A ladder is leaned against a smooth wall and it is allowed to slip on a frictionless floor. Which figure represents the track of its centre of mass ?

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining the track of the center of mass of a ladder leaning against a smooth wall and slipping on a frictionless floor, we can follow these steps:

### Step 1: Understand the System
We have a ladder leaning against a wall, with one end on the floor and the other end touching the wall. The wall is smooth, meaning there is no friction, and the floor is also frictionless. This means that the ladder can slide down without any resistance.

**Hint:** Visualize the ladder's position and the forces acting on it.

### Step 2: Identify the Forces
The forces acting on the ladder include:
- The gravitational force acting downward at the center of mass of the ladder (mg).
- The normal force exerted by the wall acting horizontally.
- The normal force exerted by the floor acting vertically.

**Hint:** Draw a free-body diagram to help visualize the forces.

### Step 3: Motion of the Center of Mass
As the ladder slips, the center of mass will move in a specific path. Since the wall is smooth and the floor is frictionless, the center of mass will not experience any horizontal force, and its vertical motion will be influenced only by gravity.

**Hint:** Consider how the center of mass behaves under the influence of gravity and the constraints of the wall and floor.

### Step 4: Analyze the Path
The center of mass of the ladder will follow a curved path as the ladder slides down. The vertical position of the center of mass will decrease while the horizontal position will change as the ladder slides down the wall.

**Hint:** Think about the trajectory of a point mass under the influence of gravity and how it relates to the ladder's position.

### Step 5: Determine the Correct Figure
From the options provided, we need to find the figure that best represents the path of the center of mass. The correct figure will show a downward curve that reflects the motion of the center of mass as it descends while the ladder slips.

**Hint:** Look for a figure that shows a downward curve, indicating the center of mass is moving downward as the ladder slides down.

### Conclusion
After analyzing the situation, we conclude that the correct representation of the track of the center of mass is option C, which shows the downward trajectory of the center of mass as the ladder slips.

**Final Answer:** The correct option is C.

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