A force `F = 2hati + 4hatj` Newton displaces the body by `s = 3hatj + 5hatk` meter in 2s. The power generated will be

Options

A. 11 W
B. 6 W
C. 22 W
D. 12 W

Answer

To solve the problem, we need to find the power generated by the force acting on the body during its displacement. Here’s a step-by-step solution:

### Step 1: Identify the Given Quantities
We are given:
- Force \( \mathbf{F} = 2 \hat{i} + 4 \hat{j} \) Newton
- Displacement \( \mathbf{s} = 3 \hat{j} + 5 \hat{k} \) meters
- Time \( t = 2 \) seconds

### Step 2: Calculate the Work Done
The work done \( W \) by a force during a displacement is given by the dot product of the force vector and the displacement vector:
\[ W = \mathbf{F} \cdot \mathbf{s} \]
Calculating the dot product:
\[ \mathbf{F} \cdot \mathbf{s} = (2 \hat{i} + 4 \hat{j}) \cdot (3 \hat{j} + 5 \hat{k}) \]
Breaking it down:
- The dot product \( 2 \hat{i} \cdot 3 \hat{j} = 0 \) (since \( \hat{i} \) and \( \hat{j} \) are perpendicular)
- The dot product \( 2 \hat{i} \cdot 5 \hat{k} = 0 \) (since \( \hat{i} \) and \( \hat{k} \) are perpendicular)
- The dot product \( 4 \hat{j} \cdot 3 \hat{j} = 12 \)
- The dot product \( 4 \hat{j} \cdot 5 \hat{k} = 0 \) (since \( \hat{j} \) and \( \hat{k} \) are perpendicular)

Thus, the total work done \( W \) is:
\[ W = 0 + 0 + 12 + 0 = 12 \text{ Joules} \]

### Step 3: Calculate the Power
Power \( P \) is defined as the work done per unit time:
\[ P = \frac{W}{t} \]
Substituting the values:
\[ P = \frac{12 \text{ Joules}}{2 \text{ seconds}} = 6 \text{ Watts} \]

### Final Answer
The power generated is \( P = 6 \) Watts.

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