Example 1
A block in the lab: the spring scale reads 39.24 N, and the normal force on the scale is 98.10 N. This is the reference value for this page. The quotient is exactly 0.4.
μ = 0.400 · share 40.0%
Friction Coefficient Calculator
Enter the friction force T and the normal force N, both in newtons. The calculator finds the friction coefficient μ = T/N from the Amontons–Coulomb definition. For T = 39.24 N and N = 98.10 N the result is 0.400. That number has no unit. The model does not compute μ from an incline angle and does not choose a static or kinetic mode.
Result
Enter the friction force T and the normal force N. The coefficient μ, the percent share, and the T/N chart appear here.
The friction force from a coefficient and a mass is computed by the T = μ·N calculator, and the acceleration of a body sliding down a slope by the incline with friction calculator.
How it works
T39.24 N
N98.10 N
μ0.400
In the Amontons–Coulomb model the friction coefficient is the quotient of the friction force and the normal force: μ = T/N. Both forces are in newtons, and the quotient has no unit. For T = 39.24 N and N = 98.10 N the calculator gives μ = 0.400. That is the textbook definition (Halliday, Resnick, and Walker; Young and Freedman): the coefficient says what fraction of the normal force the measured friction force is.
The calculator uses the numbers you enter: the friction force T and the normal force N. There is no field for mass, for an angle, or for the acceleration g. If the problem gives a mass, the normal force on a flat surface is the weight m·g, and that weight has to be known in newtons already. At g = 9.81 m/s² a mass of 10 kg gives a normal force of 98.10 N; together with a friction force of 39.24 N that returns μ = 0.400. This calculator does not perform that conversion.
What the result panel shows. Besides μ, the calculator shows the T and N you entered, the percent share T/N, and the quotient N/T when T is greater than zero. For the reference pair that is 0.400, 39.24 N, 98.10 N, a share of 40.0%, and N/T = 2.500. The coefficient is rounded to three decimal places. The bar chart computes μ from that same friction force at normal forces of 50, 75, 100, 150, 200, and 250 N.
Limits of the model. One result: the calculator does not separate μ_s and μ_k. The problem says which friction force was measured, the limit as motion starts or the friction during sliding. In this form Amontons’ law does not depend on the contact area or on the sliding speed. The calculator does not compute rolling friction, lubrication, or the angle α = arctan μ. The slide acceleration a = g(sin α − μ cos α) is computed by the incline-with-friction calculator, and the force T = μ·N from a mass is computed by the friction-force calculator.
The unit switch in the header converts newtons to pounds-force and back, at 1 lbf = 4.4482216 N. The coefficient does not change, because the same factor multiplies both forces. The worked examples in the text stay in newtons as the reference. A comma and a period in the fields are equivalent. The normal force must be greater than zero. Zero entered in the T field gives μ = 0.000. A negative number gives no result.
Examples on this page: 39.24 N and 98.10 N → 0.400; 20 N and 400 N → 0.050; 60 N and 200 N → 0.300; 8.49 N and 42.45 N → 0.200; 9,000 N and 12,000 N → 0.750. The “Fill this example” button enters those same numbers in the fields and refreshes the panel and the chart. The fields are empty when you open the page.
Formula
μ = T / N
The result has no unit. The calculator rounds it to three decimal places.
Model: the coefficient in the Amontons–Coulomb law of dry friction. The calculator divides the friction force by the normal force. It does not compute T = μ·N, it does not compute μ = tan α, and it does not compute the acceleration on an incline.
Theory, derivation, and domain
The coefficient as the quotient T/N
The friction coefficient is the number that compares the friction force with the force perpendicular to the contact surface. In the school model, and in most textbook problems, that number is the quotient μ = T/N. T and N are in newtons, 1 N = 1 kg·m·s⁻², so the units cancel. For the reference pair, 39.24 N / 98.10 N is exactly 0.4, and the panel shows 0.400. The percent share is the same quotient times 100, which is 40.0%.
The name “Amontons–Coulomb law” names the two authors who described the proportion between friction and the normal load. The symbol μ itself is a later, compact way to write the quotient. This calculator returns that quotient. It does not build a force from a mass, and it does not resolve a weight on an incline.
A short history: Amontons, Coulomb, and the symbol μ
In a memoir presented to the Academy in 1699, Guillaume Amontons described two regularities of dry friction: the resistance grows in proportion to the normal load and, to a first approximation, it does not depend on the apparent contact area. That proportion means the quotient T/N is approximately constant for a given pair of surfaces. Charles-Augustin de Coulomb developed the account in “Théorie des machines simples,” presented to the Academy in 1781 and published in the memoirs of 1785. He separated the friction that holds a body up to a limit from the friction during motion.
Leonardo da Vinci noted similar observations in the late fifteenth century, but the law entered textbooks through Amontons and Coulomb. This calculator does not rebuild Coulomb’s two thresholds. If the measured force is the limit before motion starts, the quotient is μ_s. If the force is friction during sliding, the quotient is μ_k. Both cases share one result here.
Step by step for 39.24 N and 98.10 N
Step 1. Read the friction force T = 39.24 N. That may be a spring-scale reading during steady pulling, or a force given in the problem. Step 2. Read the normal force N = 98.10 N. That is the force perpendicular to the surface, not a mass. Step 3. Divide: 39.24 / 98.10 = 0.4. Step 4. The panel rounds to three decimal places and shows 0.400. N/T = 98.10 / 39.24 = 2.5, so the panel shows 2.500. The division uses the values you entered, and the coefficient is rounded only for display.
The same pair of forces appears on the friction-force page as a result, not as input: μ = 0.4 and a mass of 10 kg on the flat, at g = 9.81 m/s², give T = 39.24 N and N = 98.10 N. Here the unknown runs the other way. If the newtons are already known, stay on this page. If the mass and the coefficient are known, open the friction-force calculator.
What this quotient does not compute
The formula μ = tan α describes a different experiment: the incline angle at which a body starts to move or slides at constant speed. There is no arctangent field here. The angle, the length of the incline, and the acceleration a = g(sin α − μ cos α) belong to the incline-with-friction calculator. Contact area does not enter the quotient either. A wide runner and a narrow runner with the same T and the same N give the same μ. That is Amontons’ assumption. Soft rubber, a tire, and a lubricated guide depart from it, and this formula has no correction for them.
An extra push, for example a hand pressed against the surface, is already part of the number N when the normal force you enter is the real one. The calculator does not add the weight to that field. The N you enter is the whole normal force in the quotient.
Units, rounding, and textbook orders of magnitude
The header switch multiplies both forces by the same factor, 1 lbf = 4.4482216 N, so μ is unchanged. The chart still computes the quotient at normal forces of 50, 75, 100, 150, 200, and 250 N; the axis labels follow the switch. The pair 15 N / 80 N has an exact quotient of 0.1875, and the panel would round it to 0.188. The examples on this page are chosen so that three decimal places show the quotient with no remainder: 0.400, 0.050, 0.300, 0.200, 0.120, 0.750, and 0.250.
Orders of magnitude in tables are not a result of this calculator. Ice is often listed near 0.05, wood on wood often between 0.2 and 0.5, and rubber on dry concrete often between 0.6 and 1.0. The pair 20 N and 400 N gives 0.050 and sits in the ice range only when those two forces were actually measured. The calculator does not choose a material.
Sources
The definition μ = T/N, the distinction between static and kinetic friction, and the history of the proportion come from textbooks and from the classic memoirs: G. Amontons, memoir on resistance in machines, Histoire de l’Académie Royale des Sciences, 1699; C.-A. de Coulomb, Théorie des machines simples, Mémoires de Mathématique et de Physique de l’Académie Royale des Sciences, 1785 (text presented in 1781); D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics, the friction chapter; H. D. Young and R. A. Freedman, University Physics, the friction chapter; R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. I, Ch. 12. The newton: BIPM, The International System of Units (SI), 9th edition. This page is educational.
How to use
- Enter the friction force T in newtons, for example 39.24.
- Enter the normal force N in newtons, for example 98.10. Zero does not pass, because the calculator divides by N.
- Click Calculate, or wait for the live result. For 39.24 N and 98.10 N the calculator shows μ = 0.400.
- Check the percent share and the quotient N/T. The chart computes the same quotient at normal forces from 50 N to 250 N.
- The Clear button removes the values you entered and hides the panel. The unit switch in the header converts newtons to pounds-force.
Symbol glossary
μ is the coefficient, T the friction force, and N the normal force. For 39.24 N and 98.10 N the result is 0.400.
- μ
- Friction coefficient with no unit. At 39.24 N and 98.10 N it is 0.400. The panel keeps three decimal places.
- T
- Friction force in newtons. It is an input, not the result. Zero gives μ = 0.000. A negative number does not pass.
- N
- Normal force perpendicular to the surface, in newtons. It must be greater than zero. This is not a mass.
- T/N
- Share of the friction force in the normal force. For the reference pair it is 40.0%. That is the same quotient as μ, written as a percent.
- N/T
- How many times the normal force exceeds the friction force. For the reference pair it is 2.500. At T = 0 the quotient does not exist, and the panel shows a dash.
Worked examples
Example 2
A crate pulled on ice: friction 20.00 N at a normal force of 400.00 N. The coefficient 0.050 sits in the textbook range for ice, if those forces were measured.
μ = 0.050 · share 5.0%
Example 3
Wood on wood: friction 60.00 N and a normal force of 200.00 N. The normal force is 3.333 times the friction force.
μ = 0.300 · share 30.0%
Example 4
A small block from a problem set: friction 8.49 N and a normal force of 42.45 N. The quotient is exactly 0.2, and N/T is 5.000.
μ = 0.200 · share 20.0%
Example 5
A carton on a floor: friction 30.00 N and a normal force of 250.00 N. The share is 12.0%, and N/T is 8.333.
μ = 0.120 · share 12.0%
Example 6
The same proportion as in the lab, at another scale: friction 40.00 N and a normal force of 100.00 N. Amontons expected a constant quotient, not a constant force.
μ = 0.400 · share 40.0%
Example 7
A large contact in a test: friction 9,000.00 N and a normal force of 12,000.00 N. The scale of the forces is different, and the quotient is still ordinary division.
μ = 0.750 · share 75.0%
Example 8
A pallet on concrete, from force readings: friction 50.00 N and a normal force of 200.00 N. N/T is 4.000. This is not the mass of the pallet.
μ = 0.250 · share 25.0%
Example 9
A model with no friction: friction 0.00 N at a normal force of 100.00 N. The coefficient is 0.000, and the quotient N/T does not exist.
μ = 0.000 · share 0.0%
Suggested uses
- Reading μ from a spring scale and from a normal force measured on a scale, when the problem gives two forces rather than an incline angle.
- Checking whether two pairs of forces give the same coefficient before that number goes into the friction-force calculator.
Frequently asked questions
What does this calculator compute, and from which formula?
The calculator finds the friction coefficient from μ = T/N, the Amontons–Coulomb definition. T is the friction force and N is the normal force, both in newtons. For T = 39.24 N and N = 98.10 N the result is 0.400. The calculator rounds μ to three decimal places.
Which units should I use for the friction force and the normal force?
Enter both forces in newtons. The result μ has no unit. After you switch the header to US units, the forces are in pounds-force and the panel also shows the value in newtons. There is no mass field: the normal force has to be known in newtons first.
Does the coefficient μ have a unit?
No. Newtons in the numerator and the denominator cancel. 0.400 is four tenths, not 0.400 N. The unit switch does not give the coefficient a unit.
Is this static friction or kinetic friction?
The calculator computes one μ from the quotient T/N. The problem says whether the measured friction force is the limit before motion starts or the friction during sliding. The calculator does not separate μ_s and μ_k, and it does not decide whether the body will start to move.
How is this different from the friction-force calculator and the incline?
Here the force T and the normal force N give μ. On the friction-force page, μ and the mass give T = μ·N. The slide acceleration a = g(sin α − μ cos α) is computed by the incline-with-friction calculator. This calculator has no angle field.
Why does a zero normal force give no result, while zero friction does?
A normal force of zero has no quotient, so the calculator shows no result. Zero entered in the T field, with a positive N, gives μ = 0.000. A negative number in either field also gives no result. Both fields need a number.
Are a comma and a period equivalent?
Yes. The calculator reads 39.24 and 39,24 as the same friction force. The fields are empty when you open the page. The hint in a field is not an entered value.
What does the chart show?
The bar chart computes μ = T/N from the friction force you entered, at normal forces of 50, 75, 100, 150, 200, and 250 N. It does not show an incline angle. The normal-force axis is that fixed list, not the value in the N field.
Does the calculator find μ from an incline angle?
No. The formula μ = tan α belongs to the angle at which a body starts to move or slides at constant speed. That is a different experiment. This calculator has no angle field and does not compute an arctangent. If the problem gives an angle rather than two forces, this quotient does not replace that angle.
What does the unit switch change?
It converts newtons to pounds-force and back, at 1 lbf = 4.4482216 N. The coefficient stays the same, because one factor multiplies both forces. The worked examples in the text stay in newtons.
How do I calculate the friction coefficient with μ = T/N?
Divide the friction force by the normal force. For 39.24 N and 98.10 N: 39.24/98.10 = 0.400.
What is μ at T = 20 N and N = 50 N?
μ = 20/50 = 0.40. The result is dimensionless because both forces are in newtons.
How does this result relate to weight on a level surface?
On a level surface N = m·g. A 10 kg mass with μ = 0.4 gives N = 98.10 N and T = 39.24 N, the same quotient 0.400.
How is this page different from the friction-force calculator?
Here the unknown is μ when T and N are known. The friction-force companion computes T = μ·N from known μ and mass.
Can μ be computed as tan α on an incline?
Not on this page. For uniform motion or the slip threshold, μ = tan α belongs to the incline-with-friction model.
Sources
The quotient, the history, and the newton come from established physics and metrology sources.
- Wikipedia — Friction
- Feynman Lectures on Physics, Vol. I, Ch. 12 — Friction
- BIPM — The International System of Units (SI)
- NIST — SI units
- Amontons, G., memoir on resistance in machines, Histoire de l’Académie Royale des Sciences, 1699.
- Coulomb, C.-A. de, Théorie des machines simples, 1785.
- Halliday, D.; Resnick, R.; Walker, J., Fundamentals of Physics — friction.
- Young, H. D.; Freedman, R. A., University Physics — friction.
Page updated November 3, 2026.