2
\begin{enumerate}[label=(\alph*)]
\item Find the dimensions of energy.

The moment of inertia, $I$, of a rigid body rotating about a fixed axis is measured in $\mathrm { kg } \mathrm { m } ^ { 2 }$.
\item State the dimensions of $I$.

The kinetic energy, $E$, of a rigid body rotating about a fixed axis is given by the formula\\
$\mathrm { E } = \frac { 1 } { 2 } \mathrm { I } \omega ^ { 2 }$,\\
where $\omega$ is the angular velocity (angle per unit time) of the rigid body.
\item Show that the formula for $E$ is dimensionally consistent.

When a rigid body is pivoted from one of its end points and allowed to swing freely, it forms a pendulum. The period, $t$, of the pendulum is the time taken for it to complete one oscillation. A student conjectures the formula\\
$\mathrm { t } = \left. \mathrm { k } ( \mathrm { mg } ) ^ { \alpha } \mathrm { r } ^ { \beta } \right| ^ { \gamma }$,\\
where

\begin{itemize}
  \item $k$ is a dimensionless constant,
  \item $m$ is the mass of the rigid body,
  \item $g$ is the acceleration due to gravity,
  \item $r$ is the distance between the pivot point and the rigid body's centre of mass.
\item Use dimensional analysis to find the values of $\alpha , \beta$ and $\gamma$.
\end{itemize}

The moment of inertia of a thin uniform rigid rod of mass 1.5 kg and length 0.8 m , rotating about one of its endpoints, is $0.32 \mathrm {~kg} \mathrm {~m} ^ { 2 }$. The student suspends such a rod from one of its endpoints and allows it to swing freely. The student measures the period of this pendulum and finds that it is 1.47 seconds.
\item Using the formula conjectured by the student, determine the value of $k$.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Mechanics A AS 2024 Q2 [11]}}