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?

Step 1: Define the moment of inertia and parameters.

Let the moment of inertia of the cylindrical object about its central axis be represented as:

$$I = \lambda m R^2$$

where:

• For a solid cylinder, $\lambda = \frac{1}{2}$
• For a thin-walled hollow cylinder, $\lambda = 1$

Step 2: Set up the equations of motion.

Let $f$ be the friction force acting at the contact point with the ground, and $a$ be the acceleration of the center of mass. Taking the direction of the applied force $F$ as positive, the equation for horizontal translation is:

$$F + f = ma$$

For rotation about the center of mass, the torque equation is:

$$fR = -I\alpha$$

where $\alpha$ is the angular acceleration. Since the object rolls without slipping, the linear and angular accelerations are related by:

$$a = \alpha R$$

Step 3: Solve for acceleration $a$ and friction force $f$.

Substitute $\alpha = \frac{a}{R}$ and $I = \lambda m R^2$ into the torque equation:

$$fR = -(\lambda m R^2)\left(\frac{a}{R}\right)$$ $$f = -\lambda m a$$

Substitute this expression for $f$ into the translational equation:

$$F - \lambda m a = ma$$ $$a = \frac{F}{m(1+\lambda)}$$

Using this acceleration, the friction force is:

$$f = -\frac{F\lambda}{1+\lambda}$$

Step 4: Analyze the cases for solid and hollow cylinders.

For a solid cylinder ($\lambda = \frac{1}{2}$):

$$a = \frac{2F}{3m}$$ $$f = -\frac{F}{3}$$

For a thin-walled hollow cylinder ($\lambda = 1$):

$$a = \frac{F}{2m}$$ $$f = -\frac{F}{2}$$

Since the acceleration $a$ depends on $\lambda$, it is different for solid and hollow cylinders. Thus, Statement (1) is false, and Statement (4) is correct.

Step 5: Determine the maximum acceleration for rolling without slipping.

For the object to roll without slipping, the magnitude of the static friction force must not exceed its maximum limit:

$$|f| \le \mu N$$

Since there is no vertical acceleration, the normal force is $N = mg$:

$$\frac{F\lambda}{1+\lambda} \le \mu mg$$ $$F \le \frac{\mu mg(1+\lambda)}{\lambda}$$

The maximum acceleration $a_{\max}$ corresponding to this maximum force is:

$$a_{\max} = \frac{F_{\max}}{m(1+\lambda)} = \frac{\mu g}{\lambda}$$

For a solid cylinder ($\lambda = \frac{1}{2}$):

$$a_{\max} = 2\mu g$$

This shows that Statement (2) is correct. Statement (3) is false because the friction force is self-adjusting and only reaches $\mu mg$ at the slipping threshold.

Answer: The correct statements are (2) and (4).