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
6v0/7 in forward direction
B
6v0/7 in backward direction
C
7v0/6 in forward direction
D
7v0/6 in backward direction

Solution

## The correct option is A $$6{v}_{0}/7$$ in forward directionInitially 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,$$f=Ma \implies a=\dfrac fM$$$$v=v_0-at\implies v=v_0-\dfrac fM t$$       ..........(1)Also,$$\tau =I\alpha$$$$f_r=\dfrac 25 Mr^2 \alpha \implies \alpha \dfrac{5f}{2Mr}$$$$w_f=w_i+\alpha t$$$$w_f=\dfrac{v_0}{2r}+\dfrac{5ft}{2Mr}\implies \dfrac 25 rw_f=\dfrac{v_0}{2}\times \dfrac 25+\dfrac{ f_t}{M}$$        ........(2)Adding (1) and (2), we get$$v+\dfrac 25 rw_f=\dfrac{v_0}{5}+v_0$$Also$$v=rw_f$$    (pure rolling)$$\implies v=\dfrac{6v_0}{7}$$ in forward direction.Physics

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