The coefficient of kinetic friction u(k) tells us how rough or smooth the frictional force is between two surfaces. It opposes the movement of the object in motion. To calculate u(k), it is necessary to know the equation for the frictional force, and Newton's law.

- Skill level:
- Moderately Challenging

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### Things you need

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- Introductory physics text

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## Instructions

- 1
Learn and understand the equation used to find the coefficient of friction. The magnitude of the force of kinetic friction is f(k) = u(k)N, where N is the normal force. The coefficient therefore is f(k)/N. It is a dimensionless number, and its value depends upon the surface. Smooth surfaces have lower values than rougher ones. For example, for a glass-upon-glass surface, it may be 0.6, and for ice-on-ice it may be 0.03. These numbers are approximate depending on whether the surfaces are clean, wet or sanded, for example.

- 2
Use the equation given in Step 1 to find the coefficient if you have a problem where you are given f(k) and N. Plug in the values and solve directly for u(k). If you are not given the frictional and normal forces, proceed to Step 3.

- 3
Calculate f(k) and N by using Newton's second law F = ma. Find the sum of all of the forces acting upon the moving object. Remember that for no acceleration, the net forces will be equal to zero. Remember also that f(k) opposes movement, and so will be opposite in sign to the force making the object move.

- 4
Draw a free body diagram to find the horizontal and vertical components of the forces acting upon the object in Step 3. A standard example is a box being pulled by a rope or a person pulling another person on a sled.

- 5
Practice Steps 3 and 4 by studying the diagram to the left, and writing out the appropriate equations. Shown is a pulling force, F, along with a frictional force, f(k), a normal force, N, and a gravitational force, mg. There is no acceleration.

The sum of all the forces: F + N + f(k) + mg = 0. The sum of all the horizontal forces: Fh -- f(k) = 0 (there is no acceleration). The sum of all of the vertical forces: Fv + N -- mg = 0 (the object is not moving up or down). Use f = u(k)N, Fv = Fsin(theta), and Fh = Fcos(theta) to solve the second and third equations simultaneously to find the coefficient of friction u(k). The answer is u(k) = Fcos(theta) / (mg -- Fsin(theta)).