Question

A small part of the rim of a flywheel breaks off while it is rotating at a constant angular speed. Then its radius of gyration will.
A) increase.
B) decrease.
C) Remain unchanged.
D) Nothing definite can be said.

Answer 1

Hint: The radius of gyration of a body is the distance to point on the body which is considered to have all mass of the body if the mass is to be concentrated. The moment of inertia is the product of mass and the square of the distance from the axis.

Complete step by step answer:
As it is given that the rim of the flywheel breaks off while it is rotating at an angular constant speed which means that the mass of the flywheel has decreased a little bit which will definitely decrease the inertia of the rim of a flywheel.
As the moment of inertia is given by $I = m{k^2}$ where $I$ is the moment of inertia, m is the mass of the body and k is the distance of the point where all the mass can be concentrated from the axis of rotation.
As the mass of the rim is decreased therefore the moment of inertia of the body will decrease but as the angular momentum is always conserved therefore as the angular momentum is conserved therefore the moment of inertia decrease causes the increase in the angular velocity and the decrease in moment of inertia of the will also cause a decrease in the radius of gyration.

Therefore, the correct option is option B.

Note:
The angular momentum of a rotating body is always conserved which means that with a decrease in the moment of inertia the angular velocity of the body will increase. The moment of inertia depends upon the mass of the body and radius of gyration.