Normal Force

The Normal Force is Caused by Interatomic Electric Forces

From a distance, when you set a box on a table it doesn't appear as if anything has changed. If you look closer, you might notice that the table bends, or deforms, a little according to how heavy the box is. On an atomic level, the weight of the box causes the box's atoms to squish against the table's atoms. The electron clouds within each object are repulsed by each other and push away from each other. The table's atoms and their bonds don't like being bent out of their natural shape, so they exert forces to get back to normal. All these tiny electric forces add together to create the normal force.

Does the Normal Force Have a Formula or Equation?

The normal force doesn't have its own specific formula or equation. Instead, we can find the normal force by using free-body diagrams and Newton's Second Law,$\Sigma F=ma$.

Solve for the Normal Force Using a Free-Body Diagram and Newton's Second Law

To solve for the normal force, we want to start by drawing a free-body diagram so we can see and account for all the forces in play. Let's look at our box on a table, pictured below:

Box sitting on a table with forces shown, StudySmarter Originals

We've drawn the forces acting on the box: the normal force,${\mathrm{F}}_n$, and the gravitational force,$F_g=\mathrm{mg}$. The normal force is sometimes also denoted as$\mathrm{N}$, but we will use${\mathrm{F}}_{\mathrm{n}}$so it doesn't get confused with Newtons.

Then, we apply the equation from Newton's Second Law. We will choose down to be negative and up to be positive. Because the box is not accelerating we will insert zero for the acceleration, so the sum of the forces equals zero:

$\begin{array}{rcl}-F_g+F_n& =& 0\\ F_n& =& F_g\end{array}$

In this case, the normal force equals the gravitational force, which is the weight of the box.

The Normal Force is a Reaction Force

The normal force is a reaction force; the surface reacts to any forces that cause an object to be pressed against it. Now, there is a common misconception that the normal force is only a reaction to the gravitational force. This misconception is easy to understand because even in our example above, the normal force equaled the weight of the box. However, what if we pressed on the box, adding another downward force? The box still wouldn't fall through the table, so the normal force must increase to match the weight of the box plus our added force. In this case, the normal force reacts to more than just the gravitational force.

This principle is even clearer if you imagine pushing horizontally against a wall, like in the image below. When you push against a wall, you don't fall through the wall, so there must be a force pushing back against you. Again, this is due to the normal force, this time in a horizontal direction. We've included the forces in play as blue arrows in the image--our push,$\mathrm{F}$, and the normal force,${\mathrm{F}}_{\mathrm{n}}$.

Push against a wall and the normal force reaction, adapted from image by Freepik

Gravity always acts downward and the normal force always acts perpendicular to the surface. So for this example, when we sum the forces horizontally (acceleration is still 0), the normal force would equal our pushing force, and gravity wouldn't be a factor at all. The normal force is an equal reaction to however much force we apply to the wall.

Normal Force Examples

We explained two very simple examples above already. Now we'll go over a couple of more examples with different variations on finding the normal force.

Normal Force on an Incline

How do we find the normal force for an object on an incline like in the figure on the left below? The most important thing to remember is that the normal force always acts perpendicular to the surface, and the gravitational force always acts straight down (gravity pulls objects straight towards the earth). You can see these principles applied in our free-body diagram in the figure on the right below.

Box sitting on an incline, StudySmarter Originals

Free-body diagram for the box on an incline, StudySmarter Originals

To solve for the normal force, we want to tilt our coordinate system to match the angle of the surface. This way the normal force acts in the y-direction and the friction force acts in the x-direction; the only force that doesn't match the coordinate system is the gravitational force. We will use the principle of superposition of forces to split the gravitational force into an x component and a y component. We can see the new coordinate system and components of the gravitational force in the figure below.

Free-body diagram with tilted axis and gravitational force split into x and y components, StudySmarter Originals

Now we can use Newton's Second Law equation in the y-direction to find the normal force. Since the box isn't accelerating in the y-direction, we can sum the forces to equal zero:

$F_n-F_{gy}=0$

Using trigonometry, we can substitute $F_g\cos \theta$ for$F_{gy}$:

$F_n=F_g\cos \theta$

For this example, the normal force is equal to the y component of the gravitational force.

Normal Force with Acceleration

All of our previous examples have had boxes standing still. If a box moves horizontally and the normal force acts vertically, the movement of the box won't affect the normal force because they are on separate axes. However, what happens if the box moves in the same direction as the normal force? Let's say our box is in an elevator. The box weighs$15\mathrm{kg}$, and the elevator accelerates down at$2\mathrm{m}/{\mathrm{s}}^2$. What is the normal force?

Free-body diagram of the box in the elevator, StudySmarter Originals

We drew our free-body diagram in the image above. Now we can use Newton's Second Law in the vertical direction to solve for the normal force, and this time we will include the downward acceleration.

$\begin{array}{rcl}{F}_n-mg& =& ma\\ F_n& =& 15\mathrm{kg}·\left(-2\mathrm{m}/{\mathrm{s}}^2\right)+15\mathrm{kg}·9.81\mathrm{m}/{\mathrm{s}}^2\\ F_n& =& 117.15\mathrm{N}\end{array}$

The normal force is$117.15\mathrm{N}$.