Question

A hollow cylinder, a spherical shell, a solid cylinder and a solid sphere are allowed to roll purely on an inclined rough surface of inclination $\theta$. The CORRECT statement(s) is/are:

• A. Frictional force will be equal for all the rolling objects, if having same mass
• B. If all the objects have same mass, the kinetic energy (K.E.) of all the objects will be same at the bottom of inclined plane
• C. Work done by the gravitational force between the top and bottom points will be same for all, if the masses are same.
• D. If cylindrical shell can roll on inclined plane, all other objects will also roll.

Answer 1

Answer: (B), (C), (D)

The correct options are
B If all the objects have same mass, the kinetic energy (K.E.) of all the objects will be same at the bottom of inclined plane
C Work done by the gravitational force between the top and bottom points will be same for all, if the masses are same.
D If cylindrical shell can roll on inclined plane, all other objects will also roll.

We know that for inclined plane condition,
$mg\sin \theta -f=ma$
Now, to satisfy the condition of pure rolling
$a=r\alpha$
and $fr=I\alpha$

On substituting, we get
$f=\frac{mg\sin \theta }{1+\frac{m{r}^2}{I}}$
Again, $f=\mu mg\cos \theta$
Therefore, on dividing both we get
$\mu =\frac{\tan \theta }{1+\frac{m{r}^2}{I}}$
Thus MOI for hollow cylinder is maximium of all the three objects therefore maximium $\mu$ is offered by the plane, so if it can roll all other objects will roll too.

Change in potential energy is same and no frictional loss in case of pure rolling, then applying energy conservation we can conclude that K.E. will be same at the bottom.