Under the action of a force F = Cx, the position of a body changes from 0 to x. The work done is

Options

A. `(1)/(2)Cx^(2)`
B. `Cx^(2)`
C. Cx
D. `(1)/(2)C x`

Explanation

To solve the problem of finding the work done under the action of a force \( F = Cx \) as the position of a body changes from \( 0 \) to \( x \), we can follow these steps:

### Step 1: Understand the Work Done Formula
The work done \( W \) by a variable force can be calculated using the integral of the force over the distance moved. The formula is given by:
\[ W = \int_{x_1}^{x_2} F \, dx \]
where \( F \) is the force and \( x_1 \) and \( x_2 \) are the initial and final positions, respectively.

### Step 2: Substitute the Force
In this case, the force is given as \( F = Cx \). We need to substitute this into the work done formula:
\[ W = \int_{0}^{x} Cx \, dx \]

### Step 3: Factor Out the Constant
Since \( C \) is a constant, we can factor it out of the integral:
\[ W = C \int_{0}^{x} x \, dx \]

### Step 4: Perform the Integration
Now we need to integrate \( x \):
\[ \int x \, dx = \frac{x^2}{2} \]
Thus, we can write:
\[ W = C \left[ \frac{x^2}{2} \right]_{0}^{x} \]

### Step 5: Evaluate the Integral
Now we evaluate the definite integral from \( 0 \) to \( x \):
\[ W = C \left( \frac{x^2}{2} - \frac{0^2}{2} \right) = C \left( \frac{x^2}{2} \right) \]

### Step 6: Final Expression for Work Done
Therefore, the work done \( W \) is:
\[ W = \frac{Cx^2}{2} \]

### Conclusion
The work done under the action of the force \( F = Cx \) as the position changes from \( 0 \) to \( x \) is:
\[ W = \frac{Cx^2}{2} \]

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