Question

A semicircular plate of mass 'm' has radius ‘r’ and centre 'c'. The centre of mass of the plate is at a distance 'x' from its centre 'c'. Its moment of inertia about an axis passing through its centre of mass and perpendicular to its plane is

Answer 1

Moment of inertia of a semi circular plate about its centre C=$\frac{mr^2}{2}$

Moment of inertia about axis passing through a distance "x" from centre=

$I_x=I_c-mx^2$

$I_x=\frac{mr^2}{2}-mx^2$

This x is the centre of mass distance which is equal to $\frac{4r}{3 \pi}$

Substituting this value of x,

we get,

$I_x=\frac{mr^2}{2}-m{(\frac{4r}{3 \pi})}^2$

$I_x=\frac{mr^2}{2}-\frac{16 m r^2}{9 \pi ^2}$