A particale moves under the effect of a force `F = Cx` from `x = 0` to `x = x_(1)`. The work down in the process is

Options

A. `Cx_(1)^(@)`
B. `Cx_(1)^(2)//2`
C. zero
D. `Cx_(1)^(3)`

Explanation

To solve the problem of finding 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 is given by the integral of the force over the distance moved. Mathematically, this is expressed as:
\[ W = \int F \, dx \]

### Step 2: Substitute the Force
In this case, the force \( F \) is given as \( F = Cx \). Therefore, we can substitute this into the work done formula:
\[ W = \int Cx \, dx \]

### Step 3: Set the Limits of Integration
The particle moves from \( x = 0 \) to \( x = x_1 \). Thus, we set the limits of integration accordingly:
\[ W = \int_{0}^{x_1} Cx \, dx \]

### Step 4: Perform the Integration
Now, we can perform the integration:
\[ W = C \int_{0}^{x_1} x \, dx \]
The integral of \( x \) is:
\[ \int x \, dx = \frac{x^2}{2} \]
Thus, we have:
\[ W = C \left[ \frac{x^2}{2} \right]_{0}^{x_1} \]

### Step 5: Evaluate the Integral
Now we evaluate the integral from \( 0 \) to \( x_1 \):
\[ W = C \left( \frac{x_1^2}{2} - \frac{0^2}{2} \right) = C \left( \frac{x_1^2}{2} \right) \]
This simplifies to:
\[ W = \frac{Cx_1^2}{2} \]

### Step 6: Conclusion
Thus, the work done by the force as the particle moves from \( x = 0 \) to \( x = x_1 \) is:
\[ W = \frac{Cx_1^2}{2} \]

### Final Answer
The correct option is \( \frac{Cx_1^2}{2} \).

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