# Equilibrium of a rigid body

Why is it necessary for a body to have the net torque acting on it be balanced along with the forces for it to be in equilibrium? Isn't torque just some special case of force like rotation is a special case of translation? Why doesn't it suffice to have only the forces balanced for the body to be in equilibrium? I know there are examples which prove otherwise, but I need a more theoretical explanation. The case where the net torque is zero even when the force is not is understandable, since torque only represents the rotary component of force.

There would have been no need of introducing the concept of torque if we were only concerned with motion of a point particle (as the name suggests, it is hypothetical physical object having mass but no size). There is no concept of rotation of a particle and thus no need to introduce the concept of torque.

The concept of torque arises when we are concerned with system of particles. Here, there is a possibility that the net force on our system is zero but still the body is somehow moving. This possibility arises because the forces may act on different particles and there may exist internal interactions as well,causing the body to rotate if it is rigid.

That is how the introduction of the concept of torque is justified.

Torque is qualitatively a twisting force. Therefore, if net torque is zero, then the body cannot rotate. And if net force is zero the body will not translate. That is why for mechanical equilibrium, both the conditions must hold .

I hope I made myself clear.

• Yes it is very much clear now. I got confused while making transition from study of point object to the system of point objects. So when a system rotates about the center of mass the net force is zero since the center of mass is in a state of rest. But that doesn't mean the forces on individual particle is zero too. And in order for them to be in a state of rest too the external forces apart from being equal and opposite should also be along the same line of action. Which is the condition of torque equillibrium. Right? Dec 28, 2017 at 16:50
• If net torque is zero the body can still rotate but its angular momentum cannot change. If net force is zero it can still move with constant velocity but its linear momentum cannot change. Dec 29, 2017 at 13:50

You can consider rotation and translation as two completely separate things. When all forces balance out, there is no translational acceleration. When all torques balance out, there is no rotational or angular acceleration. And we need both to be 0 before saying that an object is in equilibrium.

When all forces balance out it just means that the object is not moving translationally but it can still rotate and spin on the spot.

From your comment below:

if the net force on the point object becomes zero then so will the net torque

True so far that if two forces act on a point object and balance each other then their torques do also balance. Simply because they act at the same distance from the rotation point (reference point).

But for an object consisting of many points or particles, the forces do not necessarily act on the same points even though they balance out overall. We could have one force pulling up in a point on the right side and the same force pushing down in a point on the left side of the object. They cancel each other out when we look at the body as a whole. Think of the example with a bike wheel being spun that I gave in the comments.

So, this body as a whole does not translate. But the force locally at each of the points is not balanced. That point will translate locally. So they will cause torques that do not balance.

• I don't think force and torque should be considered as separate things. Consider the case of a point object. Torque here is the force component perpendicular to the position vector times the length of position vector. But if the net force on the point object becomes zero then so will the net torque since there won't be any perpendicular component if there is no force itself. But this is not the case with rigid bodies. Why is that? I know the question seems quite trivial. But I still fell there is a subtle difference when we apply these concept in the case of point objects and rigid bodies Dec 28, 2017 at 9:53
• @SiddharthPrakash What do you mean by point object? Dec 28, 2017 at 9:56
• Translation and rotation can be considered separate because one can be present without the other and vice versa. The separation between them becomes clear when you have forces balancing that are not causing the same torques. Lift your bike and spin the wheel with your hand. The wheel doesn't go anywhere - there is no translation since the bike chassis holding at the centre and your pull at the edge cancel each other out - but it rotates. The forces are equal and balance but they cause different torques that do not balance. Dec 28, 2017 at 10:06
• @SiddharthPrakash How would you apply a torque to a point object? How would you get a non-zero length for the position vector?
– JMac
Dec 28, 2017 at 15:16
• @Steeven You cleared my doubt when you gave the example of rotating wheel. I asked my self what would be the force acting on the wheel and realized it is zero since the center of mass is at rest. But still the wheel rotated and so there was force acting on individual component of the wheel. Balancing of external forces in a system of points only implies that the center of mass is at rest not the individual points. The individual points can move as long as the center of mass does not changes. Hence they are not enough to establish the condition of equilibrium in rigid bodies. Thank you Dec 28, 2017 at 21:52

A torque is actually a manifestation of a force at a distance. By balancing the forces and the torques, you are matching the magnitude of force and the location where the force acts.

In simple terms, two equal and opposite forces acting on a body at rest will cause a rotation unless their line of action coincides. If they are offset from each other, then a net torque will be present and the body will rotate.

• Zero net force, assures the center of mass does not translate.
• Zero net torque about the center of mass, assures the body does not rotate.
• I was referring to a body at rest. I will edit the answer. Dec 29, 2017 at 20:03