Static Equilibrium

We have mostly been concerned with objects in motion until now. Sometimes, however, we are more interested in avoiding motion or at least certain kinds of motion. When this is the case we are studying ``statics''. It is an important part of what civil engineers have to do but as you shall see similar techniques are important for analyzing the mechanical functionality of living organisms.

There are no new conceptual points for us to learn here. We are simply establishing a formalism for determining the conditions for a rigid body to be static. We have often applied the logic that in order for a body not to move there can be no net force acting on it. This is indeed a sufficient condition for a point particle because the only type of motion possible for it is translation. However, when we are dealing with macroscopic rigid bodies we have to augment this description because such a body not only can move through translation but it can also rotate. In order for a rigid body to be static we must therefore require that

The question arises what reference point should be associated with the torque equation. In principle we should require that vanishes for any point of reference. It turns out however that if there is no torque about one particular chosen point then given that then there will be no torque about any point I choose. This follows from the equation:

the important equation which relates torque about any chosen point of reference to torque with respect to the center of mass of the rigid body. Because the last term vanished in our case we see that irrespective of the point of reference we obtain the same torque which equals , the torque with respect to the center of mass point, . This is important to notice in problem solutions because

1. There is only a single torque equation associated with each rigid body in a statics problem. Nothing is gained by writing torque equations for several points of reference.
2. We have complete freedom in choosing our point of reference. With a judicious choice there can be significant computational simplifications. A good choice will
   • be a point about which unknown forces in the problem have no torque. Such reference points produce a torque equation with few unknowns.
   • be a point which makes it geometrically simple to calculate the torque associated with the remaining forces.

With this said it is important not to panic about choosing the ``correct'' reference point. There is no ``wrong'' choice. You will be able to work your way through a problem with any reference point chosen. A judicious choice can only make the algebraic manipulation simpler and less error prone.

Here as in other areas of physics it is interesting to note that there is a sense in which our brains already knows all this material. Take a one year old baby just learning to stand up. No chance that the child has a conscious knowledge of mechanics but nonetheless it learns how to maintain static equilibrium with little tutoring simply through trial and error. The brain evidently is capable of making the right moves based on experience rather than understanding. This is important for you to remember also when solving physics problems. Use the knowledge of physics that you have built up since you you were born to guess the solutions and to check whether your calculated solutions make any sense.

 
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