Inclined Plane Calculator

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)

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

ADA / Building Code Ramps

US ADA accessibility ramps are limited to 1:12 slope (about 4.76°), giving IMA ≈ 12:1. Take a 200 kg loaded wheelchair plus user, so 1962 N of weight. The gravity component alone is only 163 N (about 17 kgf). Add caster friction at μ = 0.05 and the real push comes to 261 N, about 27 kgf. That is why the code caps the slope rather than trusting the arithmetic: 1:12 is roughly the steepest a person can self-propel up, and friction is more than a third of the effort at that angle.

Conservation of Energy

Total work = m × g × h, regardless of the ramp angle. The ramp spreads that work over a longer distance, trading 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 × (0.3333 + 0.15 × 0.9428) ≈ 233 N (~24 kgf)
• Vertical lift would need 491 N (~50 kgf): more than twice as hard.

Ramp length is not the same as run, and mixing them up is easy

That 1.8 m is the length of the plank, measured along the slope. The run is the horizontal distance it covers, which for a 0.6 m rise is √(1.8² − 0.6²) = 1.70 m. The two differ by about 6% here and by much more on a steep ramp. Use arcsin(rise / length) if you measured the plank, and arctan(rise / run) if you measured the ground. The rise-and-run mode below wants the ground measurement, so enter 0.6 and 1.70 for this example, not 0.6 and 1.8. The calculator reports the ramp length in its answer so you can check it against the plank you actually own.

Where you encounter inclined planes in real life

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.