Under what conditions is the torque about some location equal to zero?

The force that can cause an object to twist along an axis is measured as torque. In linear kinematics, force is what causes an object to accelerate.

The torque acting about some location of the system is the cross product of the position vector, $\overrightarrow{r}$and force vector, $\overrightarrow{F}$of that location.

$$\begin{gathered} \overrightarrow{\tau }=\overrightarrow{r}\times \overrightarrow{F} \\ =r·F\sin \theta \end{gathered}$$ …… (1)

Here, $\theta$is the angle between the position and the force vector,

The torque acting about some location of the system is numerically equal to the rate of change of angular momentum of the object. Mathematically, it is expressed as

$\begin{array}{rcl}\overrightarrow{\tau }& =& \frac{d\overrightarrow{L}}{dt}\\ & =& \frac{d\left(I\overrightarrow{\omega }\right)}{dt}\\ & =& I\frac{d\overrightarrow{\omega }}{dt}\\ & & \end{array}$ …… (2)

Here,

$\overrightarrow{L}\to$Angular momentum of the system

$I\to$Moment of inertia of the system.

$\frac{d\overrightarrow{\omega }}{dt}\to$Rate of change of angular momentum of the system

Using equations (1) & (2),we can draw the conditions for which torque acting about some location of the system is zero.

Using equation (1),the torque, $\overrightarrow{\tau }=rF\sin \theta$

From the torque equation, the possible conditions for zero torque are

1. Magnitude of the position vector has to be zero, $r=0$
2. Magnitude of the force vector has to be zero, $F=0$
3. Angle between force and position vector has to be $0,\theta =0$

From equation (2), the torque is

$\begin{array}{rcl}\overrightarrow{\tau }& =& \frac{d\overrightarrow{L}}{dt}\\ & =& I\frac{d\overrightarrow{\omega }}{dt}\\ & & \end{array}$

Hence, the possible conditions for torque to be zero are,

1. The angular velocity of the system, has to be constant.
2. The rate of change of angular speed (Angular acceleration) of the system has to be zero.