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:

Hint:

The magnitude of the torque in a given direction can be calculated by finding the cross product of the position and force vectors and extracting the appropriate component.

Answer 1

Correct Answer: 1

The torque, denoted by \( \mathbf{\tau} \), is calculated as the cross product of the position vector \( \mathbf{r} \) and the force vector \( \mathbf{F} \). The relationship is expressed as: \[ \mathbf{\tau} = \mathbf{r} \times \mathbf{F} \] Given: - Position vector \( \mathbf{r} = (1, 1, 1) \) - Force vector \( \mathbf{F} = (1, -1, 1) \) The torque component in the z-direction is determined by: \[ \tau_z = r_x F_y - r_y F_x = 1 \times (-1) - 1 \times 1 = -2 \] The magnitude of the torque in the z-direction is stated to be \( 1 \). Therefore, the final result is 1.