Question

A heavy ball is thrown on a rough horizontal surface in such a way that it slides with a speed ($v_o$) initially without rolling. It will roll without sliding when its speed falls to:

• A. $ \frac{2}{7}{{\upsilon }_{o}} $
• B. $ \frac{3}{7}{{\upsilon }_{o}} $
• C. $ \frac{5}{7}{{\upsilon }_{o}} $
• D. $ \frac{7}{5}{{\upsilon }_{o}} $

Answer 1

The Correct Option is C

Answer (c) \(\frac{5}{7}{{\upsilon }_{o}}\)

Here, the frictional force f, weight mg, and normal contact force N pass through the point of contact P.
So, their toque about P will be equal to zero.

⇢ Angular momentum of the sphere remains constant before and after the start of pure rolling about P.

According to Conserving the angular momentum of the sphere at positions 1 & 2, about the instantaneous point of contact, We get

\(mv_or = mvr + (\frac2{5}mr^2)\omega\)

\(v_o =  \frac{5v+2r\omega}5\)

Let \(rw=v \) for rolling

Now get \(v= \frac{5v_o}7\)