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 find the value of \(b\) such that the work done on the particle is zero, we first recall that work done \(W\) by a force \(\mathbf{F}\) on a particle during a displacement \(\mathbf{r}\) is given by the dot product: \(W = \mathbf{F} \cdot \mathbf{r}\).

Given: \(\mathbf{F} = 2\hat{i} + b\hat{j} + \hat{k}\) and \(\mathbf{r} = \hat{i} - 2\hat{j} - \hat{k}\).

We calculate the dot product:

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

The terms \( \hat{i} \cdot \hat{j}, \hat{i} \cdot \hat{k}, \hat{j} \cdot \hat{i}, \hat{j} \cdot \hat{k}, \hat{k} \cdot \hat{i}, \hat{k} \cdot \hat{j} \) are all zero because the unit vectors are orthogonal. Thus, only the coefficients of like unit vectors are non-zero.

Now, simplifying, we have:

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

We know the work done is zero, so:

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

Thus, the value of \(b\) for which the work done is zero is \( \frac{1}{2} \).