A particle moves under the effect of a force F = Cx from x = 0 to `x = x_1` The work done in the process is

Options

A. `Cx_1^2`
B. `1/2 Cx_(1)^2`
C. `Cx_1`
D. Zero

Explanation

To solve the problem of calculating the work done by the force \( F = Cx \) as a particle moves from \( x = 0 \) to \( x = x_1 \), we can follow these steps:

### Step 1: Understand the Work Done Formula
The work done \( W \) by a force when moving an object from position \( x_1 \) to \( x_2 \) is given by the integral of the force over the distance:
\[ W = \int_{x_1}^{x_2} F \, dx \]

### Step 2: Substitute the Force into the Integral
In this case, the force \( F \) is given as \( F = Cx \). Therefore, the work done when moving from \( x = 0 \) to \( x = x_1 \) can be expressed as:
\[ W = \int_{0}^{x_1} Cx \, dx \]

### Step 3: Factor Out the Constant
Since \( C \) is a constant, we can factor it out of the integral:
\[ W = C \int_{0}^{x_1} x \, dx \]

### Step 4: Perform the Integral
Now we need to compute the integral \( \int x \, dx \):
\[ \int x \, dx = \frac{x^2}{2} \]
Now, we apply the limits from \( 0 \) to \( x_1 \):
\[ W = C \left[ \frac{x^2}{2} \right]_{0}^{x_1} = C \left( \frac{x_1^2}{2} - \frac{0^2}{2} \right) \]

### Step 5: Simplify the Expression
This simplifies to:
\[ W = C \cdot \frac{x_1^2}{2} \]

### Step 6: Final Result
Thus, the work done by the force as the particle moves from \( x = 0 \) to \( x = x_1 \) is:
\[ W = \frac{C x_1^2}{2} \]

### Conclusion
The final answer for the work done is:
\[ W = \frac{1}{2} C x_1^2 \]

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