4 \includegraphics[max width=\textwidth, alt={}, center]{a473cbb8-877f-48df-8751-c76d96396734-3_906_1538_248_301} The end \(A\) of a uniform \(\operatorname { rod } A B\), of mass \(4 m\) and length \(3 a\), is rigidly attached to a point on a uniform spherical shell, of mass \(\lambda m\) and radius \(3 a\). The end \(B\) of the rod is rigidly attached to a point on a uniform ring. The ring has centre \(O\), mass \(4 m\) and radius \(\frac { 1 } { 2 } a\). The ring and the rod are in the same vertical plane. The line \(O B A\), extended, passes through the centre of the spherical shell. \(B C\) is a diameter of the ring (see diagram). Show that the moment of inertia of this system, about a fixed horizontal axis through \(C\) perpendicular to the plane of the ring, is \(( 30 + 55 \lambda ) m a ^ { 2 }\). Given that the system performs small oscillations of period \(2 \pi \sqrt { } \left( \frac { 5 a } { g } \right)\) about this axis, find the value of \(\lambda\).

## Question 4:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $I_{\text{Sphere}} = \frac{2}{3}\lambda m(3a)^2 + \lambda m(7a)^2$ | M1 A1 | MI of sphere about $C$ |
| $I_{\text{Rod}} = \frac{1}{3}4m(3a/2)^2 + 4m(5a/2)^2$ | M1 A1 | MI of rod about $C$ |
| $I_{\text{Ring}} = 4m(\frac{1}{2}a)^2 + 4m(\frac{1}{2}a)^2$ | M1 | MI of ring about $C$ |
| $I = (55\lambda + 28 + 2)ma^2 = (30 + 55\lambda)ma^2$ **A.G.** | A1 | Combine to give MI of system |
| $Id^2\theta/dt^2 = -(\lambda\times 7 + 4\times 5/2 + 4\times\frac{1}{2})mga\sin\theta$ | M1 A1 | Equation of motion for system |
| $\omega^2 = (12 + 7\lambda)g/(30 + 55\lambda)a$ | M1 A1 | Approximate $\sin\theta \approx \theta$, find $\omega^2$ in SHM |
| $5(12 + 7\lambda) = 30 + 55\lambda$; $\lambda = 3/2$ | M1 A1 | Equate $\omega^2$ to $4\pi^2/T^2$ to find $\lambda$ |

**Total: 12 marks (6 + 6)**

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\includegraphics[max width=\textwidth, alt={}, center]{a473cbb8-877f-48df-8751-c76d96396734-3_906_1538_248_301}

The end $A$ of a uniform $\operatorname { rod } A B$, of mass $4 m$ and length $3 a$, is rigidly attached to a point on a uniform spherical shell, of mass $\lambda m$ and radius $3 a$. The end $B$ of the rod is rigidly attached to a point on a uniform ring. The ring has centre $O$, mass $4 m$ and radius $\frac { 1 } { 2 } a$. The ring and the rod are in the same vertical plane. The line $O B A$, extended, passes through the centre of the spherical shell. $B C$ is a diameter of the ring (see diagram). Show that the moment of inertia of this system, about a fixed horizontal axis through $C$ perpendicular to the plane of the ring, is $( 30 + 55 \lambda ) m a ^ { 2 }$.

Given that the system performs small oscillations of period $2 \pi \sqrt { } \left( \frac { 5 a } { g } \right)$ about this axis, find the value of $\lambda$.

\hfill \mbox{\textit{CAIE FP2 2013 Q4 [12]}}