Question

If \(\vec{F} = (5\hat{i} - 10\hat{j})\) and \(\vec{r} = (4\hat{i} - 3\hat{j})\), then the torque acting on the object will be

• A. \(\hat{i} - 2\hat{j}\)
• B. \(2\hat{i} - \hat{j}\)
• C. \(25\hat{k}\)
• D. \(-25\hat{k}\)

Hint:

Torque in 2D always comes along \(\hat{k}\) direction.

Answer 1

The Correct Option is D

Concept:
Torque is given by: \[ \vec{\tau} = \vec{r} \times \vec{F} \] Step 1: Write vectors. \[ \vec{r} = 4\hat{i} - 3\hat{j}, \quad \vec{F} = 5\hat{i} - 10\hat{j} \]
Step 2: Use determinant form. \[ \vec{\tau} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 4 & -3 & 0 \\ 5 & -10 & 0 \end{vmatrix} \]
Step 3: Expand determinant. \[ \vec{\tau} = \hat{k} \begin{vmatrix} 4 & -3 \\ 5 & -10 \end{vmatrix} \] \[ = \hat{k} (4 \times (-10) - (-3 \times 5)) \] \[ = \hat{k} (-40 + 15) = -25\hat{k} \]
Step 4: Conclusion. \[ \vec{\tau} = -25\hat{k} \]