A block of mass M hanging over a smooth and light pulley through a light string. The other end of the string is pulled by a constant force F. The kinetic of the block increases by `40J` in `1s`. State whether the following statements are true or false.
(a) The tension in the string is `Mg` .
(b) The work done the tension on the block is `40J`.
(c) the tension in the string is `F`.
(d) The work done by the force of gravity is 40J in the above 1s .

Explanation

To solve the problem, we need to analyze the situation with the block, pulley, and forces acting on the block. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Forces Acting on the Block
- The block of mass \( M \) is hanging and experiences two main forces: the gravitational force \( Mg \) acting downwards and the tension \( T \) in the string acting upwards.
- Additionally, there is a constant force \( F \) being applied to the string on the other side of the pulley.

### Step 2: Analyze the Change in Kinetic Energy
- The problem states that the kinetic energy of the block increases by \( 40 \, J \) in \( 1 \, s \).
- According to the work-energy theorem, the total work done on the block is equal to the change in kinetic energy. Therefore, the total work done by all forces acting on the block is \( 40 \, J \).

### Step 3: Evaluate Each Statement
#### Statement (a): The tension in the string is \( Mg \).
- **Analysis**: The tension \( T \) in the string is not equal to \( Mg \) because the string is being pulled with a force \( F \). The tension will equal the applied force \( F \) when the system is in motion.
- **Conclusion**: **False**.

#### Statement (b): The work done by the tension on the block is \( 40 \, J \).
- **Analysis**: The work done by the tension is part of the total work done. Since the total work done is \( 40 \, J \) and includes work done by both tension and gravity, it is incorrect to say that the work done by tension alone is \( 40 \, J \).
- **Conclusion**: **False**.

#### Statement (c): The tension in the string is \( F \).
- **Analysis**: As established, the tension \( T \) in the string equals the force \( F \) applied to the other end of the string. This is true since the pulley is smooth and light.
- **Conclusion**: **True**.

#### Statement (d): The work done by the force of gravity is \( 40 \, J \) in the above 1s.
- **Analysis**: The work done by gravity is not solely \( 40 \, J \). The total work done is \( 40 \, J \), which includes contributions from both the tension and the force of gravity. Therefore, it cannot be said that the work done by gravity alone is \( 40 \, J \).
- **Conclusion**: **False**.

### Final Summary of Statements:
- (a) False
- (b) False
- (c) True
- (d) False