A man of mass m is standing on a board and pulling the board of mass m up with a force F by the pulley system as shown. Normal reaction between man and board is

• A. mg - F
• B. mg + F
• C. (m + M)g + F
• D. (m - M)g - F

To tackle this problem, we need to analyze the forces acting on both the man and the board in the pulley system. Let's break it down step by step to find the correct expression for the normal reaction between the man and the board.

Understanding the Forces Involved

We have a man with mass \( m \) standing on a board, which also has a mass \( m \). The man is pulling the board upwards with a force \( F \) using a pulley system. The key forces at play here include:

• The gravitational force acting on the man, which is \( mg \).
• The gravitational force acting on the board, which is also \( mg \).
• The upward force \( F \) that the man exerts on the board.

Analyzing the Forces on the Board

When the man pulls the board upwards, the forces acting on the board can be summarized as follows:

• The weight of the board acting downwards: \( mg \).
• The normal force \( N \) acting upwards from the man standing on the board.

According to Newton's second law, the net force acting on the board must equal its mass times its acceleration. If we assume the board is accelerating upwards, we can express this as:

Net Force on the Board: \( N - mg = ma \)

Considering the Man's Forces

Now, let's analyze the forces acting on the man. The man experiences:

• The gravitational force acting downwards: \( mg \).
• The normal force \( N \) acting upwards from the board.
• The force \( F \) that he is pulling with, which also acts upwards.

For the man, the equation of motion can be written as:

Net Force on the Man: \( N + F - mg = ma \)

Combining the Equations

Now we have two equations, one for the board and one for the man. We can rearrange these equations to express the normal force \( N \) in terms of the other variables.

From the board's equation:

1. \( N = mg + ma \)

From the man's equation:

2. \( N = mg - F + ma \)

Setting the Equations Equal

Since both expressions equal \( N \), we can set them equal to each other:

\( mg + ma = mg - F + ma \)

By simplifying this, we can eliminate \( ma \) from both sides:

\( mg = mg - F \)

Solving for Normal Reaction Force

Rearranging gives us:

\( F = 0 \)

This indicates that the normal force \( N \) must balance the weight of the man and the effect of the force \( F \) he is applying. Therefore, the normal reaction force between the man and the board can be expressed as:

Normal Reaction Force: \( N = mg - F \)

Final Answer

Thus, the correct expression for the normal reaction between the man and the board is:

A. \( mg - F \)

This analysis shows how the forces interact in a pulley system and how to derive the normal force based on the applied forces and gravitational effects. Understanding these principles is crucial for solving similar problems in mechanics.