How to Calculate the Force of Friction

How to Calculate the Force of Friction
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Surfaces exert a frictional force that resists sliding motions, and you need to calculate the size of this force as part of many physics problems. The amount of friction mainly depends on the “normal force,” which surfaces exert on the objects sitting on them, as well as the characteristics of the specific surface you’re considering. For most purposes, you can use the formula:

F=\mu N

to calculate friction, with ​N​ standing for the “normal” force and “​μ​” incorporating the characteristics of the surface.

What Is Friction?

Friction describes the force between two surfaces when you try to move one across the other. The force resists motion, and in most cases the force acts in the opposite direction to the motion. Down at the molecular level, when you press two surfaces together, minor imperfections in each surface can interlock, and there may be attractive forces between the molecules of one material and the other. These factors make it harder to move them past each other. You don’t work at this level when you calculate the force of friction, though. For everyday situations, physicists group all of these factors together in the “coefficient” ​μ.

Calculating the Force of Friction

    The “normal” force describes the force that the surface an object is resting on (or is pressed onto) exerts on the object. For a still object on a flat surface, the force must exactly oppose the force due to gravity, otherwise the object would move, according to Newton’s laws of motion. The “normal” force (​N​) is the name for the force that does this.

    It always acts perpendicular to the surface. This means that on an inclined surface, the normal force would still point directly away from the surface, while the force of gravity would point directly downwards.

    The normal force can be simply described in most cases by:

    N= mg

    Here, ​m​ represents the mass of the object, and ​g​ stands for the acceleration due to gravity, which is 9.8 meters per second per second (m/s2), or netwons per kilogram (N/kg). This simply matches the “weight” of the object.

    For inclined surfaces, the strength of the normal force is reduced the more the surface is inclined, so the formula becomes:


    With ​θ​ standing for the angle the surface is inclined to.

    For a simple example calculation, consider a flat surface with a 2-kg block of wood sitting on it. The normal force would point directly upwards (to support the weight of the block), and you would calculate:

    N=2\times 9.8 = 19.6\text{ N}

    The coefficient depends on the object and the specific situation you’re working with. If the object isn’t already moving across the surface, you use the coefficient of static friction ​μstatic, but if it is moving you use the coefficient of sliding friction ​μslide.

    Generally, the coefficient of sliding friction is smaller than the coefficient of static friction. In other words, it’s easier to slide something that’s already sliding than to slide something that’s still.

    The materials you’re considering also affect the coefficient. For example, if the block of wood from earlier was on a brick surface, the coefficient would be 0.6, but for clean wood it can be anywhere from 0.25 to 0.5. For ice on ice, the static coefficient is 0.1. Again, the sliding coefficient reduces this even more, to 0.03 for ice on ice and 0.2 for wood on wood. Look these up for your surface using an online table (see Resources).

    The formula for the force of friction states:

    F=\mu N

    For the example, consider a wood block of 2-kg mass on a wooden table, being pushed from stationary. In this case, you use the static coefficient, with ​μstatic = 0.25 to 0.5 for wood. Taking ​μstatic = 0.5 to maximize the potential effect of friction, and remembering the ​N = 19.6 N from earlier, the force is:

    F=0.5\times19.6 = 9.8\text{ N}

    Remember that friction only provides force to resist motion, so if you start pushing it gently and get firmer, the force of friction will increase to a maximum value, which is what you have just calculated. Physicists sometimes write ​Fmax to make this point clear.

    Once the block is moving, you use ​μslide = 0.2, in this case:

    F_{slide}=\mu_{slide} N=0.2\times 19.6 = 3.92\text{ N}

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