Question

A sphere of mass $M$ and radius $r$ slips on a rough horizontal plane. At some instant it has translation velocity $v_0$ and rotational velocity about the centre $v_0/2r$. The translation velocity after the sphere starts pure rolling

• A. $6v_0/7$ in forward direction
• B. $6v_0/7$ in backward direction
• C. $7v_0/6$ in forward direction
• D. $7v_0/6$ in backward direction

Answer 1

The correct option is A $6v_0/7$ in forward direction

Initially the velocity of COM of sphere is $v_0$ and angular velocity is $w_i$

When the pure rolling starts, let the velocity of COM of sphere be $V$ and angular velocity be $w_f$

Now,

$f=Ma⟹a=\frac{f}{M}$

$v=v_0-at⟹v=v_0-\frac{f}{M}t$ ..........(1)

Also,

$\tau =I\alpha$

$f_r=\frac{2}{5}M{r}^2\alpha ⟹\alpha \frac{5f}{2Mr}$

$w_f=w_i+\alpha t$

$w_f=\frac{v_0}{2r}+\frac{5ft}{2Mr}⟹\frac{2}{5}r{w}_f=\frac{v_0}{2}\times \frac{2}{5}+\frac{f_t}{M}$ ........(2)

Adding (1) and (2), we get

$v+\frac{2}{5}r{w}_f=\frac{v_0}{5}+v_0$

Also

$v=r{w}_f$ (pure rolling)

$⟹v=\frac{6v_0}{7}$ in forward direction.