- (a) A uniform lamina of mass \(m\) is in the shape of a triangle \(A B C\). The perpendicular distance of \(C\) from the line \(A B\) is \(h\). Prove, using integration, that the moment of inertia of the lamina about \(A B\) is \(\frac { 1 } { 6 } m h ^ { 2 }\).
(b) Deduce the radius of gyration of a uniform square lamina of side \(2 a\), about a diagonal.
The points \(X\) and \(Y\) are the mid-points of the sides \(R Q\) and \(R S\) respectively of a square \(P Q R S\) of side \(2 a\). A uniform lamina of mass \(M\) is in the shape of \(P Q X Y S\).
(c) Show that the moment of inertia of this lamina about \(X Y\) is \(\frac { 79 } { 84 } M a ^ { 2 }\).