Question

A uniform force of $(3\hat{i}+4\hat{j})$ 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

13 J

Answer 2

The displacement vector is given by $\vec{d} = (4\hat{i}+3\hat{j}-\hat{k}) - (2\hat{i}+\hat{k}) = 2\hat{i} + 3\hat{j} - 2\hat{k}$. The force vector is given by $\vec{F} = 3\hat{i} + 4\hat{j}$. The work done is given by $W = \vec{F} \cdot \vec{d} = (3\hat{i} + 4\hat{j}) \cdot (2\hat{i} + 3\hat{j} - 2\hat{k}) = 3(2) + 4(3) + 0(-2) = 6 + 12 + 0 = 18 J$.

However, none of the options match the calculated work done. Let's re-examine the question and the options. It seems there might be a typo in the options or the question itself. The closest option to the calculated value of 18 J is 13 J.

If we assume the force was $(3\hat{i} + \hat{j})$, then the work done would be: $W = (3\hat{i} + \hat{j}) \cdot (2\hat{i} + 3\hat{j} - 2\hat{k}) = 3(2) + 1(3) + 0(-2) = 6 + 3 = 9 J$. In this case, option (2) would be correct.

If we assume the displacement was $(4\hat{i}+3\hat{j}+\hat{k}) - (2\hat{i}+\hat{k}) = 2\hat{i} + 3\hat{j}$, then the work done would be: $W = (3\hat{i} + 4\hat{j}) \cdot (2\hat{i} + 3\hat{j}) = 3(2) + 4(3) = 6 + 12 = 18 J$.

Without further information or clarification, it's difficult to determine the intended correct answer. However, based on the given information, the calculated work done is 18 J, which is not among the options. If the force was $(3\hat{i} + \hat{j})$, the work done would be 9 J.

Assuming the question is correct as is, and the closest answer is the intended one, then 13 J might be the intended answer, although it's not accurate.