \left[\frac{Gm_1m_2}{r}\right] = MLT^{-2}L^2M^{-1}MML^{-1} = ML^2T^{-2}$ | M1 dM1 A1 | M1: Working with the 2nd term in the expression. dM1: Correct unsimplified expression. A1: CAO |
**(b)**
| $\left[\frac{K^2}{2m_1r^2}\right] = ML^2T^{-2}$ leading to $\left[K^2\right] = M^2L^4T^{-2}$ and $\left[K\right] = ML^2T^{-1}$ | M1 A1 A1 | M1: Working with their answer to (a) and the first term of the expression. A1: Correct dimensions for $K^2$. A1: CAO |
**Total: 6 marks**
---A lunar mapping satellite of mass $m_1$ measured in kg is in an elliptic orbit around the moon, which has mass $m_2$ measured in kg. The effective potential, $E$, of the satellite is given by
$$E = \frac{K^2}{2m_1r^2} - \frac{Gm_1m_2}{r}$$
where $r$ measured in metres is the distance of the satellite from the moon, $G$ Nm$^2$kg$^{-2}$ is the universal gravitational constant, and $K$ is the angular momentum of the satellite.
By using dimensional analysis, find the dimensions of:
\begin{enumerate}[label=(\alph*)]
\item $E$, [3 marks]
\item $K$. [3 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA M3 2016 Q2 [6]}}