Question

Prove the result that the velocity v of translation of a rolling body (like a ring, disc, cylinder or sphere) af the bottom of an inclined plane of a height h is given by v² =2gh/(1+k²/R²) , using dynamical consideration (i.e. by consideration of forces and torques). Note k is the radius of gyration of the body about its symmetry axis, and R is the radius of the body. The body starts from rest at the top of the plane.

Answer 1

Consider the figure

Let R be the radius of the body and k be the radius of gyration about its axis of symmetry let the body have a man m and initial height is h

∴ I = mk² ……. (1)

The body translates without slipping.

∴ v = r ω ………(2)

Also, at any point of time, the friction does not do any work, since the point of contact to plane has zero velocity.

∴ The only work done is by gravity