Problem Statement: Body A is placed on a frictionless wedge making an angle theta with the horizon. The horizontal acceleration towards the left to be imparted to the wedge for the body A to freely fall vertically is?

Solution:

To solve this problem, we need to use the concept of resolving forces on the body and applying Newton's laws of motion.

Step 1: Draw a free-body diagram of the system.

The free-body diagram of the system is shown below:


Here, the forces acting on the body A are its weight mg, and the normal force N. The weight of the wedge is also acting vertically downwards.

Step 2: Resolve the weight of the body A into its components.

The weight of the body A can be resolved into two components as shown below:


Here, the component of the weight acting perpendicular to the wedge is balanced by the normal force N. Therefore, we only need to consider the component of the weight acting parallel to the wedge.

Step 3: Apply Newton's second law of motion.

The horizontal component of the weight of the body A is given by:

F = mg sin(theta)

According to Newton's second law of motion, the acceleration of the system is given by:

a = F / (m + M)

where m is the mass of the body A, and M is the mass of the wedge.

Step 4: Calculate the acceleration of the system.

a = (mg sin(theta)) / (m + M)

Therefore, the horizontal acceleration towards the left to be imparted to the wedge for the body A to freely fall vertically is:

a = (mg sin(theta)) / (m + M)

Conclusion:

Thus, the horizontal acceleration towards the left to be imparted to the wedge for the body A to freely fall vertically is (mg sin(theta)) / (m + M).

I think a = g sin thetha