1 \includegraphics[max width=\textwidth, alt={}, center]{b486decd-75b8-44bd-889f-2472f1163871-2_553_435_258_854} Three identical uniform rods, \(A B , B C\) and \(C D\), each of mass \(M\) and length \(2 a\), are rigidly joined to form three sides of a square. A uniform circular disc, of mass \(\frac { 2 } { 3 } M\) and radius \(a\), has the opposite ends of one of its diameters attached to \(A\) and \(D\) respectively. The disc and the rods all lie in the same plane (see diagram). Find the moment of inertia of the system about the axis \(A D\).

## Question 1:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $I_{AB} = \frac{1}{3}Ma^2 + Ma^2 = \frac{4Ma^2}{3}$ | B1 | Find MI of rod $AB$ or $CD$ about $AD$ |
| $I_{BC} = M(2a)^2 = 4Ma^2$ | B1 | Find MI of rod $BC$ about $AD$ |
| $I_{Disc} = \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{2}{3} Ma^2 = \frac{Ma^2}{6}$ | M1 A1 | Find MI of disc about $AD$ by perpendicular axes |
| $I = 2I_{AB} + I_{BC} + I_{Disc} = \frac{41}{6}Ma^2$ | M1 A1 | Find MI of system about $A$ |
| **Total: 6 marks** | | |

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\includegraphics[max width=\textwidth, alt={}, center]{b486decd-75b8-44bd-889f-2472f1163871-2_553_435_258_854}

Three identical uniform rods, $A B , B C$ and $C D$, each of mass $M$ and length $2 a$, are rigidly joined to form three sides of a square. A uniform circular disc, of mass $\frac { 2 } { 3 } M$ and radius $a$, has the opposite ends of one of its diameters attached to $A$ and $D$ respectively. The disc and the rods all lie in the same plane (see diagram). Find the moment of inertia of the system about the axis $A D$.

\hfill \mbox{\textit{CAIE FP2 2013 Q1 [6]}}