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 find the work done by the force acting on the particle, we can follow these steps:

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

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

### Step 3: Calculate the Displacement Vector
The displacement vector \(\vec{d}\) can be calculated as:
\[ \vec{d} = \vec{r_2} - \vec{r_1} \]
Substituting the values:
\[ \vec{d} = (5\hat{i} + 8\hat{j}) - (3\hat{i} + 4\hat{j}) = (5 - 3)\hat{i} + (8 - 4)\hat{j} = 2\hat{i} + 4\hat{j} \, \text{m} \]

### Step 4: Calculate the Work Done
The work done \(W\) by the force is given by the dot product of the force vector and the displacement vector:
\[ W = \vec{F} \cdot \vec{d} \]
Substituting the vectors:
\[ W = (5\hat{i} + 5\hat{j}) \cdot (2\hat{i} + 4\hat{j}) \]
Calculating the dot product:
\[ W = (5 \cdot 2) + (5 \cdot 4) = 10 + 20 = 30 \, \text{J} \]

### Final Answer
The work done by the force is:
\[ \boxed{30 \, \text{J}} \]

---