A horizontal force $F$ is applied at the center of mass of a cylindrical object of mass m and radius $R$, perpendicular to its axis as shown in the figure. The coefficient of friction between the object and the ground is $\mu$. The center of mass of the object has an acceleration $a$. The acceleration due to gravity is $g$. Given that the object rolls without slipping, which of the following statement(s) is(are) correct?

(B) For a solid cylinder, the maximum possible value of $a$ is $2\mu g$; (D) For a thin-walled hollow cylinder, $a=\frac{F}{2m}$

(A) For the same $F$, the value of $a$ does not depend on whether the cylinder is solid or hollow

(C) The magnitude of the frictional force on the object due to the ground is always $\mu mg$

JEE Advanced · 2021 · 03 Oct 2021 · Paper 1 · Rotational Mechanics

A horizontal force

F

is applied at the center of mass of a cylindrical object of mass m and radius

R

, perpendicular to its axis as shown in the figure. The coefficient of friction between the object and the ground is

μ

. The center of mass of the object has an acceleration

a

. The acceleration due to gravity is

g

. Given that the object rolls without slipping, which of the following statement(s) is(are) correct?

(A)For the same $F$ , the value of $a$ does not depend on whether the cylinder is solid or hollow
(B)For a solid cylinder, the maximum possible value of $a$ is $2 μg$ ✓ Correct
(C)The magnitude of the frictional force on the object due to the ground is always $μ m g$
(D)For a thin-walled hollow cylinder, $a = \frac{F}{2 m}$ ✓ Correct

Answer: (B), (D)

For rolling without slipping:

F - f = m a_{C}

and

f R = I_{C} α

with

a_{C} = α R

.

Combining:

a_{C} = \frac{F}{m + I _{C} / R ^{2}}

and

f = \frac{F I _{C} / R ^{2}}{m + I _{C} / R ^{2}}

.

For a thin-walled hollow cylinder,

I_{C} = m R^{2}

, so

a_{C} = \frac{F}{2 m}

✓ (option D).

For a solid cylinder,

I_{C} = \frac{m R ^{2}}{2}

. The friction constraint is

f = \frac{I _{C} a _{C}}{R ^{2}} \leq μ m g

, giving

a_{C} \leq \frac{μ m g R ^{2}}{I _{C}} = \frac{μ m g R ^{2}}{m R ^{2} /2} = 2 μg

✓ (option B).

Option (A) is incorrect because

a_{C}

depends on

I_{C}

, which differs for solid vs hollow cylinders.

Option (C) is incorrect because

f = \frac{F I _{C} / R ^{2}}{m + I _{C} / R ^{2}}

, not always

μ m g

JEE Advanced · 2021 · 03 Oct 2021 · Paper 1 · One or more correct options