Inclined Plane & Friction Calculator

About the Inclined Plane & Friction Calculator

Put a block on a slope and this calculator works out everything about it: the weight and its two parts (along the slope and into it), the normal force, how much static friction can hold, whether the block slides, its acceleration, the time and speed at the bottom, and the force you need to push it up the slope, hold it still or start it moving. A free-body diagram draws the forces to scale, a very small one just long enough to see.

Pick the two surfaces from the coefficients of friction in OpenStax University Physics (Table 6.1) or type your own, give the slope as an angle or a grade, and use Earth’s gravity, the Moon’s or a planet’s. An angle of 0° is a flat floor. With a Pro pass, copy the result or download it as a CSV, or unlock this one result. Without a pass the free preview shows the acceleration down the slope and whether it slides, with the first rows of the force table, a marked diagram and the first step of the working.

Common uses

• Checking a physics homework problem on a block, a box or a sled on a slope, step by step
• Working out the push needed to move a load up a ramp, and whether it would slide back down
• Finding the steepest slope a crate, a car or a person rests on before slipping
• Comparing how the same slope behaves on the Moon or Mars
• Drawing a free-body diagram to check your own

The formulas

• Weight W = m·g, straight down
• Along the slope W∥ = m·g·sin θ (down the slope); into the slope W⊥ = m·g·cos θ
• Normal force N = m·g·cos θ: the surface pushes back as hard as the block presses into it
• Static friction fs ≤ μs·N: as much as needed to stop sliding, up to μs·N
• Kinetic friction fk = μk·N while it slides
• It slides when m·g·sin θ > μs·m·g·cos θ, that is when tan θ > μs; the steepest slope it rests on is the angle of repose, atan μs
• Acceleration while sliding: a = g·(sin θ − μk·cos θ)
• From rest over a length d along the slope: t = √(2d ÷ a) and v = √(2·a·d); the potential energy m·g·h it loses becomes ½·m·v² plus μk·N·d of heat
• Pushing parallel to the slope: up at a steady speed F = m·g·sin θ + μk·N; to start it up, more than m·g·sin θ + μs·N; to hold a sliding block still, at least m·g·sin θ − μs·N

Static and kinetic friction

Static friction is not a fixed force: it grows to match whatever pulls the block along the surface, up to its limit μs·N. That is why a block at rest stays put on every slope up to the angle of repose and then suddenly slides.

Once the block moves, kinetic friction μk·N takes over, and it is usually smaller. So between atan μk and atan μs a block rests on its own but, once nudged, keeps sliding and speeds up. On a slope of exactly atan μk it slides at a steady speed, which is a simple way to measure μk.

Friction does not depend on the area in contact: a brick on its side and on its end slides the same way, because the same weight presses on more or less area.

Coefficients of friction

The surfaces in the list are the approximate values of OpenStax University Physics, Table 6.1: for example rubber on dry concrete (μs 1.0, μk 0.7), wood on wood (0.5, 0.3), steel on steel (0.6, 0.3 dry; 0.05, 0.03 oiled), shoes on ice (0.1, 0.05) and Teflon on steel (0.04, 0.04). Rubber on wet concrete is printed as a range, 0.5–0.7 and 0.3–0.5; the calculator uses the middle of it. Real coefficients depend on the exact materials, how clean, dry or worn they are and the speed, so for anything that matters measure your own: tilt the slope until the block just starts to slide, and μs = tan θ.

Sources

• OpenStax, University Physics Volume 1, §6.2 Friction: static friction fs ≤ μs·N, kinetic friction fk = μk·N and Table 6.1, the approximate coefficients of static and kinetic friction (CC BY 4.0)
• OpenStax, University Physics Volume 1, §5.6 Common Forces: the parts of the weight on an incline, w sin θ and w cos θ, the normal force N = m·g·cos θ and Example 5.12 (CC BY 4.0)

Limitations

• The block is a rigid body that slides without tipping or rolling; wheels, balls and cylinders need a rolling model.
• Pushes are parallel to the slope; a push at another angle changes the normal force and is not modelled.
• Coefficients of friction are approximate and change with the surfaces’ condition, temperature and speed; air resistance is ignored.
• The time and speed at the bottom assume it starts from rest and slides the whole length.
• An educational estimate: not for designing ramps, loading equipment or anything people rely on for safety.

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