1.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d4fc2ea6-3ffc-42f2-b462-9694adfe2ec1-02_826_649_244_708}
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\caption{Figure 1}
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A letter P from a shop sign is modelled as a uniform plane lamina which consists of a rectangular lamina, \(O A B D E\), joined to a semicircular lamina, \(B C D\), along its diameter \(B D\).
$$O A = E D = a , A B = 2 a , O E = 4 a \text {, and the diameter } B D = 2 a \text {, as shown in Figure } 1 .$$
Using the model,
- find, in terms of \(\pi\) and \(a\), the distance of the centre of mass of the letter P ,
from (i) \(O E\)
(ii) \(O A\)
The letter P is freely suspended from \(O\) and hangs in equilibrium. The angle between \(O E\) and the downward vertical is \(\alpha\).
Using the model, - find the exact value of \(\tan \alpha\)