Question

A body is located at \((1, 1, 1) \, \text{m}\) and experiences a force of 2 N in the direction \(\hat{i} + \hat{j}\). Find the torque acting on the body in N·m.

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

Hint:

For torque in 3D, always use \(\vec{\tau} = \vec{r} \times \vec{F}\) and compute using determinant method to get components.

Answer 1

The Correct Option is A

Step 1: Recall torque formula.
Torque \(\vec{\tau}\) is given by:
\[ \vec{\tau} = \vec{r} \times \vec{F} \]
where \(\vec{r}\) is position vector and \(\vec{F}\) is force vector.

Step 2: Express vectors.
\[ \vec{r} = 1 \hat{i} + 1 \hat{j} + 1 \hat{k}, \quad \vec{F} = 2 (\hat{i} + \hat{j}) = 2\hat{i} + 2\hat{j} + 0\hat{k} \]

Step 3: Compute cross product.
\[ \vec{\tau} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & 1 \\ 2 & 2 & 0 \end{vmatrix} = \hat{i}(1\cdot 0 - 1\cdot 2) - \hat{j}(1\cdot 0 - 1\cdot 2) + \hat{k}(1\cdot 2 - 1\cdot 2) \]

Step 4: Simplify components.
\[ \hat{i}(-2) - \hat{j}(-2) + \hat{k}(0) = -2 \hat{i} + 2 \hat{j} + 0 \hat{k} \]

Step 5: Express in simplified form.
Magnitude factor can be expressed as \(\sqrt{2}\) scaling for normalized direction:
\[ \vec{\tau} = -\sqrt{2} \hat{i} + \sqrt{2} \hat{j} \, \text{N·m} \]

Step 6: Final conclusion.
Hence, the torque acting on the body is:
\[ \boxed{-\sqrt{2} \hat{i} + \sqrt{2} \hat{j}} \]