Stable And Unstable Equilibrium: Definition

Potential Energy is often ignored. Students think that as long as you know Newton’s equations of motion, you don’t need to know anything else about the problem!
However, a simple differential of the Potential Energy can make life easier even in complicated situations.
That’s because –

(We’ll use this in just a sec)

Stable equilibrium

The classic case of a block attached to a spring –

(Pic from Kshitij JEE classes)

The Potential Energy, in this case, is given by –

Using the equation for Force (F) above, we have-

This is generally all we use to solve a spring question.

Instead of looking at the body diagram, let’s look at the Potential Energy Curve! (Plotted against distance)

Looks familiar?

The Potential energy curve, interestingly is a lot like a ball in a bowl –

(Forgive my poor skills with photoshop ;P)

In fact, the potential energy curve for this situation is very similar to that of the spring and block example above! And both of these cases behave exactly the same.

If the Potential Energy curve is like a cup, any object (like a ball) will try to get to the bottom of that cup!

Unstable Equilibrium

How about a case such as this-

Take a look at this crazy potential energy curve

This example is from Chemistry. The same concept of Potential Energy is equally useful in Physics and Chemistry.
At x(0) and x(2) a body (or chemical) would be safe. They would be in stable equilibrium. But at a point like x(1) it stands precariously at that point! Much like these rocks!

They are steady now, but hardly stable!

If they move even a little to the left or to the right, they will fall! These rocks would have a Potential energy curve exactly like the point x(1) in the previous diagram.

The curve of the potential energy can tell you a lot about how a body will behave. A body moves towards the direction of decreasing potential energy. This is shown by-

If the potential energy increases along x, the Force is negative, pulling it away from that direction!

Equilibrium is that point on the potential curve where Force is zero, i.e.-

Just like points x(1), x(2) or x(3)

At these points, the force F on the body would be zero. But if it goes even just a little bit to either side, then think of it like a ball, ready roll down.

For the mathematical explanation, involving the double differentials, check out this link. It will help in every subject.

For more fun ways to learn, check out embibe!