Inclined Plane Calculator
Calculate effort force, mechanical advantage, and friction loss on an inclined plane (ramp).
Useful for loading docks, ADA ramps, and physics homework.
Inclined Plane
A ramp lets you lift a heavy object using less force, in exchange for moving it a longer distance. The flatter the ramp, the smaller the force, but the longer the path.
Ideal Mechanical Advantage (IMA)
IMA = ramp length / ramp height = 1 / sin(θ)
| Slope | IMA |
|---|---|
| 5° | 11.5:1 |
| 10° | 5.76:1 |
| 15° | 3.86:1 |
| 30° | 2:1 |
| 45° | 1.41:1 |
| 60° | 1.15:1 |
| 90° (vertical) | 1:1 |
Effort to Push Up the Ramp (frictionless)
F_effort = m × g × sin(θ)
For a 100 kg crate on a 15° ramp:
- F_effort = 100 × 9.81 × sin(15°) = 254 N (≈ 26 kgf)
That is far easier than the full 981 N (100 kgf) of vertical lifting.
Adding Friction
If μ is the kinetic coefficient of friction between the load and the ramp surface:
F_effort = m × g × (sin(θ) + μ × cos(θ))
For the same crate with μ = 0.20:
- F_effort = 100 × 9.81 × (0.2588 + 0.20 × 0.9659) = 444 N
Friction can easily double the required effort.
Common Friction Coefficients
| Surfaces | μ_kinetic |
|---|---|
| Steel on dry steel | 0.4–0.6 |
| Wood on wood | 0.2–0.5 |
| Rubber tire on asphalt | 0.6–0.8 |
| Skis on snow | 0.04 |
| Greased surfaces | 0.05 |
| ADA-compliant ramp + casters | 0.05–0.10 |
ADA / Building Code Ramps
US ADA accessibility ramps are limited to 1:12 slope (about 4.76°), giving IMA ≈ 12:1. A 200 kg loaded wheelchair plus user (1962 N) requires only ~163 N (~17 kgf) of horizontal push — well within human capability.
Conservation of Energy
Total work = m × g × h, regardless of the ramp angle. The ramp redistributes that work over a longer distance — you trade peak force for total path length. Friction adds extra heat-dissipated work, which is why steeper ramps with friction are far less efficient.
Worked Example: Loading a Pickup
Pushing a 50 kg motorcycle up a 1.8 m ramp into a 0.6 m bed:
- Angle: arcsin(0.6 / 1.8) = 19.5°
- Frictionless effort: 50 × 9.81 × sin(19.5°) ≈ 164 N (~17 kgf)
- With μ = 0.15: 50 × 9.81 × (0.334 + 0.15 × 0.943) ≈ 234 N (~24 kgf)
- Vertical lift would need 491 N (~50 kgf): twice as hard.
Where you encounter inclined planes in real life
| Application | Typical angle | Why |
|---|---|---|
| Loading ramps for trucks | 10–15° | Balance between cart-pushing force and ramp length needed |
| Wheelchair ramps (US ADA) | max 4.76° (1:12 slope) | Manageable for unassisted wheelchair users |
| Highway grades | typically 5–7% (3–4°) | Trucks can climb without major speed loss; brake-fade limits descents |
| Mountain road maximum | 8–10% (5–6°) | Beyond this, runaway-truck ramps become a regular feature |
| Skateboard quarter-pipes | 30–45° | Speed-vs-control trade-off for rider |
| Roof pitch (US residential) | 14–34° (3:12 to 8:12) | Sheds water without becoming too steep to walk on |
The shallower you go, the easier the push, but the longer the ramp must be to gain the same height. Architects sizing a ramp pick the angle by working backward from the available horizontal space, not the desired effort.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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