Question

A particle is displaced from point \( P(3 \text{ m}, 4 \text{ m}, 5 \text{ m}) \) to a point \( Q(2 \text{ m}, 3 \text{ m}, 4 \text{ m}) \) under a constant force \( \vec{F} = (3\hat{i} + 4\hat{j} + 5\hat{k})\text{N} \). The work done by the force in this process is

• A. +10 J
• B. +4 J
• C. -8 J
• D. -12 J

Hint:

Work is negative when the force and displacement are in opposite directions (angle $> 90^\circ$).

Answer 1

The Correct Option is D

Step 1: Concept
Work done ($W$) by a constant force is the dot product of force ($\vec{F}$) and displacement ($\vec{d}$).
Step 2: Displacement Calculation
$\vec{d} = \vec{r}_Q - \vec{r}_P = (2-3)\hat{i} + (3-4)\hat{j} + (4-5)\hat{k}$
$\vec{d} = -\hat{i} - \hat{j} - \hat{k}$
Step 3: Work Done Calculation
$W = \vec{F} \cdot \vec{d} = (3\hat{i} + 4\hat{j} + 5\hat{k}) \cdot (-\hat{i} - \hat{j} - \hat{k})$
$W = (3 \times -1) + (4 \times -1) + (5 \times -1)$
$W = -3 - 4 - 5 = -12 \text{ J}$
Step 4: Conclusion
The work done is -12 J.
Final Answer:(D)