Question

A force \( \mathbf{F} = 2\hat{i} + b\hat{j} + \hat{k} \) is applied on a particle and it undergoes a displacement \( \mathbf{r} = \hat{i} - 2\hat{j} - \hat{k} \). What will be the value of \( b \), if the work done on the particle is zero?

• A. \( \frac{2}{3} \)
• B. \( \frac{1}{2} \)
• C. 0
• D. \( \frac{1}{3} \)

Hint:

The work done by a force is calculated using the dot product of the force vector and displacement vector. If the work is zero, set the dot product equal to zero and solve for the unknown.

Answer 1

The Correct Option is B

To determine the value of \(b\) for which the work done on the particle is zero, we utilize the formula for work done \(W\) by a force \(\mathbf{F}\) on a particle undergoing a displacement \(\mathbf{r}\), which is given by the dot product: \(W = \mathbf{F} \cdot \mathbf{r}\).

The given force is \(\mathbf{F} = 2\hat{i} + b\hat{j} + \hat{k}\) and the displacement is \(\mathbf{r} = \hat{i} - 2\hat{j} - \hat{k}\).

The dot product is calculated as follows:

\[\begin{align*} W &= (2\hat{i} + b\hat{j} + \hat{k}) \cdot (\hat{i} - 2\hat{j} - \hat{k}) \\ &= (2\hat{i} \cdot \hat{i}) + (2\hat{i} \cdot (-2\hat{j})) + (2\hat{i} \cdot (-\hat{k})) \\ &\quad + (b\hat{j} \cdot \hat{i}) + (b\hat{j} \cdot (-2\hat{j})) + (b\hat{j} \cdot (-\hat{k})) \\ &\quad + (\hat{k} \cdot \hat{i}) + (\hat{k} \cdot (-2\hat{j})) + (\hat{k} \cdot (-\hat{k})) \\ &= 2 \times 1 + 0 + 0 + 0 - 2b + 0 + 0 + 0 -1. \end{align*}\]

The terms involving the dot products of orthogonal unit vectors (e.g., \( \hat{i} \cdot \hat{j} \)) are zero. Therefore, only the products of coefficients of like unit vectors are non-zero.

Simplifying the expression for \(W\) yields:

\[W = 2 - 2b - 1.\]

Given that the work done is zero, we set the expression equal to zero:

\[2 - 2b - 1 = 0 \implies 1 - 2b = 0 \implies 2b = 1 \implies b = \frac{1}{2}.\]

Consequently, the value of \(b\) that results in zero work done is \( \frac{1}{2} \).