Friction Formula

Team Physics -
Created by: Team Physics -, Last Updated: May 7, 2024

Friction Formula

What is Friction Formula?

Friction is a fundamental concept in physics that describes the force resisting the motion of an object when it is sliding or rolling over another surface. The friction formula quantifies this force and is vital for understanding how objects interact in real-world scenarios. This formula is given by

𝐹 = πœ‡ Γ— 𝑁
  • 𝐹​ represents the frictional force.
  • ΞΌ is the coefficient of friction.
  • 𝑁 is the normal force exerted by the surface against the object.

The coefficient of friction, ΞΌ, is a dimensionless number that characterizes the interaction between the materials in contact. It varies based on the materials and their condition. The normal force N is influenced by the weight of the object, which brings us to the weight formula, given by

π‘Š = π‘š Γ— 𝑔
  • W is the weight
  • m is the mass of the object
  • g is the acceleration due to gravity, approximately 9.8m/sΒ² on Earth.

The concept of friction and its mathematical expression were developed over time by several scientists, but the first systematic study was conducted by Leonardo da Vinci and later refined by Guillaume Amontons. Amontons is often credited with the rediscovery and further explanation of the laws of friction in the late 17th century. The modern formula for friction, linking it directly to the normal force, was established through the work of various physicists who recognized the proportional relationship between these forces.

Applications of Friction Formula

  1. Vehicle Braking Systems: Mechanics use the friction formula to design brakes that efficiently stop vehicles by calculating the necessary friction between brake pads and wheels.
  2. Tire Tread Design: Tire manufacturers apply the friction formula to create treads that offer optimal grip with road surfaces under varying conditions, enhancing safety.
  3. Sports Equipment: Designers optimize the materials and surfaces of sports equipment, like tennis rackets and basketball shoes, using the friction formula to improve performance and control.
  4. Conveyor Belts: Engineers use the friction formula to ensure that conveyor belts have the right amount of grip to transport goods without slipping.
  5. Climbing Gear: The friction formula is critical in designing climbing gear that can safely hold the weight of climbers against rock faces.

Example Problems of Friction Formula

Problem 1: Calculating Frictional Force on a Flat Surface

Question: A 10 kg box is resting on a wooden floor. If the coefficient of friction between the box and the floor is 0.4, calculate the frictional force exerted when an attempt is made to move the box.


The normal force (N) is equal to the gravitational force acting on the box, which is the product of mass (m) and gravity (g).

Here, 𝑁 = π‘š Γ— 𝑔 = 10 kg Γ— 9.8 m/sΒ² = 98 N.

Using the friction formula 𝐹=πœ‡Γ—π‘,

where πœ‡ = 0.4, 𝐹 = 0.4 Γ— 98 N = 39.2 N.

Therefore, the frictional force is 39.2 N.

Problem 2: Determining the Coefficient of Friction on an Incline

Question: A 5 kg block slides down a 30Β° inclined plane with a constant velocity. Calculate the coefficient of friction between the block and the incline.


Since the block moves with constant velocity, the net force along the incline is zero, indicating that the frictional force equals the component of gravity along the incline.

The component of gravity along the incline is

π‘š Γ— 𝑔 Γ—sin⁑(πœƒ)=5 kg Γ— 9.8 m/sΒ² Γ— sin⁑(30Β°) = 24.5 N.

The normal force is

π‘š Γ— 𝑔 Γ— cos⁑(πœƒ) = 5 kg Γ— 9.8 m/sΒ² Γ— cos⁑(30Β°) β‰ˆ 42.44 N.

Using the friction formula 𝐹 = πœ‡ Γ— 𝑁 and knowing 𝐹=24.5 N,

πœ‡ = 𝐹 / 𝑁 = 24.5 / 42.44 β‰ˆ 0.577.

Therefore, the coefficient of friction is approximately 0.577.

Problem 3: Maximum Load Without Slipping

Question: What is the maximum weight that can be placed on a 15 kg sled on a horizontal ice surface before it begins to slip, if the coefficient of friction between the sled and the ice is 0.1?


The normal force is the weight of the sled plus the maximum load, 𝑁=(π‘šβ‚›β‚—β‚‘β‚” + π‘šβ‚—β‚’β‚β‚”) Γ— g.

The maximum friction force before slipping is 𝐹 = πœ‡ Γ— N.

Rearranging for π‘šβ‚—β‚’β‚β‚”β€‹,

π‘šβ‚—β‚’β‚β‚” = 𝐹 / (πœ‡Γ—π‘”) βˆ’π‘šβ‚›β‚—β‚‘β‚”β€‹.

Assuming F needs to be at least the weight of the sled to prevent slipping, 𝐹 = 0.1Γ—(15 kg + π‘šβ‚—β‚‘β‚β‚”) Γ— 9.8 m/sΒ².

For the sled to begin slipping, the force due to the load will equal the maximum frictional force that can be countered:

π‘šβ‚—β‚’β‚β‚” Γ— 𝑔 = 𝐹

π‘šβ‚—β‚’β‚β‚” Γ— 9.8 = 0.1 Γ— (15 + π‘šβ‚—β‚’β‚β‚”) Γ— 9.8

Simplifying, we cancel out g and solve for π‘šβ‚—β‚’β‚β‚”β€‹:

π‘šβ‚—β‚’β‚β‚” = 0.1 Γ— (15+π‘šβ‚—β‚’β‚β‚”)


mₗₒₐₔ βˆ’ 0.1 Γ— mₗₒₐₔ ​= 1.5

0.9 Γ— π‘šβ‚—β‚’β‚β‚” = 1.5

mₗₒₐₔ = 1.50.9 β‰ˆ 1.67 kg


What is the Force of Friction?

The force of friction is the resistance force generated by two surfaces interacting and moving against each other.

Is Friction Actually a Force?

Yes, friction is a force that opposes the relative motion between two surfaces in contact.

How to Calculate Normal Force?

Calculate normal force by multiplying the mass of the object by the acceleration due to gravity. The formula is N=π‘šΓ—π‘”.

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