Question

A solid disc and a ring, both of radius 10 cm are placed on a horizontal table simultaneously, with initial angular speed equal to $10\pi$ rad $s^{-1}$. Which of the two will start to roll earlier ? The co-efficient of kinetic friction is ${\mu }_k=0.2$.

Answer 1

The motion of the two objects is caused by the frictional force acting on the objects=${\mu }_{k}mg$

Thus acceleration due to frictional force=${\mu }_{k}g$

Thus linear velocity attained in time t,$v=u+at=0+{\mu }_{k}gt={\mu }_{k}gt$

Also a torque would act due to friction causing a rotation about the center.

Thus $\tau =I\alpha$

$⟹fr=I\alpha$

Thus angular acceleration, $\alpha =\frac{fr}{I}=\frac{{\mu }_{k}mgr}{I}$

Thus angular velocity attained after time t, $\omega ={\omega }_0-\alpha t$

When rolling starts, $v=r\omega$

$⟹{\mu }_{k}gt=r({\omega }_0-\frac{{\mu }_{k}mgr}{I}t)$

$⟹t=\frac{r{\omega }_0}{{\mu }_{k}g+\frac{{\mu }_{k}mg{r}^2}{I}}$

Since $I$ for ring is greater than that for solid disc, ring takes longer time to achieve pure rolling. Thus the solid disc starts to roll earlier.