Question

A horizontal force $F$ is applied at the centre 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?

• 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 $2 \mu g$
• C. The magnitude of the frictional force on the object due to the ground is always $\mu mg$
• D. For a thin-walled hollow cylinder, $a=\frac{F}{2 m}$

Answer 1

The Correct Option is D

Given:

• A force \( F \) is applied horizontally at the center of mass of a rolling cylindrical object of mass \( m \) and radius \( R \)
• Coefficient of friction = \( \mu \), and the object rolls without slipping
• Acceleration due to gravity = \( g \)
• We need to find the linear acceleration \( a \) of the center of mass

Important condition: No slipping means the point of contact has zero relative velocity, hence:

\[ a = R \alpha \quad \text{(where } \alpha \text{ is angular acceleration)} \]

Translational motion:
Net force on the center of mass: \[ F - f = m a \tag{1} \]

Rotational motion about center:
\[ f R = I \alpha \Rightarrow f R = I \cdot \frac{a}{R} \tag{2} \]

For a thin-walled hollow cylinder:
Moment of inertia about the center is: \[ I = m R^2 \] Substituting in equation (2): \[ f R = m R^2 \cdot \frac{a}{R} \Rightarrow f = m a \tag{3} \]

Now substitute (3) into (1): \[ F - m a = m a \Rightarrow F = 2 m a \Rightarrow a = \frac{F}{2m} \]

✓ Correct Answer: Option (D): For a thin-walled hollow cylinder, \( a = \dfrac{F}{2m} \)