Friction Force Calculator

Definition of dry friction and the Amontons–Coulomb law

Dry friction is the resistance between two solid bodies whose surfaces touch. In the model used in class and in most textbook problems, that resistance is proportional to the force pressing the surfaces together: T = μ·N. The coefficient μ has no unit. It depends on the pair of materials and on the condition of the surfaces, and in this model it does not depend on the apparent contact area. The name “Amontons–Coulomb law” records the two authors who gave the proportion the form still used in class. The symbol μ itself is not named after one person.

The SI unit of force is the newton: 1 N = 1 kg·m·s⁻². The calculator reports T and N in newtons. After the header is switched to US units, the same quantity also appears in pounds-force, with 1 lbf = 4.4482216 N. The examples in the text stay in newtons, so they match the SI reading on the panel.

History: Amontons, Coulomb, and the coefficient μ

Guillaume Amontons, in a memoir presented to the Academy in 1699, described two regularities of dry friction: the resistance grows in proportion to the normal load and, to a first approximation, does not depend on the apparent area of contact. Charles-Augustin de Coulomb developed that account in Théorie des machines simples, presented to the Academy in 1781 and published in the Academy’s memoirs for 1785. He separated static friction, which holds a body up to a limit, from friction in motion, and he treated kinetic friction in the simple model as only weakly dependent on sliding speed.

The coefficient μ is a later, compact way to write the ratio T/N. Leonardo da Vinci recorded similar observations at the end of the fifteenth century, but the law entered textbooks through Amontons and Coulomb. This calculator computes the product μ·N. It does not reproduce Coulomb’s full account with separate thresholds for rest and for motion.

Derivation of N = mg cos α and T = μ·N

Step 1. On a flat surface the body does not accelerate perpendicular to the table. The weight m·g is balanced by the normal force N, so N = m·g. The calculator uses g = 9.81 m/s² and has no field for g. For 10 kg, N = 98.10 N. Step 2. In this model the friction force is the product T = μ·N. At μ = 0.4 the result is 39.24 N. The panel rounds forces to two decimal places; the multiplication uses the full value before that rounding is shown.

Step 3. On an incline of angle α, measured from the horizontal, the weight splits into a perpendicular component m·g·cos α and a parallel component m·g·sin α. If no extra force presses the body into the plane, the normal force is the perpendicular component: N = m·g·cos α. At 30°, with μ = 0.4 and a mass of 10 kg, cos 30° = 0.8660, N = 84.96 N, T = 33.98 N, and the parallel component is 49.05 N. Step 4. The calculator shows both components and the product μ·N. It does not compute the difference m·g·sin α − T, and it does not turn that difference into an acceleration.

Assumptions: one μ, a true normal force, no extra loads

The formula N = m·g·cos α holds when the only forces perpendicular to the surface are the weight and the normal force. A rope that pulls at an angle, a hand that presses a block down, or a cart with an extra load all change N. There is no field for that extra force: the mass you enter is the mass of the body, and the angle is the angle of the incline. If the problem already gives the normal force in newtons, this formula still starts from the mass and the angle. The coefficient is then computed with the friction-coefficient calculator, from μ = T/N.

The angle must lie between 0° and 90°, inclusive. A blank field is zero, which is a flat surface. At 90° the cosine vanishes, the normal force on a vertical “incline” is zero, and T = 0, unless something else presses the body against the wall. Gravity is fixed at 9.81 m/s². That is not 9.80665 m/s², and it is not a local g. On the Moon, or in an elevator, this result does not apply.

Static and kinetic friction, and the limit of motion

Static friction does not have a single value. It adjusts to the applied force, up to a limit of about μ_s·N. Only after that limit does the body start to move, and kinetic friction is then approximately μ_k·N, directed opposite the sliding velocity. Tables often list μ_k below μ_s. This calculator has one μ field and always returns the product of that coefficient and N. It does not know the pulling force, so it cannot say whether the body stays at rest.

On an incline the comparison is visible on the panel, and it is still not a verdict. The parallel component m·g·sin α and the force T sit side by side. If the μ you enter is a kinetic coefficient and the body is already sliding, the acceleration is a = g(sin α − μ cos α), and a separate calculator computes it. If μ is a static coefficient, the body starts down the incline only when m·g·sin α exceeds μ·N, which is when tan α exceeds μ. This calculator does not announce that inequality as a result.

Units, orders of magnitude, and friction nearby

Orders of magnitude from tables, not from a measurement on this page: ice is about 0.05, wood on wood is often 0.2 to 0.5, rubber on concrete is about 0.6, and dry asphalt under a tire is often quoted near 0.7. A real tire also depends on slip and on how the rubber deforms, so 0.7 × 1400 kg × 9.81 m/s² = 9,613.80 N is the order of the Coulomb model, not the specification of a particular tire. The same force, 39.24 N, comes from 0.4 and 10 kg and from 0.05 and 80 kg, because the product μ·m is the same.

Contact area does not enter this formula. A wide runner and a narrow runner with the same μ and the same normal force give the same T. That is Amontons’ assumption, not a law for every rubber. A soft tread and a lubricated slide depart from it. This page has no correction for those cases.

Sources

The definition, the split between static and kinetic friction, and the formula T = μ·N come from general-physics 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 chapter on friction; H. D. Young and R. A. Freedman, University Physics, the chapter on friction; R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. I, Chapter 12. The newton: BIPM, The International System of Units (SI), 9th edition. This page is for educational purposes.