An annular disk of mass $M$, inner radius $a$ and outer radius $b$ is placed on a horizontal surface with coefficient of friction $\mu$, as shown in the figure. At some time, an impulse $j_0\hat{x}$ is applied at a height $h$ above the center of the disk. If $h = h_m$ then the disk rolls without slipping along the $x$-axis. Which of the following statement(s) is(are) correct?

Step 1: Relate linear impulse to velocity.

The linear impulse $j_0$ in the $x$-direction determines the velocity $v$ of the center of mass.

$$Mv = j_0$$ $$v = \frac{j_0}{M}$$

Step 2: Relate angular impulse to angular velocity.

The angular impulse about the center of mass due to the force applied at height $h$ determines the angular velocity $\omega$.

$$I\omega = j_0 h$$ $$\omega = \frac{j_0 h}{I}$$

Step 3: Write the moment of inertia of the annular disk.

The moment of inertia $I$ of the annular disk about its central axis is given by its mass and radii.

$$I = \frac{1}{2}M(a^2 + b^2)$$

Step 4: Apply the condition for rolling without slipping.

For pure rolling along the horizontal surface, the velocity at the contact point must be zero.

$$v = \omega b$$

Step 5: Solve for the height $h_m$.

Substitute the expressions for $v$ and $\omega$ into the rolling condition.

$$\frac{j_0}{M} = \frac{j_0 h_m b}{I}$$ $$h_m = \frac{I}{Mb}$$ $$h_m = \frac{a^2 + b^2}{2b}$$

Step 6: Evaluate Statement (1).

For a solid disk, the inner radius $a$ approaches zero.

$$\lim_{a \to 0} h_m = \frac{b}{2}$$

This matches Statement (1), so Statement (1) is correct.

Step 7: Evaluate Statement (2).

Setting $h_m = b$ requires a specific relation between the radii.

$$\frac{a^2 + b^2}{2b} = b$$ $$a = b$$

This is not true for a general annular disk where $a < b$, so Statement (2) is incorrect.

Step 8: Evaluate Statement (3).

Calculate the angular velocity $\omega$ when the impulse is applied at height $h_m$.

$$\omega = \frac{j_0 h_m}{I}$$ $$\omega = \frac{j_0}{Mb}$$

Since this does not depend on $a$, Statement (3) is correct.

Step 9: Evaluate Statement (4).

When $\mu = 0$ and $h = 0$, there is no torque acting on the disk.

$$\omega = 0$$ $$v = \frac{j_0}{M} \neq 0$$

The disk slides without rolling, so Statement (4) is correct.

Answer: Statements (1), (3), and (4) are correct.