A positon dependent force ,`F=8-4x+3x^(2)N` acts on a small body of mass 2 kg and displaces it from x=0 to x=5 m. The work done in joule is

Options

A. 35
B. 70
C. 115
D. 270

Explanation

To calculate the work done by the force \( F = 8 - 4x + 3x^2 \) as the body moves from \( x = 0 \) to \( x = 5 \) meters, we will follow these steps:

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

### Step 2: Set up the integral
In this case, \( F = 8 - 4x + 3x^2 \), \( x_1 = 0 \), and \( x_2 = 5 \). Thus, we can set up the integral as:
\[ W = \int_{0}^{5} (8 - 4x + 3x^2) \, dx \]

### Step 3: Calculate the integral
We will now calculate the integral:
\[ W = \int_{0}^{5} (8 - 4x + 3x^2) \, dx \]
Breaking this down:
\[ W = \int_{0}^{5} 8 \, dx - \int_{0}^{5} 4x \, dx + \int_{0}^{5} 3x^2 \, dx \]

Calculating each term separately:
1. \(\int_{0}^{5} 8 \, dx = 8x \bigg|_{0}^{5} = 8(5) - 8(0) = 40\)
2. \(\int_{0}^{5} 4x \, dx = 4 \cdot \frac{x^2}{2} \bigg|_{0}^{5} = 2x^2 \bigg|_{0}^{5} = 2(25) - 2(0) = 50\)
3. \(\int_{0}^{5} 3x^2 \, dx = 3 \cdot \frac{x^3}{3} \bigg|_{0}^{5} = x^3 \bigg|_{0}^{5} = 125 - 0 = 125\)

### Step 4: Combine the results
Now, substituting these results back into the expression for work:
\[ W = 40 - 50 + 125 \]
Calculating this gives:
\[ W = 40 - 50 + 125 = 115 \, \text{Joules} \]

### Final Answer
The work done by the force as the body moves from \( x = 0 \) to \( x = 5 \) meters is \( \boxed{115} \) Joules.

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