Inclined plane calculator

What is the formula for acceleration on an inclined plane?

A block on a slope at angle θ is pulled down the slope by m·g·sin θ and held back by kinetic friction μk·m·g·cos θ, so a = g(sin θ − μk cos θ), and the mass cancels out. With no friction it is simply a = g sin θ (OpenStax University Physics, Volume 1, section 6.2). OpenStax Example 6.14: a snowboarder on a 13° slope with μk = 0.20 accelerates at 0.29 m/s². The frictionless skier of Example 5.12 on a 25° slope gets 4.14 m/s². Here g is the exact standard value, 9.80665 m/s² (32.174 ft/s²); OpenStax rounds to 9.8, which changes the last digit at most.

How do I find the normal force and the friction force on a ramp?

The normal force is N = m·g·cos θ, always smaller than the weight on a slope; the part of the weight pulling the block down the slope is m·g·sin θ; kinetic friction is f = μk·N. A 50 lb block on a 30° wood-on-wood slope (μs = 0.5, μk = 0.3, OpenStax Table 6.1) presses on the surface with 43.3 lbf, is pulled along the slope with 25 lbf, feels 13.0 lbf of kinetic friction and accelerates at 7.73 ft/s² (2.36 m/s²). Mass in pounds is treated as weight in lbf at standard gravity, so the force columns read the same as the pounds you type.

At what angle does a block start to slide, and what is the angle of repose?

A resting block slides when the pull along the slope beats the largest static friction, m·g·sin θ > μs·m·g·cos θ, that is when tan θ > μs. The steepest slope it can sit on is the angle of repose θ = arctan μs (OpenStax shows this in a problem to section 6.2). Sliding at a constant speed needs tan θ = μk (section 6.2): a parcel on waxed wood with μk = 0.100 slides steadily at 5.71°, and OpenStax Example 6.11 finds μk = 0.082 for a 62 kg skier who feels 45.0 N of friction on a 25° slope. With μs = 0.5 (wood on wood) the angle of repose is 26.6°.

How much force does it take to push something up a ramp?

Pushing parallel to the slope at constant speed takes F = m·g(sin θ + μk cos θ): the pull of gravity along the slope plus friction. Without friction that is m·g·sin θ, so the 2.0 kg block on the smooth 30° ramp in OpenStax section 5.6 needs 9.8 N. To get it moving from rest, replace μk with μs. To accelerate, add m·a. OpenStax checks this with a 250 kg gun safe pushed by 1150 N up a 17° ramp: 1.7 m/s² with no friction and 1.3 m/s² with 120 N of friction. Ignoring friction, a longer, shallower ramp lowers the force but not the work done against gravity.

How long does it take to slide down a slope, and how fast does it go?

Starting from rest with a constant acceleration a, the time to cover a distance d is t = √(2d/a) and the final speed is v = a·t = √(2ad). A problem in OpenStax section 6.1 describes a roller coaster car on a 30.0 m track inclined at 20.0°, with no friction: a = 3.35 m/s², so it needs 4.2 s to reach the bottom. Enter the distance, or the time, in the form and the calculator gives the other numbers. If the answer says the block does not start by itself, there is nothing to time: it needs a push first.

What does this calculator leave out?

Air drag, rolling (a wheel or a ball does not slide, and its acceleration is lower than g sin θ), and the fact that real friction coefficients vary with surface, speed and moisture: the OpenStax table gives only one or two digits, and the values for your surfaces may differ a lot. The block is treated as a point mass on a flat plane and the push is assumed to be parallel to the slope. For a wheelchair ramp built to ADA rules, use the ramp calculator.

Sources: OpenStax University Physics, Volume 1, sections 5.6 (Example 5.12, problems), 6.1 and 6.2 (Table 6.1, Examples 6.11 and 6.14, problems); the exact value of standard gravity, 9.80665 m/s², is by definition (CGPM 1901), and the pound is exactly 0.45359237 kg. Related: friction calculator, ramp calculator, slope calculator and force calculator.