A force `F_(y)=(3x+2)" N"` is acting on a body. The work done by this force if it tends to displace the body from x = 0 m to x = 4 m will be

Options

A. 32 J
B. 16 J
C. 0 J
D. `(12x+8)" J"`

Answer

To find the work done by the force \( F_y = (3x + 2) \, \text{N} \) as it displaces a body from \( x = 0 \, \text{m} \) to \( x = 4 \, \text{m} \), 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:
\[ W = \int_{x_1}^{x_2} F(x) \, dx \]
where \( F(x) \) is the force as a function of position \( x \), and \( x_1 \) and \( x_2 \) are the initial and final positions, respectively.

### Step 2: Set Up the Integral
In this case, the force is given by:
\[ F_y = 3x + 2 \]
We need to calculate the work done from \( x = 0 \) to \( x = 4 \):
\[ W = \int_{0}^{4} (3x + 2) \, dx \]

### Step 3: Calculate the Integral
Now, we can compute the integral:
\[ W = \int_{0}^{4} (3x + 2) \, dx = \int_{0}^{4} 3x \, dx + \int_{0}^{4} 2 \, dx \]

Calculating each part:
1. For \( \int 3x \, dx \):
\[ \int 3x \, dx = \frac{3x^2}{2} \]
Evaluating from 0 to 4:
\[ \left[ \frac{3(4)^2}{2} \right] - \left[ \frac{3(0)^2}{2} \right] = \frac{3 \cdot 16}{2} = 24 \]

2. For \( \int 2 \, dx \):
\[ \int 2 \, dx = 2x \]
Evaluating from 0 to 4:
\[ \left[ 2(4) \right] - \left[ 2(0) \right] = 8 - 0 = 8 \]

### Step 4: Combine the Results
Now, we can add the results of both integrals:
\[ W = 24 + 8 = 32 \, \text{J} \]

### Final Answer
The work done by the force as it displaces the body from \( x = 0 \, \text{m} \) to \( x = 4 \, \text{m} \) is:
\[ \boxed{32 \, \text{J}} \]