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 given by the cross product of position vector \( \mathbf{r} \) and the force \( \mathbf{F} \): \[ \tau = \mathbf{r} \times \mathbf{F} \] where \( \mathbf{r} = \langle 1, 1, 1 \rangle \) and \( \mathbf{F} = \langle 1, -1, 1 \rangle \). The torque in the \( z \)-direction is the \( z \)-component of the cross product: \[ \tau_z = \hat{k} \cdot (\mathbf{r} \times \mathbf{F}) \] Using the determinant formula for the cross product, we get: \[ \tau_z = \hat{k} \cdot \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & 1 \\ 1 & -1 & 1 \end{vmatrix} \] After solving, we find: \[ \tau_z = 1 \] Thus, the correct answer is \( 1 \).