Question

A particle moves from a point $(-2 \hat{i} + 5 \hat{j})$ to $( 4 \hat{j} + 3 \hat{k})$ when a force of $(4 \hat{i} + 3 \hat{j} ) N$ is applied. How much work has been done by the force ?

• A. 8 J
• B. 11 J
• C. 5 J
• D. 2 J

Answer 1

The Correct Option is C

The work done by a force is calculated using the formula:

W = \vec{F} \cdot \vec{d}

where:

• \vec{F} is the force vector
• \vec{d} is the displacement vector

First, we need to determine the displacement vector \vec{d} from the initial to the final position.

The initial position vector is (-2 \hat{i} + 5 \hat{j}) and the final position vector is (4 \hat{j} + 3 \hat{k}).

The displacement vector \vec{d} is calculated as:

\vec{d} = [(0 \hat{i} + 4 \hat{j} + 3 \hat{k})] - [(-2 \hat{i} + 5 \hat{j} + 0 \hat{k})]

Simplifying, we get:

\vec{d} = (0 + 2)\hat{i} + (4 - 5)\hat{j} + (3 - 0)\hat{k}

\vec{d} = 2\hat{i} - 1\hat{j} + 3\hat{k}

The force vector is given as \vec{F} = (4 \hat{i} + 3 \hat{j}).

The work done is therefore:

W = \vec{F} \cdot \vec{d} = (4 \hat{i} + 3 \hat{j}) \cdot (2 \hat{i} - 1 \hat{j} + 3 \hat{k})

This can be expanded and solved as:

W = 4 \times 2 + 3 \times (-1) + 0

W = 8 - 3 + 0

W = 5

Hence, the work done by the force is 5 J.

This matches the given correct answer, which is 5 J.