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 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 \( \boxed{\frac{1}{2}} \).

Answer 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 \( \dfrac{1}{2} \).

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Final Answer:

\[ \boxed{b = \dfrac{1}{2}} \]