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?
Show Hint
\( \frac{2}{3} \)
\( \frac{1}{2} \)
- 0
- \( \frac{1}{3} \)
The Correct Option is B
Approach Solution - 1
The work done \( W \) by a force \( \mathbf{F} \) on a particle moving through a displacement \( \mathbf{r} \) is given by the dot product: \[ W = \mathbf{F} \cdot \mathbf{r}. \] Given: \[ \mathbf{F} = 2\hat{i} + b\hat{j} + \hat{k}, \quad \mathbf{r} = \hat{i} - 2\hat{j} - \hat{k}. \] The dot product \( \mathbf{F} \cdot \mathbf{r} \) is: \[ W = (2\hat{i} + b\hat{j} + \hat{k}) \cdot (\hat{i} - 2\hat{j} - \hat{k}). \] Using the properties of the dot product: \[ W = 2(1) + b(-2) + 1(-1) = 2 - 2b - 1 = 1 - 2b. \] For the work to be zero: \[ 1 - 2b = 0 \quad \Rightarrow \quad b = \frac{1}{2}. \] Thus, the value of \( b \) is \( \boxed{\frac{1}{2}} \).
Approach Solution -2
Step 1: Formula for work done.
Work done by a constant force is given by the dot product:
\[
W = \mathbf{F} \cdot \mathbf{r}.
\]
Step 2: Substitute the given vectors.
\[
\mathbf{F} = 2\hat{i} + b\hat{j} + \hat{k}, \quad
\mathbf{r} = \hat{i} - 2\hat{j} - \hat{k}.
\]
\[
W = (2)(1) + (b)(-2) + (1)(-1).
\]
Simplify:
\[
W = 2 - 2b - 1 = 1 - 2b.
\]
Step 3: Use the given condition.
Work done \( W = 0 \), so:
\[
1 - 2b = 0.
\]
\[
b = \frac{1}{2}.
\]
Hence, the correct value of \( b \) is \( \dfrac{1}{2} \).
Final Answer:
\[ \boxed{b = \dfrac{1}{2}} \]
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