Inclined Plane Calculator
The forces on a slope: what pulls down it, what presses into it, whether it slides, and the force to hold or raise it.
Results
What this tool does
A slope splits weight in two. Part of it presses the object into the surface, which is what friction has to work with; the rest pulls it down the slope, which is what friction has to resist. Everything about ramps follows from that split. Put in the angle, the mass and the two coefficients, and this page gives both components, the friction available, whether the object stays put, and — if it does not — how fast it accelerates. It also gives the force needed to hold it and to push it up, which is the number a ramp is usually built to reduce.
Formula
along the plane: mg sin θ · normal: mg cos θ · it slides when tan θ > μₛ
Variables
| Symbol | Meaning | Unit |
|---|---|---|
mass | Mass | kg |
angle | Angle of the slope | ° |
static | Static friction coefficient | — |
kinetic | Kinetic friction coefficient | — |
push | Force pushing it up the slope | N |
gravity | Gravitational acceleration | m/s² |
AC | Acceleration | m/s² |
SL | Does it slide? | — |
AL | Pull down the slope | N |
NF | Normal force | N |
MS | Most friction can hold | N |
KF | Friction once it is sliding | N |
NE | Force left over | N |
HF | Force needed to hold it still | N |
PU | Force to push it up at steady speed | N |
MF | Smallest coefficient that would hold it | — |
RA | Angle at which it starts to slide | ° |
MA | How much easier than lifting it straight up | x |
WT | Weight | N |
Worked example
- Mass20 kg
- Angle of the slope30 °
- Static friction coefficient0.4
- Kinetic friction coefficient0.3
- Force pushing it up the slope0 N
- Gravitational acceleration9.80665 m/s²
- Acceleration2.3555 m/s²
- Does it slide?Yes
- Pull down the slope98.0665 N
- Normal force169.8562 N
- Most friction can hold67.9425 N
- Friction once it is sliding50.9568 N
- Force left over47.1097 N
- Force needed to hold it still30.1240 N
- Force to push it up at steady speed149.0233 N
- Smallest coefficient that would hold it0.5774
- Angle at which it starts to slide21.801 °
- How much easier than lifting it straight up2.0000 x
- Weight196.1330 N
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
- The default values are typical reference figures, not measurements of your situation. Replace them with your own data whenever you have it.
Frequently asked questions
Why does the mass not change whether it slides?
Because mass appears on both sides and cancels. The pull down the slope is mg sin θ and the most friction can resist is μ mg cos θ, so the object slides exactly when tan θ exceeds μ — no mass in sight. A grand piano and a matchbox on the same ramp with the same surfaces either both stay or both go. Mass does change the forces and the acceleration once it is moving; it just does not change the decision.
What is the angle of repose?
The steepest the slope can get before the object starts to slide on its own — the angle whose tangent equals the static coefficient. It is also the easiest way to measure that coefficient: tilt the surface slowly until the object moves, note the angle, and the tangent of it is μ. Loose materials show the same thing in reverse: pour sand and the cone it forms settles at its own angle of repose, which is why a gravel heap always has the same slope regardless of how much gravel you pour.