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, where ‌N‌ is the normal force, and the coefficient of friction ‌μ‌ for specific surfaces. scales this normal force appropriately.

What Is Friction?

Friction describes the resistive contact force between two surfaces when you try to move one across the other. This 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 of friction ‌μ.

There are two types of friction with different relationships to the normal force and the surfaces in contact: static friction and kinetic friction. The distinction here depends on a property that the coefficient of friction is different for two surfaces if there is (or is not) motion between them.

Static Friction

When two objects or surfaces are not moving relative to one another, they experience the force of static friction. Static friction acts on these two stationary bodies according to the coefficient of static friction between the two contact surfaces. This follows a similar friction equation as above with a slightly different relationship:

f_s \leq \mu_s N

Here, the force of the static friction is less than or equal to coefficient of friction and normal force. Static friction is not always a constant force, it actually responds to be equal and opposite the applied force. Think of continuing to apply force to heavy box on the ground. It only starts to slip once the applied force exceeds the upper limit of static friction. Before this point, the static friction simply resists the applied force to create zero net force.

Kinetic Friction

When two surfaces or objects are already in motion past one another, they experience kinetic friction. There is a resistive force as they move defined by the normal force and the coefficient of kinetic friction:

f_k = \mu_k N

The coefficient of kinetic friction is lower than the coefficient for static friction for the same two surfaces. Think of the motion of kinetic friction as almost skipping across the irregularities and points of contact with the other surface, leading to a lower average force of kinetic friction.

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 horizontal surface, the force must exactly oppose the force due to gravity, otherwise the object would move, according to Newton’s second law 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 plane, 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 newtons 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:

N = mg \cos \theta

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 ‌μs‌, but if it is moving you use the coefficient of kinetic friction μk.

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).

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 of 0.25 to 0.5 for wood. Let our static friction coefficient be 0.5 to maximize the potential effect of friction, and remembering the ‌N‌ = 19.6 N from earlier, the force is:

F_s = 0.5 \times 19.6 \text{ N} = 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 the coefficient of kinetic friction to be 0.2, in this case:

F_k = 0.2 \times 19.6 \text{ N} = 3.92 \text{ N}

Other Considerations

These calculations consider a simplified picture of friction. There are many other factors that influence the static and kinetic friction force. Friction does not usually behave simply, and the coefficients of friction often change depending on external conditions or deviation from a constant speed in the case of kinetic friction.

There are different calculations for rolling friction with wheels or bearings, fluid friction – in the case of two fluids or a fluid and a solid, or dry friction/wet friction scenarios. In the real world, air resistance (a form of friction with the molecules in the air), differences in smooth surfaces and rough surfaces, and friction in the case of oscillating systems becomes incredibly complicated.

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