Question

The coordinates of a particle with respect to origin in a given reference frame is \( (1, 1, 1) \) meters. If a force of \( \mathbf{F} = \hat{i} - \hat{j} + \hat{k} \) acts on the particle, then the magnitude of torque (with respect to origin) in the z-direction is:

Answer 1

Correct Answer: 2

The torque \( \mathbf{\tau} \) exerted by the force \( \mathbf{F} = \hat{i} - \hat{j} + \hat{k} \) on a particle at \( \mathbf{r} = (1 \, \text{m}, 1 \, \text{m}, 1 \, \text{m}) \) with respect to the origin is calculated using the cross product \( \mathbf{\tau} = \mathbf{r} \times \mathbf{F} \).
The vectors are: \( \mathbf{r} = \hat{i} + \hat{j} + \hat{k} \) and \( \mathbf{F} = \hat{i} - \hat{j} + \hat{k} \).
The cross product is computed via the determinant:

• \(\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & 1 \\ 1 & -1 & 1 \end{vmatrix}\)

Expanding the determinant yields:

• \(\hat{i}(1 \cdot 1 - 1 \cdot (-1)) = 2\hat{i}\)
• \(\hat{j}( -(1 \cdot 1 - 1 \cdot 1)) = 0\hat{j}\)
• \(\hat{k}(1 \cdot (-1) - 1 \cdot 1) = -2\hat{k}\)

Thus, the torque vector is \( \mathbf{\tau} = 2\hat{i} + 0\hat{j} - 2\hat{k} \).
The magnitude of the torque in the z-direction is \( |-2| = 2 \).