2 A rod, of length \(x \mathrm {~m}\) and moment of inertia \(I \mathrm {~kg} \mathrm {~m} ^ { 2 }\), is free to rotate in a vertical plane about a fixed smooth horizontal axis through one end. When the rod is hanging at rest, its lower end receives an impulse of magnitude \(J\) Ns, which is just sufficient for the rod to complete full revolutions. It is thought that there is a relationship between \(J , x , I\), the acceleration due to gravity \(g \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and a dimensionless constant \(k\), such that $$J = k x ^ { \alpha } I ^ { \beta } g ^ { \gamma }$$ where \(\alpha , \beta\) and \(\gamma\) are constants.
Find the values of \(\alpha , \beta\) and \(\gamma\) for which this relationship is dimensionally consistent.
[0pt] [6 marks]

# Question 2:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Dimensions of $J$: $[\text{N s}] = \text{kg\,m\,s}^{-1}$ | B1 | Correct dimensions of impulse |
| Dimensions of $x$: m; $I$: $\text{kg\,m}^2$; $g$: $\text{m\,s}^{-2}$ | B1 | Correct dimensions of each quantity |
| Mass: $1 = \beta$ | M1 A1 | Equating mass indices |
| Length: $1 = \alpha + 2\beta + \gamma$ | M1 | Equating length indices |
| Time: $-1 = -2\gamma$ so $\gamma = \dfrac{1}{2}$ | A1 | Correct $\gamma$ |
| $\beta = 1$, $\alpha = 1 - 2(1) - \frac{1}{2} = -\dfrac{3}{2}$ | A1 | $\alpha = -\dfrac{3}{2}$, $\beta = 1$, $\gamma = \dfrac{1}{2}$ |

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2 A rod, of length $x \mathrm {~m}$ and moment of inertia $I \mathrm {~kg} \mathrm {~m} ^ { 2 }$, is free to rotate in a vertical plane about a fixed smooth horizontal axis through one end.

When the rod is hanging at rest, its lower end receives an impulse of magnitude $J$ Ns, which is just sufficient for the rod to complete full revolutions.

It is thought that there is a relationship between $J , x , I$, the acceleration due to gravity $g \mathrm {~m} \mathrm {~s} ^ { - 2 }$ and a dimensionless constant $k$, such that

$$J = k x ^ { \alpha } I ^ { \beta } g ^ { \gamma }$$

where $\alpha , \beta$ and $\gamma$ are constants.\\
Find the values of $\alpha , \beta$ and $\gamma$ for which this relationship is dimensionally consistent.\\[0pt]
[6 marks]\\

\hfill \mbox{\textit{AQA M3 2014 Q2 [6]}}