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: Understanding the Question:
Work done is the scalar product of force and displacement.
Step 2: Key Formula or Approach:
\( W = \vec{F} \cdot \vec{d} \).
Displacement vector \( \vec{d} = \vec{r}_Q - \vec{r}_P \).
Step 3: Detailed Explanation:
Force vector \( \vec{F} = 3\hat{i} + 4\hat{j} + 5\hat{k} \).
Displacement vector \( \vec{d} = (2-3)\hat{i} + (3-4)\hat{j} + (4-5)\hat{k} = -\hat{i} - \hat{j} - \hat{k} \).
Work done:
\[ W = (3\hat{i} + 4\hat{j} + 5\hat{k}) \cdot (-\hat{i} - \hat{j} - \hat{k}) \] \[ W = (3)(-1) + (4)(-1) + (5)(-1) = -3 - 4 - 5 = -12 \text{ J} \] Step 4: Final Answer:
The work done is -12 J.