# How do action-reaction pairs keep a book sitting on a table in equilibrium?

Apr 19, 2018

The question of equilibrium of a book sitting on a table is quite a subtle one. I am afraid this is going to be a really long answer ...

#### Explanation:

First up - let me clear the air on a standard misconception. The pair of forces that act on the book are not an action-reaction pair! One of the two forces that act on the book is the force of gravity - the force with which the earth pulls at the book downwards. This "action" definitely has a "reaction" - but that acts on the earth! This is one of the central and often misunderstood parts of the the third law of motion - action and reaction always acts on different bodies.

The two forces that do act on the book are

• gravity, and
• the normal force exerted by the table

The latter is often called the normal reaction force, which, in my opinion, is a horrible name, on two counts. Firstly, it makes no sense to call this force a reaction. All forces occur in action-reaction pairs, and it is just convention that dictates which one of such a pair is designated as the action, and which the reaction. Secondly, the name is responsible for entrenching the completely wrong notion that this is the reaction to the weight.

So, if the normal force is not a reaction to the weight - how come it is equal and opposite to it, just like a reaction is supposed to be? To understand this, we must take a look at what causes this force. Imagine the situation just at the moment the book is placed on the table. Prior to that instant, the table was not exerting an upward force on the book. Right at that instant, too - the upward force on the book has not yet come into being. This means that there is only one force on the book - its weight. The book then responds the only way a body subjected to an unbalanced force can - it accelerates in the direction of the force!

The book moves down - but the table is is the way! The book moves down nonetheless - bending the table in the process. The book thus exerts a deforming force on the table, and the reaction to this is the restoring force that the table exerts on the book. It is this restoring force that we call the normal reaction. Being a reaction to the deforming force that the book exerts on the table, it is, indeed, equal and opposite to it. But the question still remains - what makes it equal and opposite to the weight of the book?

The answer to this is simple. As long as the normal force exerted by the table on the book does not balance the book's weight, the book will experience a net downward force - and it will continue to accelerate downwards. This will bend the table even more - and with the increase in deformation, comes an increase in the magnitude of the restoring force. This process will continue until the normal force becomes equal and opposite to the weight. At this point the forces cease to change, and the book remains in equilibrium.

Actually, things are a bit more complicated than that! By the time the normal force grows to match the weight, the book has already acquired some speed - so it overshoots. This causes the normal force to grow further, exceeding the weight. So now, the net force is upwards, slowing down the book. It is easy to see that the book will move up and down a few times until dissipative forces will slow it down enough to halt it in the equilibrium position.

All this is over in a very short time. This is why we do not often notice this. Again, the deformation of the table in the process may be too tiny to notice unless someone is paying close attention. Of course, if you were to place the book on a soft bed instead of a table the deformation would be plainly visible.

By the way, we have been saying that the book deforms the table. One could equally well say that the table deforms the book, and the force that the book exerts on the table (the one that we have been calling the deforming force so far) is actually a restoring force that the book exerts in an attempt to relieve this deformation. This happens whenever two hard bodies try to move through each other. This mutual deformation is plainly visible in those fast action snapshots of a ball hitting a bat that you can easily find on the net.

The normal force then is a reaction - but not to the weight. It is equal and opposite to the weight because it adjusts itself until it becomes that way. On the other hand, the force that the book exerts on the table is equal and opposite to the normal force - this time because it really is the reaction to it! This means that the force the book exerts on the table is actually equal and in the same direction as the weight. You must realize that it is this force that we usually perceive as the "weight "of an object. When we hold something in our hands, our nerves have no way of feeling the force the earth exerts on the object - they can only respond to the force that the object exerts on them! Indeed, the force that a spring balance, say, will measure when you are "weighing" an object is precisely this force - not really the weight. Since it is equal and parallel to the weight, this confusion does not cause any problems (but see the next paragraph).

Note that if the book-table combination was in a lift (elevator to the Americans!) which starts to accelerate down, the book will stop bending the table any further when the two share a common
downward acceleration - that of the lift. Here the normal force does not have to grow to the extent that it stops the book accelerating altogether - just enough so that under the action of the net force it accelerates down at the same rate as the lift. This is why the upward normal force is less than the weight in this case. But then again, so is the downward force that the book exerts on the table! This is what we really mean when we say that an object on a downward accelerating lift loses weight - the earth does not attract it any less, it just pushes down on the lift floor less hard!

The extreme case is when a body is in free fall. Since it is accelerating at the same rate as everything surrounding it - it does not deform anything. so it does not push at anything - and nothing pushes back!

Let's come back to our original problem of the book resting quietly on a table fixed to the ground. Of course, one may not be satisfied by the answer that the book is kept still on the table by restoring forces that are caused by deformation - we may want to ask what causes this restoring force in the first place! At a molecular level, the molecules of the book (more precisely, those at the lower surface of the book in contact with the table) move a bit closer to those on the upper surface of the table. Now, molecules have an equilibrium distance - push them any closer and they repel. This repulsion is ultimately the result of inter-molecular electromagnetic interactions. It is the net effect of the tiny repulsions of individual pairs of molecules that add up to the restoring force on the book.

Just pause to think about it. When you are standing on the ground, every molecule in your body is pulled down towards the center of the earth by the net attractive effect of every molecule on the earth. What balances this force is just the electromagnetic repulsion of the much, much smaller number of pairs of molecules right where the soles of your shoes touch the ground. It goes to show that despite being the one force that we experience the most everyday, how imaginably weak gravitation really is compared to the other fundamental force that we experience in daily life - that of electromagnetism!