Question

A force of \(-F\hat{i}\) acts at the origin of the coordinate system. The torque about the point \((0,1,-1)\) is:

• A. \(F(\hat{j}+\hat{k})\)
• B. \(-F(\hat{j}+\hat{k})\)
• C. \(F(\hat{i}+\hat{k})\)
• D. \(-F(\hat{i}+\hat{j})\)

Hint:

Torque about a point is calculated using position vector from that point to the point of application of force.

Answer 1

The Correct Option is B

Step 1: Torque formula.
\[ \vec{\tau} = \vec{r} \times \vec{F} \]

Step 2: Position vector from point \((0,1,-1)\) to origin.
\[ \vec{r} = (0,0,0) - (0,1,-1) \]
\[ \vec{r} = 0\hat{i} - \hat{j} + \hat{k} \]

Step 3: Force vector.
\[ \vec{F} = -F\hat{i} \]

Step 4: Apply cross product.
\[ \vec{\tau} = ( -\hat{j}+\hat{k}) \times (-F\hat{i}) \]

Step 5: Expand the cross product.
\[ \vec{\tau} = F(\hat{j}\times \hat{i}) - F(\hat{k}\times \hat{i}) \]
\[ \hat{j}\times \hat{i} = -\hat{k}, \quad \hat{k}\times \hat{i} = \hat{j} \]

Step 6: Simplify.
\[ \vec{\tau} = -F\hat{k} - F\hat{j} \]
\[ \vec{\tau} = -F(\hat{j}+\hat{k}) \]

Step 7: Final conclusion.
\[ \boxed{-F(\hat{j}+\hat{k})} \] Hence, correct answer is option (B).