What this calculator does
A mass on a slope is pulled straight down by gravity, but only the component along the slope drives it forward. The rest presses it into the surface, and that pressing force is what generates friction.
As the slope steepens the driving component grows while the pressing component shrinks, so friction weakens exactly when it is most needed. That is why there is a critical angle beyond which nothing stays put.
The formula
The normal force is mass times gravity times the cosine of the angle, and friction is that multiplied by the friction coefficient. Acceleration is gravity times the sine of the angle minus the friction coefficient times its cosine, and comes out as zero when friction wins.
| Term | Meaning |
|---|---|
| Normal force | The force perpendicular to the slope surface, which determines how much friction is available. |
| Angle of repose | The steepest angle at which an object stays put, where the tangent of the angle equals the friction coefficient. |
| Coefficient of friction (μ) | The ratio of friction force to normal force for the two surfaces in contact. |
The inputs explained
| Field | What to enter |
|---|---|
| Mass (kg) | The mass on the slope, in kilograms. |
| Incline angle (°) | The incline angle in degrees, measured from horizontal. |
| Coefficient of friction | The coefficient of friction between the object and the surface. |
When to use it
Checking whether a load will slide
Comparing the driving force against available friction settles whether something on a ramp needs restraining.
Designing a ramp
A shallower ramp needs less force to push a load up but more distance, which is the usual trade-off of any simple machine.
Understanding the angle of repose
Loose material piles up to exactly the angle at which friction just balances gravity, which this makes concrete.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
At what angle does a load start to slide?
The same mass and surface at a range of slope angles.
| Incline angle | Acceleration down the incline | Normal force |
|---|---|---|
| 10° | 0 m/s² (held by friction) | 96.58 N |
| 20° | 1.511 m/s² | 92.15 N |
| 30° | 3.205 m/s² | 84.93 N |
| 45° | 5.547 m/s² | 69.34 N |
Questions
Why does the normal force fall as the slope steepens?
Because less of the weight acts perpendicular to the surface. On a vertical wall none of it does, which is why friction against a vertical surface provides no support at all.
At what angle does an object start to slide?
When the tangent of the angle exceeds the friction coefficient. For μ = 0.2 that is about 11.3 degrees, which is why the 10 degree case above is still held.
Does mass affect whether it slides?
No. Mass appears in both the driving force and the friction force, so it cancels. A heavy crate and a light one on the same surface begin sliding at exactly the same angle.
Why is static friction higher than sliding friction?
Because surfaces in contact settle and interlock at rest. That is why something often needs a shove to start moving and then slides more easily once it is going.
For the friction force on its own, see the friction force calculator. For the force and acceleration relationship, see the force, mass and acceleration calculator.