A rigid uniform bar $AB$ of length $L$ is slipping from its vertical position on a frictionless floor (as shown in the figure). At some instant of time, the angle made by the bar with the vertical is $\theta$. Which of the following statements about its motion is/are correct?

Step 1: Analyze the forces acting on the bar.

The forces acting on the uniform bar $AB$ of mass $m$ and length $L$ are the gravitational force $mg$ acting vertically downwards at the midpoint, and the normal force $N$ acting vertically upwards at the contact point $B$.

Since the floor is frictionless, there are no horizontal forces.

$$F_x = 0$$

Step 2: Analyze the motion of the midpoint.

Since there is no horizontal force, the horizontal position of the center of mass (midpoint) remains constant.

$$x_{\text{CM}} = 0$$

Thus, the midpoint falls vertically downward. This confirms statement (1) is correct.

Step 3: Determine the trajectory of point $A$.

Let the vertical line of fall of the midpoint be the y-axis, so $x_{\text{CM}} = 0$.

The coordinates of point $A$ at any angle $\theta$ with the vertical are given by:

$$x_A = -\frac{L}{2}\sin\theta$$ $$y_A = L\cos\theta$$

Eliminating $\theta$ yields the equation of the trajectory:

$$\left(\frac{x_A}{L/2}\right)^2 + \left(\frac{y_A}{L}\right)^2 = 1$$

This is the equation of an ellipse, not a parabola. This confirms statement (2) is incorrect.

Step 4: Calculate the torque about the contact point $B$.

The torque about the contact point $B$ is due solely to the gravitational force acting at the midpoint.

$$\tau_B = mg \left(\frac{L}{2}\sin\theta\right)$$ $$\tau_B \propto \sin\theta$$

This confirms statement (3) is correct.

Step 5: Calculate the displacement of the midpoint.

The initial height of the midpoint is $\frac{L}{2}$, and its height at angle $\theta$ is $\frac{L}{2}\cos\theta$.

The vertical displacement $\Delta y_{\text{CM}}$ is:

$$\Delta y_{\text{CM}} = \frac{L}{2} - \frac{L}{2}\cos\theta$$ $$\Delta y_{\text{CM}} = \frac{L}{2}(1 - \cos\theta)$$ $$\Delta y_{\text{CM}} \propto (1 - \cos\theta)$$

This confirms statement (4) is correct.

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