7. In this question you may use, without proof, the formula for the centre of mass of a uniform sector of a circle, as given in the formulae book.
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\caption{Figure 3}
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The uniform lamina \(A B C D E\), shown shaded in Figure 3, is formed by joining a rectangle to a sector of a circle.
- The rectangle \(A B C E\) has \(A B = E C = a\) and \(A E = B C = d\)
- The sector \(C D E\) has centre \(C\) and radius \(a\)
- Angle \(E C D = \frac { \pi } { 3 }\) radians
The centre of mass of the lamina lies on EC.
- Show that \(a = \sqrt { 3 } d\)
The lamina is freely suspended from \(B\) and hangs in equilibrium with \(B C\) at an angle \(\beta\) radians to the downward vertical.
- Find the value of \(\beta\)