1.3 Pseudo torque

1.1 Torque or moment of a force
1.2 Relation between torque and moment of inertia
1.3 Pseudo torque
1.4 Torque equation
1.5 Principal of moments

2.1 For better understanding angular momentum is classified as following four types,

3.1 Conservation of angular momentum

5.1 Total kinetic energy of a rigid body in a combined translational and rotational motion

8.1 Geometrical method
8.2 Relative velocity method

When the motion of the body is observed from a non-inertial frame of reference having an acceleration $a$ in a fixed direction
with respect to an inertial frame of reference. Then the pseudo force is applied to the body about the center of mass in the opposite direction. This pseudo force produces a pseudo torque about an axis of rotation.

Consider a pendulum of length $L$ and mass $m$ moving upwards in a lift with an acceleration $a$.

Torque about point $O$ is given by, $$\begin{equation} \begin{aligned} {\overrightarrow \tau _O} = \left( {\overrightarrow L \times \overrightarrow T } \right) + \left( {\overrightarrow L \times m\overrightarrow g } \right) + \left( {\overrightarrow L \times m\overrightarrow a } \right) \\ {\tau _O} = 0 + mg\sin \theta - ma\sin \theta \\ {\tau _O} = mg\sin \theta + ( - ma\sin \theta ) \\\end{aligned} \end{equation} $$

The torque of pseudo force $(ma)$ is known as pseudo torque.

Hence, pseudo torque is, $\tau = ( - ma\sin \theta )$

Note: Always, apply the pseudo force about the center of mass and calculate the torque about an axis of rotation.