A uniform force of `(5hati +5hatj)N` acts on particle of mass 1 kg. The particle moves from `r_1 = (3hati + 4hatj)m` to `r_2 = (5hati + 8hatj) m` under this force. The work done by this force is

Options

A. `20`J
B. `25`J
C. `30`J
D. `50`J

Answer

To solve the problem, we need to calculate the work done by the force acting on the particle as it moves from one position to another. The work done by a force is given by the formula:

\[ W = \vec{F} \cdot \vec{d} \]

where \( W \) is the work done, \( \vec{F} \) is the force vector, and \( \vec{d} \) is the displacement vector.

### Step-by-Step Solution:

1. **Identify the Force Vector:**
The force vector is given as:
\[ \vec{F} = 5 \hat{i} + 5 \hat{j} \, \text{N} \]

2. **Identify the Initial and Final Position Vectors:**
The initial position vector \( \vec{r_1} \) is:
\[ \vec{r_1} = 3 \hat{i} + 4 \hat{j} \, \text{m} \]
The final position vector \( \vec{r_2} \) is:
\[ \vec{r_2} = 5 \hat{i} + 8 \hat{j} \, \text{m} \]

3. **Calculate the Displacement Vector:**
The displacement vector \( \vec{d} \) is calculated by subtracting the initial position vector from the final position vector:
\[ \vec{d} = \vec{r_2} - \vec{r_1} = (5 \hat{i} + 8 \hat{j}) - (3 \hat{i} + 4 \hat{j}) \]
Simplifying this gives:
\[ \vec{d} = (5 - 3) \hat{i} + (8 - 4) \hat{j} = 2 \hat{i} + 4 \hat{j} \, \text{m} \]

4. **Calculate the Work Done:**
Now, we can calculate the work done using the dot product of the force vector and the displacement vector:
\[ W = \vec{F} \cdot \vec{d} = (5 \hat{i} + 5 \hat{j}) \cdot (2 \hat{i} + 4 \hat{j}) \]
Using the properties of the dot product:
\[ W = (5 \cdot 2) + (5 \cdot 4) = 10 + 20 = 30 \, \text{J} \]

5. **Conclusion:**
The work done by the force is:
\[ W = 30 \, \text{J} \]