Normal Force

The normal force is the force that keeps us from falling to the center of the earth. Every surface or object we stand on exerts a force back on us. Otherwise, we would fall through the object/surface due to the force of gravity. The normal force is a reaction force, and as such has no formula specific to it. We'll discuss these ideas further in this article, as well as work through some examples of how to calculate the normal force.

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    Normal Reaction Force - Definition and Meaning

    The normal force is the push that a surface (or object) exerts back on an object that comes into contact with it.

    The normal force always acts perpendicular to, and away from, the surface. The name "normal" literally means perpendicular. This principle is very important to remember when solving problems that involve the normal force. The normal force is a type of contact force--two objects or surfaces have to touch for there to be a normal force. The normal force is present in cases as simple as a box sitting on a table. The gravitational force on the box pulls the box down toward the earth, but something keeps it from falling through the table--this is the 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,ΣF=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:

    Normal force box on table example StudySmarterBox sitting on a table with forces shown, StudySmarter Originals

    We've drawn the forces acting on the box: the normal force,Fn, and the gravitational force,Fg=mg. The normal force is sometimes also denoted asN, but we will useFnso 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:

    -Fg+Fn=0Fn=Fg


    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,F, and the normal force,Fn.

    Normal force reaction against wall StudySmarterPush 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.

    Normal force incline example StudySmarterBox sitting on an incline, StudySmarter Originals

    Normal force example free-body diagram on incline StudySmarterFree-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.

    Normal force example on incline StudySmarterFree-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:

    Fn-Fgy=0

    Using trigonometry, we can substitute Fgcosθ forFgy:

    Fn=Fgcosθ

    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 weighs15 kg, and the elevator accelerates down at2 m/s2. What is the normal force?

    Normal force example in an elevator StudySmarterFree-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.

    Fn-mg=maFn=15 kg·-2 m/s2+15 kg·9.81 m/s2Fn=117.15 N

    The normal force is117.15 N.

    Normal Force - Key takeaways

    • The normal force is the force a surface exerts back on an object that comes into contact with it. It is a reaction force to all forces that cause the object to press against the surface-- not just the gravitational force.
    • The normal force always acts perpendicular to and away from the surface.
    • The normal force is caused by the interatomic electric forces between the object and the surface. The electron clouds of each push against each other to keep the surfaces from running into each other.
    • There is no specific formula for normal force. We use free-body diagrams and Newton's Second Law of Motion to find the normal force.
    Frequently Asked Questions about Normal Force

    What is the normal force?

    The normal force is the push that a surface (or object) exerts back on an object that comes into contact with it.

    How do you find the normal force?

    One can find the normal force by using a free-body diagram and Newton's Second Law of Motion. These tools are used to solve for the normal force acting on an object based on the other forces acting on it.

    What is a normal force example?

    An example of the normal force is the force a person feels when that person pushes against a wall.

    What is the cause of the normal force?

    Interatomic electric forces are the cause of the normal force. When two objects come into contact with each other, the electron clouds within each object are repulsed by each other and push away from each other. All these tiny forces added together are called the normal force.

    Why is normal force important in physics?

    In physics, the normal force is important because, without it, an object would fall through a surface or another object. The force has to exist to account for the solidity of objects.

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    Test your knowledge with multiple choice flashcards

    Is the normal force a contact force or a long-range force?

    True or false: The normal force is an action-reaction pair with the gravitational force.

    Which of the following scenarios does not involve normal force?

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