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

The work \( W \) performed by a force \( \mathbf{F} \) over a displacement \( \mathbf{r} \) is computed via the dot product: \[ W = \mathbf{F} \cdot \mathbf{r}. \] The provided force and displacement vectors are: \[ \mathbf{F} = 2\hat{i} + b\hat{j} + \hat{k}, \quad \mathbf{r} = \hat{i} - 2\hat{j} - \hat{k}. \] Calculating the dot product \( \mathbf{F} \cdot \mathbf{r} \): \[ W = (2\hat{i} + b\hat{j} + \hat{k}) \cdot (\hat{i} - 2\hat{j} - \hat{k}). \] Applying the dot product properties: \[ W = 2(1) + b(-2) + 1(-1) = 2 - 2b - 1 = 1 - 2b. \] For the work to be zero: \[ 1 - 2b = 0 \quad \Rightarrow \quad b = \frac{1}{2}. \] Consequently, \( b \) equals \( \boxed{\frac{1}{2}} \).