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 \( z \)-direction is:

Hint:

To calculate torque, use the cross product of position and force vectors. The magnitude of the torque in any direction can be obtained by using the appropriate unit vector.

Answer 1

The torque \( \tau \) is defined as the cross product of the position vector \( \mathbf{r} \) and the force \( \mathbf{F} \):\[\tau = \mathbf{r} \times \mathbf{F}\]Given \( \mathbf{r} = \langle 1, 1, 1 \rangle \) and \( \mathbf{F} = \langle 1, -1, 1 \rangle \).The torque in the \( z \)-direction, \( \tau_z \), is the \( z \)-component of this cross product:\[\tau_z = \hat{k} \cdot (\mathbf{r} \times \mathbf{F})\]Calculating the cross product using the determinant method:\[\tau_z = \hat{k} \cdot \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & 1 \\ 1 & -1 & 1 \end{vmatrix}\]The result of this calculation is:\[\tau_z = 1\]Therefore, the torque in the \( z \)-direction is \( 1 \).