A body of mass 2 kg is acted upon by two forces `vecD_(1)=(2hati+3hatj)N and vecF_(2)(-2hatk+3hatj)N.` If the body is displaced from `A(3m, -2m, 1m)` to B `(-1m, +2m,-3m),` then work done on the body is

Options

A. 40 J
B. `-40 J`
C. 24 J
D. `-24J`

Answer

To find the work done on the body, we need to follow these steps:

### Step 1: Calculate the Net Force Acting on the Body
The body is acted upon by two forces:
- \( \vec{F_1} = (2 \hat{i} + 3 \hat{j}) \, \text{N} \)
- \( \vec{F_2} = (-2 \hat{k} + 3 \hat{j}) \, \text{N} \)

To find the net force \( \vec{F}_{\text{net}} \), we add these two forces together:
\[ \vec{F}_{\text{net}} = \vec{F_1} + \vec{F_2} = (2 \hat{i} + 3 \hat{j}) + (-2 \hat{k} + 3 \hat{j}) \]
\[ = 2 \hat{i} + (3 + 3) \hat{j} - 2 \hat{k} = 2 \hat{i} + 6 \hat{j} - 2 \hat{k} \]

### Step 2: Calculate the Displacement Vector
The displacement vector \( \vec{d} \) is given by the difference between the final position vector \( \vec{r}_B \) and the initial position vector \( \vec{r}_A \).

The coordinates of point A are \( (3 \, \text{m}, -2 \, \text{m}, 1 \, \text{m}) \) and point B are \( (-1 \, \text{m}, 2 \, \text{m}, -3 \, \text{m}) \).

Thus, the position vectors are:
\[ \vec{r}_A = 3 \hat{i} - 2 \hat{j} + 1 \hat{k} \]
\[ \vec{r}_B = -1 \hat{i} + 2 \hat{j} - 3 \hat{k} \]

The displacement vector \( \vec{d} \) is:
\[ \vec{d} = \vec{r}_B - \vec{r}_A = (-1 \hat{i} + 2 \hat{j} - 3 \hat{k}) - (3 \hat{i} - 2 \hat{j} + 1 \hat{k}) \]
\[ = (-1 - 3) \hat{i} + (2 + 2) \hat{j} + (-3 - 1) \hat{k} = -4 \hat{i} + 4 \hat{j} - 4 \hat{k} \]

### Step 3: Calculate the Work Done
The work done \( W \) is given by the dot product of the net force and the displacement vector:
\[ W = \vec{F}_{\text{net}} \cdot \vec{d} \]
Substituting the values:
\[ \vec{F}_{\text{net}} = 2 \hat{i} + 6 \hat{j} - 2 \hat{k} \]
\[ \vec{d} = -4 \hat{i} + 4 \hat{j} - 4 \hat{k} \]
Calculating the dot product:
\[ W = (2 \hat{i} + 6 \hat{j} - 2 \hat{k}) \cdot (-4 \hat{i} + 4 \hat{j} - 4 \hat{k}) \]
\[ = (2 \cdot -4) + (6 \cdot 4) + (-2 \cdot -4) \]
\[ = -8 + 24 + 8 = 24 \, \text{J} \]

### Final Answer
The work done on the body is \( \boxed{24 \, \text{J}} \).

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