Determine the radius of gyration of a circular ring of radius r about a line perpendicular to the ring's plane and passing through one of its particles.

Step-by-step solution

The moment of inertia of a circular ring of mass M and radius r about an axis passing through its centre and perpendicular to its plane is I_ cm = M r^2 . Using the parallel axis theorem, the moment of inertia of the ring about an axis passing through a particle on its circumference and perpendicular to its plane is given by: I = I_ cm + M d^2 Here, the distance between the parallel axes is d = r . I = M r^2 + M r^2 = 2 M r^2 The radius of gyration k is defined by I = M k^2 . Equating the two expressions for I : M