# Why, in a horizontal plane, is normal force equal and opposite to gravity even though they are not a reaction pair?

I know that normal force and gravity are not a pair of forces, However, when an object is in equilibrium normal force and gravity are equal and opposite. Why is that? Is there any reason other than that they have to be in order for the system to be in equilibrium? Is there any other relationship between normal force and gravity?

• Because they have to be for it to be in equilibrium? – Conifold Dec 7 '17 at 2:22
• What is a "pair of forces"? – jjack Dec 8 '17 at 22:47

when an object is in equilibrium normal force and gravity are equal and opposite. Why is that?

That's not necessarily true. Consider a 500 N box resting on the (horizontal) floor,(in equilibrium in an inertial reference frame) attached to a vertical rope with a tension of 100 N. The normal force the floor exerts on the box is only 400 N. And we know it's 400 N because the box is in vertical equilibrium, so that $$\Sigma F_{\mathrm{vertical}}= ma_{\mathrm{vertical}}=0.$$

And, yes, the normal force is taken to be whatever is needed for the observed net acceleration.

There is no specific, unique relationship between weight and the normal force.

This is a very primary question, really as you said, they are not action and reaction pairs, although they are same sometimes. When a box is on a horizontal table the vertical forces acting on the box are * the Mg * Normal force by the table

As the equilibrium of the system they should be same to each. But you think that a situation which the box is pulling downwards when it on a table, then exactly the normal force and the Mg is not same due to the equilibrium.

Consider a situation which the box is on a inclined plane. Then you can understand it more....

The floor up on an object just hard enough to keep the object from penetrating the floor. That is, the floor pushes back just as hard as the object is pushed against it.

The object stays in contact with the floor, with vertical component of velocity $0$. This means the vertical component of acceleration is $0$, and the vertical component of force is $0$. The upward reaction force from the floor is the same as the downward force from gravity.

Honestly, the true answer to this question is that they are actually not equal in general. This is confusing, since in those first-year physics problem we indeed always say they are equal; but bear with me.

Let's think about this box on a table - indeed it doesn't move, so by Newton's first law the gravitational and normal forces have to be equal, as you said. Notice two things about this observation: first, it comes from the first law, not the third, and second and more importantly, it is an experimental observation - we know those two forces have to be equal simply because we know, from our everyday life and observation, that boxes stay on tables. So this is not a law of physics at all, and as such you shouldn't be able to derive it from Newton's third law (or from any law) - as you mentioned. It is at this point simply an experimental observation, which we generalize in order to help us solve physics problems.

Now let's imagine increasing the box's mass. At some point, the box will become so massive, that the table will break and the box will fall down. You see? This is again, a second experimental observation, and it tells us that our first proposition (that the normal force is equal to gravity) is false - these forces are not equal in general, which is how I began my answer. At some point there's a maximum normal force, and when it is passed, the forces are not equal anymore, and the table breaks. Thus we see that our first experimental observation is in fact not true, but it is true in some cases - that is to say, when the box is not too massive. In order to solve problems, we will usually assume this - that the box is light enough for the forces to be equal, and thus our experimental observation is true and can be used. But notice: it is still an experimental observation and not a law of nature, and it is only useful sometimes - in other cases, it falls apart.

Now, the analysis in the above paragraph is usually called a first-order model. If we now think more carefully about increasing the mass of the box gradually, we will see that actually, the table doesn't just break at one single critical point - instead, at some mass the box starts to distort the shape of the table, making it go down bit-by-bit; until at some point it can't support the stress anymore and breaks apart. We may try an experiment to see how the shape of the table changes with the mass of the box, and deduce the normal force from that. This would be a higher order model, but it is still an experimental observation.

So this is the annoying, but true, answer: the normal force is not, in general, equal to the gravitational force; they are only equal in some cases, and we can utilize that to solve problems and model the world around us. The fact that they're equal in those cases comes not from any law but from experiment; and thus you can't expect to derive it on the basis of Newton's third law. THe fact that they're equal doesn't have anything to do with Newton's third law and doesn't come from it, and it doesn't come from the first law, either - it's just that we use the first law (but definitely not the third) and our experimental observations, to deduce that they must be equal in the cases we see. But again, this is an experimental and not a theoretical result.

Of course, in the end, we do want physics to be complete, and thus we want to be able to show that the normal force is approximately equal to the gavitational force in those restricted cases, from our theory - from our known laws of physics. And inded the theoretical reaon for that is known - but it probably involves more physics than you're ready for at this level. I will give you a hint though: it has to do with the electrical repulsion between the nucleii of the atoms that make up the box, and the nucleii of the atoms that make up the table. In general, any force in our everyday lives, except gravity and bouyancy (and perhaps also the tidal force, depending on how you look at it) is electromagnetic in origin - that is, originates from electric (or magnetic) forces.