Friction & Inclined Plane Calculator
Find out whether something slides down a slope and how fast. Work out friction F = μN on flat ground or a ramp, the forces along and across the slope, the acceleration, the angle of repose and the push needed to move a load up or down.
How to Use
- Pick a mode: On a slope to see if a load slides, Push up or down for the force to move it, Flat ground for F = μN, or Find μ or angle to work back from a test.
- Enter the mass and the slope angle, choosing kg, g, lb or tonnes and degrees or radians.
- Choose a pair of surfaces to fill in typical static and kinetic coefficients, or type your own μs and μk.
- Read the answer in the first readout: the acceleration, the push needed, or the solved value in its highlighted field. The diagram draws the weight, normal force and friction to scale.
- Press a preset to load an example, and check Show Work for every step from the weight to the result.
Worked Example
A box on a 30° ramp. A 10 kg box, μs = 0.5 and μk = 0.4. Its weight is 10 × 9.80665 = 98.07 N. The part pulling it down the slope is 98.07 × sin 30° = 49.03 N, and the normal force is 98.07 × cos 30° = 84.93 N. Static friction can give at most 0.5 × 84.93 = 42.46 N, less than 49.03 N, so it slides (equivalently, tan 30° = 0.5774 is more than 0.5). Sliding friction is 0.4 × 84.93 = 33.97 N, leaving 15.06 N, so a = 15.06 ÷ 10 = 1.506 m/s².
A crate that stays put. 25 kg on a 20° slope with μs = 0.5: the slope pulls with 83.85 N and friction can hold up to 115.2 N, so nothing moves; friction supplies just the 83.85 N needed. Tilt it past arctan 0.5 = 26.57° and it goes.
The common mistake: using the full weight as the normal force. On a slope the surface only pushes back with mg cos θ. Writing N = mg for the 10 kg box gives friction 0.4 × 98.07 = 39.23 N and an acceleration of 0.9807 m/s², when the right answer is 1.506 m/s². The steeper the slope, the bigger the error: at 60° the normal force is only half the weight.
Show Work
Formulas
Typical coefficients of friction, from the table in Serway & Jewett’s Physics for Scientists and Engineers. Real surfaces vary a lot with finish, dirt, moisture and temperature, so treat these as starting points and measure when it matters.
| Surfaces | Static μs (typical) | Kinetic μk (typical) | Slips at (arctan μs) |
|---|---|---|---|
| Rubber on dry concrete | 1.0 | 0.8 | 45° |
| Steel on steel (dry) | 0.74 | 0.57 | 36.5° |
| Aluminium on steel | 0.61 | 0.47 | 31.38° |
| Copper on steel | 0.53 | 0.36 | 27.92° |
| Wood on wood | 0.25–0.5 | 0.2 | 14.04°–26.57° |
| Glass on glass | 0.94 | 0.4 | 43.23° |
| Metal on metal (lubricated) | 0.15 | 0.06 | 8.53° |
| Waxed wood on wet snow | 0.14 | 0.1 | 7.97° |
| Ice on ice | 0.1 | 0.03 | 5.71° |
| PTFE (Teflon) on PTFE | 0.04 | 0.04 | 2.29° |
From Leonardo’s Notebooks to Coulomb’s Law of Friction
Leonardo da Vinci tested friction with blocks pulled across tables in the late 15th century and wrote down the two rules still taught today: friction grows in proportion to the load, and it does not depend on the area in contact. His notebooks went unpublished, and the French physicist Guillaume Amontons found the same rules again and presented them to the Paris Academy of Sciences in 1699, which is why they are often called Amontons’ laws.
The inclined plane itself was understood earlier. The Flemish engineer Simon Stevin proved the law of the slope in 1586 with his “wreath of spheres”, a loop of balls draped over a triangle that cannot turn by itself, which shows that the pull along a slope is the weight times the height over the length, our mg sin θ. Galileo then rolled balls down gentle slopes to slow free fall enough to measure it.
Charles-Augustin de Coulomb, better known for the law of electric charges, turned friction into engineering. His essay on simple machines, Théorie des machines simples, won the Paris Academy’s prize in 1781 and separated the friction that holds a body at rest from the smaller friction that acts once it slides, the μs and μk used here.
About This Tool
This calculator handles the friction problems that come up most: whether a load on a slope stays put or slides, how fast it speeds up, the push needed to move it up or down a ramp at a steady speed, plain F = μN on level ground, and working a coefficient back from a tilt test or a measured acceleration. It draws the free-body diagram to scale, with the weight, its two parts, the normal force and friction, and shows each step from the weight to the answer. Gravity is the standard 9.80665 m/s² and pushes are taken parallel to the slope.
Everything runs in your browser; nothing you enter is sent anywhere.
Related tools: Lever, Pulley & Mechanical Advantage Calculator, Force Calculator, and Kinematics Calculator.
Frequently Asked Questions
What is the formula for friction?
Friction is F = μN: the coefficient of friction times the normal force pressing the surfaces together. On level ground N is the weight, so a 50 kg box with μ = 0.3 needs 0.3 × 50 × 9.80665 = 147.1 N (33.07 lbf) to keep it sliding.
How do I know if an object will slide down a slope?
It slides when tan θ is more than the static coefficient μs. A crate with μs = 0.5 on a 20° slope stays put, because tan 20° = 0.364; the slope pulls with 83.85 N on a 25 kg crate and friction can hold up to 115.2 N. It starts to slide at arctan 0.5 = 26.57°, the angle of repose.
How fast does a block speed up when it slides down a slope?
The acceleration is a = g(sin θ − μk cos θ). For a 30° slope and μk = 0.4 that is 9.80665 × (0.5 − 0.4 × 0.866) = 1.506 m/s², whatever the mass. With no friction it would be g sin 30° = 4.903 m/s².
How much force does it take to push a box up a ramp?
Pushing parallel to the ramp at a steady speed you need F = mg(sin θ + μk cos θ). For 40 kg up a 15° ramp with μk = 0.3 that is 215.2 N, compared with 101.5 N with no friction and 392.3 N to lift it straight up. Only 47.18% of your effort goes into raising the load.
What is the difference between static and kinetic friction?
Static friction is the most the surfaces can resist before anything moves; kinetic friction acts once it is sliding, and is usually lower. Typical textbook values for rubber on dry concrete are μs = 1.0 and μk = 0.8, so a car sliding on locked wheels from 100 km/h stops in about 49.18 m, against 39.34 m at the static limit.
How do I use the Friction & Inclined Plane Calculator?
Simply type your numbers and read the result, which refreshes the instant you change something. There is nothing to submit and nothing to wait for.
Is it free? Does it work without internet?
Yes to both. It is free with no sign-up, and once the page has loaded it keeps working even with no internet.
Where does my data go?
Nowhere — every calculation runs on your own device. Nothing you enter is uploaded, logged, or stored.
Common Use Cases
Loading ramps
Pushing a 40 kg crate up a 15° ramp with μk = 0.3 takes 215.2 N, about the effort of holding up 21.94 kg, against 392.3 N to lift it.
Measuring friction
Tilt a board until a block just slips. If that happens at 25°, μs = tan 25° = 0.4663, whatever the block weighs.
Skiing and sledging
A waxed ski on wet snow (μk ≈ 0.1) on a 20° slope speeds up at 2.433 m/s², reaching 12.16 m/s after 5 s if nothing else slows it.
Braking and skids
A sliding object slows at μg. With μk = 0.8 a car skidding from 100 km/h needs 49.18 m to stop.
Physics homework
A block slides down a 35° incline with an acceleration of 2 m/s². Rearranging a = g(sin θ − μk cos θ) gives μk = 0.4512.
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