Radius of Gyration — Prelims Questions

A solid cylinder of mass $M$ and radius $R$ rolls without slipping on a horizontal surface. If its radius of gyration about its own axis is $K$, what is the ratio of its rotational kinetic energy to its total kinetic energy?

A thin circular ring and a circular disc, both of the same mass $M$ and radius $R$, are rotating about an axis passing through their centers and perpendicular to their planes. Which one has a larger radius of gyration?

A thin uniform rod of length $L$ and mass $M$ has a radius of gyration $K_1$ about an axis perpendicular to its length and passing through its center. If the axis is shifted to one end of the rod, perpendicular to its length, its new radius of gyration is $K_2$. What is the ratio $K_1/K_2$?

The radius of gyration of a body depends on:

A solid sphere of mass $M$ and radius $R$ has a radius of gyration $K_S$ about its diameter. A hollow sphere of the same mass $M$ and radius $R$ has a radius of gyration $K_H$ about its diameter. What is the relationship between $K_S$ and $K_H$?

A uniform square plate of mass $M$ and side $a$ has a radius of gyration $K$ about an axis passing through its center and perpendicular to its plane. What is the value of $K$?