A force`F = (2hat(i)+5hat(j)+hat(k))N` is acting on a particle. The particle is first displacement from (0, 0, 0) to (2m, 2m, 0) along the path x = y and then from (2m, 2m, 0) to (2m, 2m, 2m) along the path x = 2m, y = 2 m. The total work done in the complete path is

Options

A. 12 J
B. 8 J
C. 16 J
D. 10 J

Answer

To find the total work done by the force \( \mathbf{F} = (2\hat{i} + 5\hat{j} + \hat{k}) \, \text{N} \) on the particle as it moves along the specified path, we can break the problem into two segments of the path and calculate the work done in each segment separately.

### Step 1: Calculate Work Done in the First Segment

1. **Identify the path**: The particle moves from \( (0, 0, 0) \) to \( (2, 2, 0) \) along the path where \( x = y \).
2. **Parameterize the path**: We can use the parameter \( t \) such that:
\[ x = t, \quad y = t, \quad z = 0 \quad \text{for } t \in [0, 2] \]
3. **Differentiate to find \( d\mathbf{r} \)**:
\[ d\mathbf{r} = \left( \frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt} \right) dt = (1, 1, 0) dt \]
4. **Calculate the work done \( W_1 \)**:
\[ W_1 = \int_0^2 \mathbf{F} \cdot d\mathbf{r} = \int_0^2 (2\hat{i} + 5\hat{j} + \hat{k}) \cdot (1\hat{i} + 1\hat{j} + 0\hat{k}) dt \]
\[ = \int_0^2 (2 \cdot 1 + 5 \cdot 1 + 0) dt = \int_0^2 7 \, dt = 7t \bigg|_0^2 = 14 \, \text{J} \]

### Step 2: Calculate Work Done in the Second Segment

1. **Identify the path**: The particle moves from \( (2, 2, 0) \) to \( (2, 2, 2) \).
2. **Parameterize the path**: Here, we can use the parameter \( t \) such that:
\[ x = 2, \quad y = 2, \quad z = t \quad \text{for } t \in [0, 2] \]
3. **Differentiate to find \( d\mathbf{r} \)**:
\[ d\mathbf{r} = (0, 0, 1) dt \]
4. **Calculate the work done \( W_2 \)**:
\[ W_2 = \int_0^2 \mathbf{F} \cdot d\mathbf{r} = \int_0^2 (2\hat{i} + 5\hat{j} + \hat{k}) \cdot (0\hat{i} + 0\hat{j} + 1\hat{k}) dt \]
\[ = \int_0^2 (0 + 0 + 1) dt = \int_0^2 1 \, dt = t \bigg|_0^2 = 2 \, \text{J} \]

### Step 3: Calculate Total Work Done

Now, we can add the work done in both segments:
\[ W_{\text{total}} = W_1 + W_2 = 14 \, \text{J} + 2 \, \text{J} = 16 \, \text{J} \]

### Final Answer
The total work done in the complete path is \( \boxed{16 \, \text{J}} \).