Equilibrium

Equilibrium is a very generic concept in that it is not specific to Physics. We can have equilibrium in several other fields as well, be it in Chemistry, Economics, Politics et cetera. Essentially whenever there is a force of some sort or some agent that can bring about a change in a status quo then we can bring in the concept of equilibrium. So what does it mean for a system to be in equilibrium?

We try and keep our balance when we walk. We do that naturally but it took us a good number of falls and trips as a toddler to be able to master this complex action. As we move one leg forward, we have to keep our balance so as not to fall over. Walking on a tight rope is even more challenging.

Maintaining equilibrium on a tight rope

So, maintaining one’s balance is a tricky thing but we take it for granted because, almost like breathing and blinking, it has become second nature. When we are standing still, we are balanced, hopefully, and being in such a state, we can say that we are in equilibrium. But what does that mean, hence, to be in equilibrium?

For a system to be in equilibrium means that there are no net forces acting on it to make it change its current state. If the system in question is stationary, relative to the observer, and if there are no net external forces on it then it is in static equilibrium. If the system was in motion but at constant speed relative to the observer, and if there are no net external forces acting on it then it is said to be in mechanical equilibrium. By no net external forces, we mean that there could be more than one forces acting on the body but that the net effect of the forces cancel each other out. It is like pushing the palm of your right hand against the palm of your left hand with equal force so that your hands do not move either left or right. Even though there are forces acting, they cancel each other out. If the left hand was pushing harder, the net force on your right hand will not be zero and your hands would be moving in the right direction.

Walking, therefore, is about making sure the forces are balanced. Otherwise we would fall. When our left leg swings forward, our right arm swings back to try and maintain that balanced state. True, we can walk without swinging our arms but you will notice that this is slightly more difficult than when swinging our arms. When carrying a heavy bag over one shoulder, for example, we have to counteract this extra weight or force on one side by trying to lean a bit more on the opposite side. Again, this is all about us trying to balance forces, trying to maintain equilibrium. Standing still while holding a loaded shopping bag on our right hand side is a state of static equilibrium. Walking at constant speed while carrying the bag would be a state of mechanical equilibrium.

So far so good. Equilibrium is a balancing act. But there is a point that needs to be emphasised here. It is often misconceived that being in equilibrium is the same as being stable. Now, stability and equilibrium are two different things. It is easy to confuse those two concepts as they are often used in the same context but they are not the same even though they are attributes of the same object. An apple resting on a table is in equilibrium and is also stable. Equilibrium and stability are two attributes of the apple but are not the same thing. Just like the colour and the shape of the apple are two of its attributes but are not the same thing. What does it mean, therefore, to be stable? How is stability different from equilibrium?

Whereas equilibrium has to do with balancing forces, stability has to do with an object’s centre of gravity and the surface area of its base. Stability has to do with shape and mass of an object while equilibrium has to do with balancing the weight of the object according to its shape and its mass. This is a simple way of understanding those two concepts. Let’s consider a few examples in order to illustrate what stability is.

Suppose you have a coin resting on a table. If it is lying down flat on its head (or tail) then it seems pretty stable. It’s going to stay there until someone or you push it or pick it up or tilts the table. But a little shake of the table is not going to make it move. However, if the coin was standing on its rim then the slightest disturbance is enough to make it topple over. In this case, we say that the coin is less stable than in the first one. But how so? It is the same coin, the same table, the same set up. This is where the centre of gravity of the coin and its base come into play. The base of the coin is the surface of the coin which is in contact with the table. The centre of gravity of the coin is the point where all the weight of the coin is acting from. For a uniform and circular coin, we can safely assume that the centre of gravity is at the centre of the coin, the centre of that circle, inside the coin. When the coin is flat on its face, its centre of gravity relative to the table is very low. When its stood on its side, the centre of gravity is slightly higher. The lesson to be learnt here is that, the more stable you want something to be, the lower its centre of gravity has to be relative to the ground. That’s one point.

Now, the pointed end of a pencil will illustrate the second point. Let us suppose that the pencil is uniform even though it has a pointed end such that its centre of gravity is right at the centre of the graphite tube in the pencil, right in the middle. With the pencil stood on its broader, flatter surface, it is fairly stable. However, turn it upside down and try and make it stand on its pointed tip. It’s suddenly become less stable. The centre of gravity has not changed its position relative to the table; the centre of gravity stays at the same relative height. Yet, the pencil becomes unstable. What we have here is this: the centre of gravity, which is the point where all the weight is acting, is acting vertically down. With the broader surface in contact with the table, the centre of gravity has a larger surface area over which inside which it’s acting. With only the pointed tip touching the table, the centre of gravity is now restricted over a much smaller area. This means that, as soon as the centre of gravity, that vertical line that represents the weight acting downwards, as soon as that line goes out of that small area, the whole object becomes unstable. With a large surface area, the vertical line has more room around it in order for the object to remain stable. The larger the surface area in contact, or what we call the base, the more stable the object is. That’s point number two.

Balancing a pencil on its pointed tip

To maximise stability, we therefore have to ensure that the centre of gravity is as low as possible and that the base is as wide as possible. Like a martial artist, with your legs slightly further apart and crouching a little bit, you immediately take a much more stable stance, ready to kick some…

So, stability and equilibrium are different things but they can be related. For there can be three different kinds of equilibrium. Stable, unstable and neutral equilibrium. When a system is in stable equilibrium then it will always come back to its original state if its displaced from it. That is, as long as the external forces cancel each other, the system will remain where it is but if it so happens to be disturbed or if the external forces become unbalanced then the system will restore its original state once the external forces cancel each other again. Think of a freely hanging bob or as we physicist like to call it, a pendulum. When it is at rest, the bob hanging vertically down, then it is in a state of stable equilibrium. For if we displace it from its rest position it will swing back and forth to eventually come to rest back in its original position. That’s clear and simple. You may also consider a large fruit bowl and an orange. Initially the orange is happily sat at the bottom of the bowl. It is in a state of stable equilibrium and is content with its own existence as long as it is unaware that it’ll be blitzed or squashed for someone’s breakfast juice. So, the orange will remain at the trough of the bowl, completely at rest as long as there are no net external forces on it. However, if it were moved towards the upper rim of the bowl and let go then it will roll back and forth in the bowl to finally come to rest again in its initial position. That shows that the orange was, and is, in a state of stable equilibrium.

Now, in contrast with what stable equilibrium is, unstable equilibrium implies that the system does not come back to its original position after the net external forces cancel each other out. It might start off in a state of equilibrium, all forces perfectly balanced, but, if it were nudged or disturbed, it will not restore its original position. Think of the pencil balancing on its pointed tip. We can, perhaps after a large number of trial and error, manage to keep a pencil standing on its tip. It will be in equilibrium. But, because the slightest shake or the briefest breeze can make it topple and not come back to its standing stance, it is in a state of unstable equilibrium.

Finally, neutral equilibrium is about the system being indifferent to the external forces acting on it, so to speak. A coin lying flat on its face is one such example. While it’s happily at rest on the table with its face down it is in a state of equilibrium because all external forces balances out. But if it is pushed, if you strike it so that it accelerates and slides and then comes back to rest again just like a carrom piece gliding on a carrom board then while it is accelerating and decelerating, the coin is not in equilibrium. The external forces on it are not cancelling out because if it is accelerating or decelerating then there must be some external force causing this to happen. So, the point is, when the coin comes back to rest, it will be in a different location on the table (or carrom board) but it will keep its initial configuration of having its face down flat on the table. It would be as if it hasn’t changed its initial state but it’s now situated in a different location. Another example is a football on the floor. While it is motionless on the floor it is in neutral equilibrium. Kick it gently, it’ll roll and come to rest again and take its initial configuration but will now be in a different place, in some corner of the room. So, to the ball or the coin, the most that’s happened is that they’ve changed location but not their initial state or configuration. Unlike the pencil that topples or the orange that always comes back to the same location inside the fruit bowl, the coin is neither in unstable or stable equilibrium. It is in neutral equilibrium.

So there you have it, the difference between equilibrium and stability and the different kinds of equilibrium. As I had mentioned earlier, this is not restricted to physical objects or Physics. When political party P becomes unstable and cannot rule any longer they might topple and a new party, Q, then takes the reign. So here, you have a situation where party P is in unstable equilibrium. A small revolt from the population is enough to displace them from their position of power and because they do not come back to power that shows that they were in a state of unstable equilibrium. When economic forces shake financial markets a little bit, they might deviate from their normal trend but if they restore their initial state then that shows that they were in stable equilibrium. We can find many other examples of where different types of equilibrium exist in non-physical form. What is important here is that we can at least make some sense of what is going in those situations with the knowledge we have from the concepts in Physics. Once again, this goes to show that Physics does help to make sense of the world rather than make it more complicated than it already is.

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  1. Pingback: Bloodhound « electrolights

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