Library / Foundations: Physics / FPH-08
Foundations: Physics
Limiting friction on a slope
Maximum friction force on an object on an incline.
- Ref
- FPH-08
- Rev
- v1
- Field
- Foundations: Physics
- Result
- Fmax [N]
R is the normal reaction. In the standard rough-contact model limiting friction equals μR. Resolve along perpendicular axes. α is the slope angle.
Friction can resist up to a certain limit, set by how hard the surfaces are pressed together and how rough they are.
Calculator
- Down-slope weight component mg sin α82.89 N
- Angle of friction arctan μ21.8 °
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The object slides when the down-slope component mg sin α exceeds μR, i.e. when tan α > μ.
- Friction does not depend on contact area in this simple model.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N) | Local step | Change in result |
|---|---|---|---|---|---|---|
| m Mass | 1 to 500 kg | 3.555 | 1,778 | 1,774 | +10% | +10 % |
| μ Coefficient of friction | 0.05 to 1 | 8.888 | 177.8 | 168.9 | +undefined% | — |
| α Slope angle | 0 to 60 ° | 39.23 | 78.45 | 39.23 | +10% | −2.13 % |
Sweep of Slope angle
| Slope angle (°) | Limiting friction (N) |
|---|---|
| 0 | 78.45 |
| 7.5 | 77.78 |
| 15 | 75.78 |
| 22.5 | 72.48 |
| 30 | 67.94 |
| 37.5 | 62.24 |
| 45 | 55.47 |
| 52.5 | 47.76 |
| 60 | 39.23 |
Sweep one variable
Variables and how each one contributes
Limiting friction
What the equation gives you. Everything else feeds it.
Coefficient of friction
Roughness of the contact. Directly proportional.
Mass
Mass of the object. Directly proportional.
Slope angle
Angle of the slope to the horizontal. Steeper slopes lower the normal reaction and so the friction.
Worked example with the values above
- Normal reaction: R = mg cos α = 20 × 9.80665 × 0.9063 = 177.8 N.
- Multiply by μ = 0.4.
- Fmax = 71.1 N.
Assumptions and limits
- Dry friction with a constant coefficient.
- Object about to slip or at rest.
Common mistakes
- Using mg instead of the normal reaction on a slope.
- Forgetting friction may be less than its limit.
Statistics and uncertainty
The coefficient μ is the most uncertain input, often varying ±20–30% with surface condition, so results are indicative.
Where it is used
- Vehicle traction on slopes.
- Stability of equipment on ramps.
Maths you need
- Resolving forces: components along and across a slope
- Weight: W = mg