Determine the moment of inertia for a uniform semicircular wire of mass M and radius r , taken about an axis passing through its centre and perpendicular to its plane.

Step-by-step solution

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 = dI = 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 dm The total mass of the s