AIIMS 2005 Physics Motion of Rigid Bodies MCQ Question

To understand the problem of a ladder leaning against a smooth wall and slipping on a frictionless floor, it is essential to analyze the motion and trajectory of its center of mass.

Explanation of the Correct Answer (Option C)

When a ladder is leaned against a smooth wall and slips on a frictionless floor, it undergoes a specific motion characterized by the presence of two forces: gravitational force acting downward and the normal force exerted by the wall acting horizontally.

As the ladder slips, it rotates about its base (the point in contact with the ground). The center of mass of the ladder will trace out a path that reflects this rotational motion.

1. Path of the Center of Mass: The center of mass of the ladder moves in a parabolic trajectory. This is because as the ladder slips, both its height (vertical position) and horizontal position change simultaneously. The vertical displacement decreases while the horizontal displacement increases, resulting in a curve.

2. Mathematical Representation: Let the length of the ladder be $L$. If we denote the angle the ladder makes with the floor as $\theta$, the coordinates of the center of mass (assuming the base of the ladder is at the origin) can be represented as:

   • Horizontal position: $x = \frac{L}{2} \cos(\theta)$
   • Vertical position: $y = \frac{L}{2} \sin(\theta)$

   As the angle $\theta$ decreases (ladder slips down), $x$ increases while $y$ decreases, which describes a parabolic path.

Thus, the correct representation of the trace of the center of mass is depicted in Option C, which shows a parabolic trajectory.

Why Other Options Are Incorrect

1. Option A (1): This option likely represents a linear path or a straight line, which is not accurate for the motion of the center of mass of the ladder. The slipping motion is inherently parabolic due to the simultaneous changes in horizontal and vertical positions.

2. Option B (2): Similar to Option A, this option may show a different type of curve that does not reflect the nature of the motion. It might depict an incorrect curvature or a shape that does not match the path traced by the center of mass.

3. Option D (4): This option may represent a different trajectory that is not consistent with the physics of the situation. It could represent a circular or non-parabolic shape which does not accurately describe how the center of mass moves as the ladder slips.

Conclusion

In summary, the center of mass of the ladder, when it slips down a frictionless floor while leaning against a smooth wall, follows a parabolic trajectory due to the nature of its motion. The correct answer is Option C, and the rationale lies in the understanding of how the ladder's center of mass behaves under the influence of gravity and the constraints of the wall and floor. This combination leads to a distinct parabolic path, which is modeled by the simultaneous changes in height and base position of the ladder.