# For rotational equilibrium of a rigid object, about which point does the torque of the body have to be zero?

In the definition of rotational equilibrium for a rigid body, the net torque about a point has to be zero.

Now, about which point will the torque be calculated? Will it be the centre of mass or centre of gravity? Or, can it be calculated about any point?

• It must be zero about all points. Commented May 16, 2022 at 10:15

The torque has to be zero about all points -- if it was nonzero for some point, the object would start to rotate around that point.

On the other hand, you often see that the torque is calculated around some specific point, and if it vanishes, the body is declaredto be in equilibrium. How does that fit together?

Here's a hint (in a simple case):

• Assume a rigid body on which some forces $$\vec F_i$$ are acting at certain points $$\vec r_i$$.
• Assume the forces add up to zero.
• Assume the torques around a specific point $$\vec p$$ add up to zero.
• Can you show thet the torques around some other point also add up to zero?

If you have shown that, you can conclude that it is sufficient for equibrium to show that the overall force and the overall torque around one point vanish. (Hence, you can pick a convenient point for the torque calculation, i.e. one where some torque is zero or some torques cancel out immediately.)

• Got it! But if the condition that net force equals zero is not met, then from the calculation it seems that torque is zero about some point/s and non-zero about others. How to interpret such behaviour? Commented Jun 4, 2022 at 12:41
• @HabibullahKhan If the nmet force is nonzero, you will have a translation and possibly a rotation. For that split, it is best to take the cneter of mass as the reference point, as that ist the point that will move according to the overall force. Commented Jun 10, 2022 at 7:06

For rotational equilibrium of an object, the sum of the torques about any point on the object must be zero.

Hope this helps.