Question

A constant force of \[ \mathbf{F} = (8\hat{i} - 2\hat{j} + 6\hat{k}) \text{ N} \] acts on a body of mass 2 kg, displacing it from \[ \mathbf{r_1} = (2\hat{i} + 3\hat{j} - 4\hat{k}) \text{ m to } \mathbf{r_2} = (4\hat{i} - 3\hat{j} + 6\hat{k}) \text{ m}. \] The work done in the process is:

• A. \( 72 \) J
• B. \( 88 \) J
• C. \( 44 \) J
• D. \( 36 \) J

Hint:

The work done by a force is calculated using the dot product \( W = \mathbf{F} \cdot \mathbf{d} \). Compute the displacement vector first, then take the dot product with the force vector.

Answer 1

The Correct Option is B

Step 1: Compute the Displacement Vector The displacement vector \( \mathbf{d} \) is given by: \[ \mathbf{d} = \mathbf{r_2} - \mathbf{r_1} \] \[ = (4\hat{i} - 3\hat{j} + 6\hat{k}) - (2\hat{i} + 3\hat{j} - 4\hat{k}) \] \[ = (4 - 2)\hat{i} + (-3 - 3)\hat{j} + (6 + 4)\hat{k} \] \[ = 2\hat{i} - 6\hat{j} + 10\hat{k} \] Step 2: Compute the Work Done Work done is given by the dot product: \[ W = \mathbf{F} \cdot \mathbf{d} \] \[ = (8\hat{i} - 2\hat{j} + 6\hat{k}) \cdot (2\hat{i} - 6\hat{j} + 10\hat{k}) \] Expanding the dot product: \[ W = (8 \times 2) + (-2 \times -6) + (6 \times 10) \] \[ = 16 + 12 + 60 \] \[ = 88 \text{ J} \] Thus, the work done in the process is: \[ \mathbf{88} \text{ J} \]