Question

A force of \( (4\hat{i} + 2\hat{j} + \hat{k}) \) N is acting on a particle of mass \( 2 \) kg displaces the particle from a position of \( (2\hat{i} + 2\hat{j} + \hat{k}) \) m to a position of \( (4\hat{i} + 3\hat{j} + 2\hat{k}) \) m. The work done by the force on the particle in joules is:

• A. \( 21 \) J
• B. \( 11 \) J
• C. \( 14 \) J
• D. \( 18 \) J

Hint:

To find the work done by a force, use the dot product formula: \( W = \mathbf{F} \cdot \mathbf{d} \). The dot product is computed by multiplying corresponding components of the vectors and summing them.

Answer 1

The Correct Option is B

Step 1: Calculate the displacement vector
The displacement vector \( \mathbf{d} \) is given by: \[ \mathbf{d} = (x_2 - x_1) \hat{i} + (y_2 - y_1) \hat{j} + (z_2 - z_1) \hat{k} \] Substituting the given values: \[ \mathbf{d} = (4 - 2) \hat{i} + (3 - 2) \hat{j} + (2 - 1) \hat{k} \] \[ \mathbf{d} = 2\hat{i} + \hat{j} + \hat{k} \] Step 2: Compute Work Done
The work done \( W \) is calculated using the dot product: \[ W = \mathbf{F} \cdot \mathbf{d} \] \[ W = (4\hat{i} + 2\hat{j} + \hat{k}) \cdot (2\hat{i} + \hat{j} + \hat{k}) \] Expanding the dot product: \[ W = (4 \times 2) + (2 \times 1) + (1 \times 1) \] \[ W = 8 + 2 + 1 = 11 \text{ J} \] Step 3: Conclusion
Thus, the work done by the force is: \[ \mathbf{11 \text{ J}} \]