Rolling Motion and Acceleration of Cylinder under Horizontal Force

To determine the correct statements, let us analyze the dynamics of a cylindrical object of mass $m$, radius $R$, and moment of inertia $I$ rolling without slipping on a horizontal surface under the action of a horizontal force $F$ applied at its center of mass.

1. Equations of Motion

Let the center of mass have a linear acceleration $a$ to the right, and let the cylinder have an angular acceleration $\alpha$ in the clockwise direction.

1. Translational Motion: Applying Newton's second law along the horizontal direction: $F - f = m a \quad \text{--- (1)}$ where $f$ is the force of static friction acting to the left at the point of contact with the ground.

2. Rotational Motion: Taking torque about the center of mass: $\tau = f R = I \alpha \quad \text{--- (2)}$

3. Condition for Pure Rolling (Rolling Without Slipping): $a = R \alpha \quad \text{--- (3)}$

From Equation (2) and Equation (3), we get the magnitude of static friction: $f = \frac{I \alpha}{R} = \frac{I a}{R^2} \quad \text{--- (4)}$

Substituting Equation (4) into Equation (1): $F - \frac{I a}{R^2} = m a \implies F = \left(m + \frac{I}{R^2}\right) a$

Thus, the linear acceleration of the center of mass is given by: $a = \frac{F}{m + \frac{I}{R^2}} \quad \text{--- (5)}$

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2. Analysis of the Options

• Option (A):

  • For a solid cylinder, $I = \frac{1}{2} m R^2$: $a_{\text{solid}} = \frac{F}{m + \frac{1}{2}m} = \frac{2F}{3m}$
  • For a thin-walled hollow cylinder, $I = m R^2$: $a_{\text{hollow}} = \frac{F}{m + m} = \frac{F}{2m}$ Since $a_{\text{solid}} \neq a_{\text{hollow}}$, the value of acceleration $a$ for a given force $F$ depends on whether the cylinder is solid or hollow. Therefore, Option (A) is incorrect.

• Option (B): For a solid cylinder ($I = \frac{1}{2}m R^2$), the frictional force from Equation (4) is: $f = \frac{\frac{1}{2}m R^2 a}{R^2} = \frac{1}{2} m a$ For pure rolling without slipping, static friction must satisfy: $f \le f_{\text{max}} = \mu N = \mu m g$ $\frac{1}{2} m a \le \mu m g \implies a \le 2\mu g$ Hence, the maximum possible acceleration for pure rolling of a solid cylinder is $2\mu g$. Therefore, Option (B) is correct.

• Option (C): The friction force $f = \frac{I a}{R^2} = \left(\frac{I}{I + m R^2}\right) F$ is static in nature and adjusts its value according to $F$. It is equal to $\mu m g$ only when the cylinder is on the verge of slipping. Therefore, Option (C) is incorrect.

• Option (D): For a thin-walled hollow cylinder, $I = m R^2$. Substituting $I$ into Equation (5): $a = \frac{F}{m + \frac{m R^2}{R^2}} = \frac{F}{2m}$ Therefore, Option (D) is correct.

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Conclusion

The correct statements are (B) and (D).