“What goes around comes around” is a popular saying that refers to doing good or bad and receiving a similar treatment in return. This common phrase has a very deep physical background, since it relates to one of the most important laws in classical mechanics: Newton’s third law of motion.

According to this law, for every force applied there is a reaction force equal in magnitude but opposite in direction. This means that when you hold your cell phone in your hand, besides its weight —which is a force acting downwards towards the center of the Earth—, there has to be a reaction force coming from your hand upwards, which balances its weight out. This is obvious if you think your cell phone will not fall down to the ground unless you stop exerting this reaction force on it.

In general, when an object comes in contact with a surface, which we will call our *surface of interest*, a reaction force arises from it towards the object. Think about a car standing on a slope. If the car is not sinking into the ground it means its weight is being counteracted by an upwards force. If the surface is not flat, like in this scenario, the reaction force has two components: one parallel to the street, and one perpendicular to it. The former is known as *friction*, and the latter as the *normal force*, where “normal” refers to it being perpendicular to the surface.

This is a key concept when dealing with free body diagrams and its calculation is relatively simple through the use of basic trigonometric functions. Let’s discover how.

## How to calculate the normal force

**Draw a free body diagram of the body coming in contact with the surface of interest and identify all the forces acting on it.****Determine the angles of these forces with respect to a reference line perpendicular to the surface of interest.****Extract the components of said forces parallel to the reference line using basic trigonometric functions.****Sum all of the resulting components together with the proper signs. If the result’s sign indicates it points towards the surface of interest, there is a normal force of the same magnitude acting on the body.****Draw the normal force vector, which direction is simply perpendicular to the surface of interest and opposite to the net force calculated in step 4.**

## What is the normal force

Newton’s third law of motion states that every action has a reaction. In the context of forces this implies that anytime you push an object, the object “pushes” you back with a force equal in magnitude but with an opposite direction.

Newton’s third law is easy to understand in the following situation: imagine you are holding a flower vase on the palm of your hand. Its weight is a force which points downwards and acts on your hand. We can call this force the *action*. For you to hold the vase and not let it fall down to the ground, you need to exert an equal force onto it, but directed upwards. We can call this, the *reaction*. This way, both forces are balanced and we can say the net force in this case is zero.

Now, imagine you put the flower vase on a table. Its weight (action) is still in place, pushing downwards, so the table must be also exerting a force upwards to counteract it (reaction). In this scenario, just as before, the vase’s weight acts on the table, and the table’s reaction acts on the vase. Since the table has a flat surface, the reaction force pointing upwards is parallel to the vase’s weight and, at the same time, perpendicular to the table. Take a look at this image to clarify this situation:

The word “normal” comes from the Latin word for a carpenter’s square. This is a very common tool composed of two perpendicular metal stripes, which is used to draw right angles. In physics, we call “normal” anything that forms an angle of 90° with a reference surface or plane. This is why we call the reaction force coming from a surface in contact with any object the *normal force*, which is usually denoted as Fn.

Now, thanks to Newton we know that for every action there must be a reaction. But, where does this reaction come from? When you are holding the vase in your hand, it is easy to understand how the reaction force is generated: you are activating your muscles to exert this force upwards and counteract the vase’s weight. Thanks to your senses, your body knows just how much force to apply to sustain the vase in the air. But, what happens when you put the vase on the table? How does the table know how much force to apply upwards?

Well, in reality the table’s reaction is the sum of the different interactions among the hundreds of millions of atoms and molecules that compose it. These are binded by different types of bonds, which act similar to a spring. When you compress a spring, it will push you back with the same force. Similarly, if you expand it, it will contract in the opposite direction. A similar thing happens between the atoms or molecules that form the table. These interactions, when added together, are what we perceive as the macroscopic reaction of a surface to any force, including the normal force.

## How to find the normal force

Let’s go back to our previous example and try to generalize it. If for some reason the table you put the vase on were not on a flat position, how does the normal force look like then? We have *defined *that the normal force is always perpendicular to the surface of interest, and this still applies even when that surface has a slope. Nevertheless, the weight of the vase will still point downwards in the same direction as before since it is a consequence of the Earth’s gravitational attraction on the vase’s mass. The following image illustrates this case:

In this scenario the normal force is no longer parallel to the vase’s weight, so we can’t just simply assume both forces are equal in magnitude. Now, only a portion of the weight is acting perpendicular to the table’s surface, and another portion is acting parallel to it. Let’s try and find the normal force in this case.

The case described by the previous image is a typical problem in physics, called an inclined plane or ramp. You will usually have a body with a known mass sitting on top of the ramp. In order to know how forces distribute in this case, we need to know how much it is tilted or its slope. This is usually represented by an angle. The next image summarizes our ramp:

Here, the inclination angle is , the body’s weight is FW and the resulting normal force is FN, which is drawn perpendicular to the ramp. If you take a closer look, you will see we have placed a coordinate system with its origin in the interface of the body and the ramp. This allows us to define which forces will be counted as “positive” and which as “negative”. Remember this refers to the fact that forces possess both magnitude and direction. This way, opposing forces, like the weight of the vase and the resulting normal force in our previous example (flat table) will have different signs, even when their magnitudes are the same.

For the sake of this example, let’s say everything pointing in the same direction as the x-axis or the y-axis will be positive. Consequently, everything pointing in the same direction as the negative x-axis or the negative y-axis will be negative. As you can see, the result of placing our coordinate system like this is that the normal force acts solely in the direction of the positive y-axis. Therefore, we expect this vector to have only a component in said direction.

The image shows how the body’s weight forms angle with the negative y-axis of our coordinate system. If you want to know how to obtain this angle in an inclined plane, go ahead and read the section called “Inclination angle in ramps”. Now that we know this angle, we can calculate the component of FW perpendicular to the surface of the ramp —which is the one related to the resulting normal force— using basic trigonometry. If we take the cosine of the inclination angle, we get: cos =FWy/FW. Solving for FWy, we get:

FWy=FWcos | (1) |

This result indicates that the portion of the body’s weight that acts perpendicularly to the ramp’s surface is a function of its inclination angle. Since the normal force is the reaction to this portion of the weight, and it points in the opposite direction (as all reactions forces do), then the normal force is:

FN=-FWy= -FWcos | (2) |

Now that we have obtained the normal force of a body sitting on top of a ramp as a function of its inclination angle, let’s do a thought experiment! If the angle were zero, which means the body would sit on a flat surface, then FWy=FW. This leads to FN=-FW, which was the result of our initial thought experiment of placing the vase on top of a flat table. Remember the negative sign indicates both forces act in opposite directions, while their magnitudes are equal.

## Examples

Let’s exercise a bit! Imagine you are pulling a 10 kg object up a hill which has an inclination of 30°. You are exerting a force of 50 N with an angle of 40° with respect to the ramp. Calculate the normal force acting on the object.

First, let’s draw a free body diagram of this situation:

The weight of the object equals its mass times the acceleration due to gravity, which is 9,8 m/s2. In this case, we get:

FW=9,8m/s2×10 kg =98 N | (3) |

As in our previous examples, the weight has a component parallel to the x-axis and another one parallel to the y-axis. Since we want to find the normal force, which is perpendicular to the ramp’s surface, we will focus on the weight’s component parallel to the y-axis. This is because, with the selected coordinate system, the normal force is aligned with the y-axis. The weight’s component that acts in this direction points to the negative end of the y-axis, and is therefore negative:

FWy = -98Ncos 30°= -84,9 N | (4) |

Your pulling force also has two components: one parallel to the x-axis and one parallel to the y-axis. Again, we will focus on the latter. Since this component points towards the positive y-axis, it is positive:

FPull-y =50Nsin 40°=32,1 N | (5) |

We now have all the forces acting along the y-axis of our coordinate system. Since we have more forces involved than just the body’s weight, we need to check if there is a net force being exerted on the surface at all. If the net force acts away from the surface (towards the positive y-axis), then it means the body will rise and lose contact with it, which implies no normal force or *reaction *will be generated.

The net force along the y-axis, Fy-net, is then:

Fy-net=FW+FPull-y= -84,9 N+32,1 N= -52,8 N | (6) |

The negative sign of the result indicates that the net force in our free body diagram acts towards the negative y-axis, a.k.a. towards the surface of the ramp. This confirms the body will not rise and it will stay in contact with the surface, thus generating a normal force. This reaction equals the force exerted on the surface but has the opposite direction, meaning it will have the same magnitude and the opposite sign:

FN= -Fy-net =52,8 N | (7) |

Since the result is positive, it means our normal force points in the direction of the positive y-axis, exactly as we expected! Take a look at the following diagram to clarify this exercise:

## Inclination angle in ramps

The following image summarizes the angles involved when a body sits on top of a ramp with an inclination angle of . All right angles are shown in yellow.

By examining the image, we can conclude that +=90°. This is because the dotted line that forms is perpendicular to the ramp. On the other hand, =90°-, since the weight vector forms a right triangle with the ramp and the black dotted line. This way, by combining these two equations, we can obtain the value of :

=90°-=90°-(90°-)= | (8) |

## Other helpful sources

If you want to grasp the concept of normal force as a reaction, use this very simple simulator by Andrew Duffy. You will be able to adjust both the weight of a body and an upward force acting on it, and see the resulting normal force. Think of this experiment as you lifting a box from the ground. What happens to the normal force if the box is heavier? What happens if you pull the box harder up?