How to Calculate Acceleration With Friction

By Ariel Balter
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Newton’s second law, F=ma, states that when you apply a force F to an object with a mass m, it will move with an acceleration a = F/m. But this often appears to not be the case. After all, it's harder to get something moving across a rough surface even though F and m might stay the same. If I push on something heavy, it might not move at all. The resolution to this paradox is that Newton’s law is really ΣF = ma, where Σ means you add up all the forces. When you include the force of friction, which may be opposing an applied force, then the law holds correct at all times.

Pushing a Wooden Crate on a Wood Floor

Step 1

Draw a square sitting on a line representing a crate resting on the floor.

Step 2

From a point in the center of the box draw an arrow downward and label it F_grav = -mg, meaning mass times the acceleration due to gravity (g = 9.81 meters per second squared). For this example, m = 50 kilograms, so F_grav = -491 Newtons. The negative sign means it’s pointing down.

Step 3

From that same point, draw an arrow upward and label it N = -F_grav = mg = 491 Newtons. This is the normal force on the block that keeps it from falling through the floor. The normal force is equal to weight and points in the opposite direction.

Step 4

Draw a horizontal arrow and label it F_applied. This is the force applied to the block. For this example, F_applied = 150 Newtons.

Step 5

Look up the coefficient of kinetic friction μ from the table in reference 3 for wood on wood. In this case, it gives μ = 0.2.

Step 6

Calculate the friction force using F_friction = μN = 98.2 Newtons.

Step 7

Add up the net horizontal force on the crate. Remember that the friction force always opposes the motion, so ΣF = F_applied - F_friction = 150 - 98.2 = 51.8 Newtons.

Step 8

Divide the net force by the mass to get the acceleration: a = ΣF/m = 1.04 meters per second squared. Without friction, the acceleration would be 3.0 meters per second squared.

About the Author

Ariel Balter started out writing, editing and typesetting, changed gears for a stint in the building trades, then returned to school and earned a PhD in physics. Since that time, Balter has been a professional scientist and teacher. He has a vast area of expertise including cooking, organic gardening, green living, green building trades and many areas of science and technology.