Question

A uniform force of $(3 \hat{i} + \hat{j})$ newton acts on a particle of mass 2kg. Hence the particle is displaced from position$ (2\hat{i} + \hat{k} )$ meter to position $(4\hat{i} + 3\hat{j} - \hat{k})$ meter. The work done by the force on the particle is :

• A. 15 J
• B. 9 J
• C. 6 J
• D. 13 J

Answer 1

The Correct Option is B

To find the work done by the force on the particle, we will use the work-energy principle, where work done is given by the dot product of the force vector and the displacement vector.

Step 1: Identify the Force Vector

The force acting on the particle is given by:

\(\vec{F} = 3 \hat{i} + \hat{j}\) Newton.

Step 2: Determine the Displacement Vector

The initial position of the particle is:

\((2\hat{i} + \hat{k})\) meter.

The final position of the particle is:

\((4\hat{i} + 3\hat{j} - \hat{k})\) meter.

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

\(\vec{d} = (4 - 2)\hat{i} + (3 - 0)\hat{j} + (-1 - 1)\hat{k}\)

Simplifying, we get:

\(\vec{d} = 2\hat{i} + 3\hat{j} - 2\hat{k}\) meter.

Step 3: Calculate the Work Done

Work done W is given by the dot product:

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

Substitute the given vectors:

\(W = (3 \hat{i} + \hat{j}) \cdot (2\hat{i} + 3\hat{j} - 2\hat{k})\)

Calculate the dot product:

• 3\hat{i} \cdot 2\hat{i} = 6
• \hat{j} \cdot 3\hat{j} = 3
• \hat{k} term does not contribute as \hat{k} \cdot \hat{k} = 0

Thus,

\(W = 6 + 3 = 9 \, \text{Joules}\)

Conclusion

The work done by the force on the particle is 9 Joules. Therefore, the correct answer is 9 J.