Determine the moment of inertia for a uniform semicircular wire of...

Consider an elemental mass $dm$ on the semicircular wire.

The distance of this elemental mass from the axis passing through the centre and perpendicular to the plane is equal to the radius $r$ of the semicircle.

The moment of inertia of this elemental mass about the given axis is $dI = (dm)r^2$.

Integrating over the entire semicircular wire, the total moment of inertia is $I = \int dI = \int r^2 dm$.

Since the distance $r$ is constant for all points on the wire, it can be taken out of the integral:

$I = r^2 \int dm$

The total mass of the semicircular wire is given as $\int dm = M$.

Therefore, $I = Mr^2$.