A particle of mass 0.1 kg is subjected to a force which is directly proportional to the distance x. the work done by the force is directly proportional to

Options

A. x
B. `x^(3)`
C. `x^(3//2)`
D. `x^(2)`

Explanation

To solve the problem, we need to analyze the relationship between the force acting on the particle and the work done by that force.

### Step-by-Step Solution:

1. **Understanding the Force Relationship**:
The problem states that the force \( F \) is directly proportional to the distance \( x \). This can be expressed mathematically as:
\[ F = kx \]
where \( k \) is a constant of proportionality.

2. **Work Done by the Force**:
The work done \( W \) by a force when it moves an object through a distance \( S \) is given by the formula:
\[ W = F \cdot S \]
In this case, since the force is proportional to the distance \( x \), we can substitute \( F \) into the work formula:
\[ W = (kx) \cdot S \]

3. **Relating Work Done to Distance**:
If we consider the displacement \( S \) to be the same as the distance \( x \) (i.e., the particle moves from \( 0 \) to \( x \)), we can rewrite the work done as:
\[ W = kx \cdot x = kx^2 \]

4. **Conclusion on Proportionality**:
From the equation \( W = kx^2 \), we can see that the work done \( W \) is directly proportional to the square of the distance \( x \). Therefore, we can conclude that:
\[ W \propto x^2 \]

### Final Answer:
The work done by the force is directly proportional to \( x^2 \).

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