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

Answer: (C)

$\frac{5}{7}{{\upsilon }_{o}}$

Answer 2

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 $v= \frac{5v_o}7$