Enter the coefficient of friction (μ) and the incline angle (θ) into the calculator to determine the normalized maximum friction force on the incline, expressed as a fraction of the object’s weight: Ca = Ff,max / W. (Note: the coefficient of friction itself does not change with angle for the same pair of surfaces.) 

Coefficient of Friction at an Angle (Normalized Friction Force) Calculator

Enter any 2 values to calculate the missing variable

Coefficient of Friction at an Angle Formula

The following formula is used to calculate the normalized maximum friction force on an incline (as a fraction of the object’s weight). For an object on an incline, N = W cos(θ), so Ff,max = μN = μW cos(θ). Dividing by W gives: 

Ca = C * COS(a)
  • Where Ca is the normalized maximum friction force (Ca = Ff/W)
  • C is the coefficient of friction (μ) 
  • a is the incline angle (θ) 

To calculate Ca, multiply the coefficient of friction by the cosine of the incline angle. (This does not mean the material coefficient of friction changes with angle; it is the available friction force that changes because the normal force changes.)

How to Calculate Coefficient of Friction at an Angle?

The following two example problems outline how to calculate Ca (the normalized maximum friction force) at an angle.

Example Problem #1:

  1. First, determine the coefficient of friction. In this example, the coefficient of friction is given as 0.50.
  2. Next, determine the incline angle. For this problem, the incline angle is given as 40°.
  3. Finally, calculate Ca using the equation above: 

Ca = C * COS(a)

Inserting the values from above and solving the equation: 

Ca = 0.50 * cos(40deg) = 0.3830

Example Problem #2: 

Using the same process as above, first define the variables required by the formula. In this case, these values are:

coefficient of friction = 0.80

incline angle = 35°

Entering these values yields:

Ca = 0.80 * COS(35deg) = 0.6553