- \hspace{0pt} [In this question you may quote, without proof, the formula for the distance of the centre of mass of a uniform circular arc from its centre.]
\begin{figure}[h]
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\caption{Figure 1}
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Five pieces of a uniform wire are joined together to form the rigid framework \(O A B C O\) shown in Figure 1, where
- \(O A , O B\) and \(B C\) are straight, with \(O A = O B = B C = r\)
- arc \(A B\) is one quarter of a circle with centre \(O\) and radius \(r\)
- arc \(O C\) is one quarter of a circle of radius \(r\)
- all five pieces of wire lie in the same plane
- Show that the centre of mass of arc \(A B\) is a distance \(\frac { 2 r } { \pi }\) from \(O A\).
Given that the distance of the centre of mass of the framework from \(O A\) is \(d\),
show that \(\mathrm { d } = \frac { 7 r } { 2 ( 3 + ) }\)
The framework is freely pivoted at \(A\).
The framework is held in equilibrium, with \(A O\) vertical, by a horizontal force of magnitude \(F\) which is applied to the framework at \(C\).
Given that the weight of the framework is \(W\)find \(F\) in terms of \(W\)