| Exam Board | Edexcel |
|---|---|
| Module | M5 (Mechanics 5) |
| Year | 2013 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Compound pendulum oscillations |
| Difficulty | Challenging +1.8 This is a challenging M5 compound pendulum problem requiring: (a) parallel axis theorem applied to a composite body with removed sections, involving careful bookkeeping of multiple moments of inertia; (b) deriving the equation of motion for a physical pendulum and applying small angle approximations. The algebraic manipulation is substantial and error-prone, and the geometry requires careful analysis of the center of mass location. This is significantly harder than standard mechanics questions but follows established M5 techniques without requiring novel insight. |
| Spec | 6.04b Find centre of mass: using symmetry6.05f Vertical circle: motion including free fall |
A uniform circular lamina has radius $2a$ and centre $C$. The points P, Q, R and S on the lamina are the vertices of a square with centre $C$ and $CP = a$. Four circular discs, each of radius $\frac{a}{2}$, with centres P, Q, R and S, are removed from the lamina. The remaining lamina forms a template $T$.
The radius of gyration of $T$ about an axis through $C$, perpendicular to $T$, is $k$.
**(a)** Show that $k^2 = \frac{55a^2}{24}$
(7 marks)
The template $T$ is free to rotate in a vertical plane about a fixed smooth horizontal axis which is perpendicular to $T$ and passes through a point on its outer rim.
**(b)** Write down an equation of rotational motion for $T$ and deduce that the period of small oscillations of $T$ about its stable equilibrium position is
$$2\pi\sqrt{\frac{151a}{48g}}$$
(8 marks)
---5.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{d3e55cec-05f7-4db3-8eb5-5d0adca38d4c-09_723_707_214_621}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
A uniform circular lamina has radius $2 a$ and centre $C$. The points $P , Q , R$ and $S$ on the lamina are the vertices of a square with centre $C$ and $C P = a$. Four circular discs, each of radius $\frac { a } { 2 }$, with centres $P , Q , R$ and $S$, are removed from the lamina. The remaining lamina forms a template $T$, as shown in Figure 1.
The radius of gyration of $T$ about an axis through $C$, perpendicular to $T$, is $k$.
\begin{enumerate}[label=(\alph*)]
\item Show that $k ^ { 2 } = \frac { 55 a ^ { 2 } } { 24 }$
The template $T$ is free to rotate in a vertical plane about a fixed smooth horizontal axis which is perpendicular to $T$ and passes through a point on its outer rim.
\item Write down an equation of rotational motion for $T$ and deduce that the period of small oscillations of $T$ about its stable equilibrium position is
$$2 \pi \sqrt { } \left( \frac { 151 a } { 48 g } \right)$$
\end{enumerate}
\hfill \mbox{\textit{Edexcel M5 2013 Q5 [15]}}