- \hspace{0pt} [In this question, you may assume that the centre of mass of a circular arc, radius \(r\), with angle at centre \(2 \alpha\), is a distance \(\frac { r \sin \alpha } { \alpha }\) from the centre.]
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\caption{Figure 5}
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A thin non-uniform metal plate is in the shape of a sector \(O A B\) of a circle with centre \(O\) and radius \(a\). The angle \(A O B = \frac { \pi } { 2 }\), as shown in Figure 5.
The plate is modelled as a non-uniform lamina.
The mass per unit area of the lamina, at any point \(P\) of the lamina, is modelled as \(k ( O P ) ^ { 2 }\), where \(k = \frac { 4 \lambda } { \pi a ^ { 4 } }\) and \(\lambda\) is a constant.
Using the model,
- find the mass of the plate in terms of \(\lambda\),
- find, in terms of \(a\), the distance of the centre of mass of the plate from \(O\).