Challenging +1.3 This is a standard compound pendulum problem requiring multiple techniques: deriving moment of inertia by integration (bookwork), applying parallel axis theorem, using energy conservation, and showing SHM for small angles. While it involves several steps and careful calculation with multiple components, each individual technique is well-practiced in M4. The structure is predictable and follows standard compound pendulum methodology without requiring novel insight.
Show, by integration, that the moment of inertia of a uniform rod of mass \(m\) and length \(2 a\) about an axis through its centre and perpendicular to the rod is \(\frac { 1 } { 3 } m a ^ { 2 }\).
A pendulum of length 1 m is made by attaching a uniform sphere of mass 2 kg and radius 0.1 m to the end of a uniform rod AB of mass 1.2 kg and length 0.8 m , as shown in Fig. 3. The centre of the sphere is collinear with A and B .
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Find the moment of inertia of the pendulum about an axis through A perpendicular to the rod.
The pendulum can swing freely in a vertical plane about a fixed horizontal axis through A .
The pendulum is held with AB at an angle \(\alpha\) to the downward vertical and released from rest. At time \(t , \mathrm { AB }\) is at an angle \(\theta\) to the vertical. Find an expression for \(\dot { \theta } ^ { 2 }\) in terms of \(\theta\) and \(\alpha\).
Hence, or otherwise, show that, provided that \(\alpha\) is small, the pendulum performs simple harmonic motion. Calculate the period.
3 (i) Show, by integration, that the moment of inertia of a uniform rod of mass $m$ and length $2 a$ about an axis through its centre and perpendicular to the rod is $\frac { 1 } { 3 } m a ^ { 2 }$.
A pendulum of length 1 m is made by attaching a uniform sphere of mass 2 kg and radius 0.1 m to the end of a uniform rod AB of mass 1.2 kg and length 0.8 m , as shown in Fig. 3. The centre of the sphere is collinear with A and B .
\begin{figure}[h]
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\includegraphics[alt={},max width=\textwidth]{8aab7e54-a204-481b-8f09-4bf4ca4e115d-3_442_291_717_886}
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\caption{Fig. 3}
\end{center}
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(ii) Find the moment of inertia of the pendulum about an axis through A perpendicular to the rod.
The pendulum can swing freely in a vertical plane about a fixed horizontal axis through A .\\
(iii) The pendulum is held with AB at an angle $\alpha$ to the downward vertical and released from rest. At time $t , \mathrm { AB }$ is at an angle $\theta$ to the vertical. Find an expression for $\dot { \theta } ^ { 2 }$ in terms of $\theta$ and $\alpha$.\\
(iv) Hence, or otherwise, show that, provided that $\alpha$ is small, the pendulum performs simple harmonic motion. Calculate the period.
\hfill \mbox{\textit{OCR MEI M4 2007 Q3 [24]}}