Example 1
α = 30°, μ = 0 → a = g/2.
Incline with friction calculator
Type angle α [°] and μ. The calculator computes a = g(sin α − μ cos α): 30° and 0.2 is about 3.21 m/s². When μ is at least tan α, a is not positive and you do not start from rest.
Result
Angle α (degrees), Friction coefficient μ, Ramp length L (optional) and g (default 9.81). The result shows up here.
How it works
With kinetic friction a = g(sin α - μ cos α). At 30° and μ = 0.2: a = 9.81 × (0.5 - 0.2 × 0.866) ≈ 3.21 m/s², less than 4.9 m/s² with no friction. At μ = 0 and 30° you return to g/2. At 45°, μ = 0.2 and L = 12 m, a ≈ 5.55 m/s², t ≈ 2.08 s, vₖ ≈ 11.5 m/s. At 25°, μ = 0.15 and L = 20 m, a ≈ 2.81 m/s², t ≈ 3.77 s.
At 10° and μ = 0.3: tan 10° ≈ 0.176, so μ is larger and a ≤ 0. From rest the board holds. At 15° and μ = 0.7, a is often ≤ 0 as well. At 20° and μ = 0.35, a is near zero, about 0.13 m/s². At 30°, μ = 0.1 and g = 1.62 m/s², a ≈ 0.67 m/s². Mass does not enter: in this model a does not depend on m.
Fields: α in degrees, μ with no unit, L optional in meters, empty g = 9.81. Blank L leaves a alone. Typed μ = 0 returns to the smooth incline. α and μ both need a number before 3.21 m/s² can appear. A comma and a period are the same μ: 0,2 and 0.2. The angle is in degrees, not radians.
T = μ N alone is on friction force: normal and μ, no slide time. The smooth board is on the no-friction page. The coefficient from T and N is μ = T/N.
t and vₖ with L assume a start from rest and a > 0. When a ≤ 0, the calculator does not build a slide time. If the block is already moving up the board, that is another sign and another calculator.
Type 30 and 0.2, click Calculate, and match a ≈ 3.21 m/s². Then 10 and 0.3 and see that you do not start from rest. The header symbol does not sprinkle sand on the board.
No μ: smooth incline. T alone: friction force.
How to use
- Type angle α in degrees, for example 30.
- Type μ, for example 0.2. That is a number with no unit.
- L is optional, for example 12 m, if you want t and vₖ when a > 0.
- Click Calculate. 30° and 0.2 give a ≈ 3.21 m/s². 10° and 0.3 give a ≤ 0.
- With no μ, open the smooth incline. T alone is on friction force.
Formula
a = g(sinα − μ cosα)
If a ≤ 0: no slide from rest.
With L and a > 0: t = √(2L/a), vk = √(2a L)
a, μ, and α on a frictional slide
Inclined plane with friction: a = g(sin α − μ cos α). 30° and μ = 0.2 is about 3.21 m/s². When μ is at least tan α, a rest start does not go.
- a
- Acceleration along the ramp [m/s²]. 9.81 × (0.5 − 0.2 × 0.866) ≈ 3.21.
- μ
- Friction coefficient, no unit. 0.2. μ = 0 returns to g sin α.
- α
- Ramp angle [°], not radians. 30 or 45. The field wants degrees.
- L
- Optional length [m]. 12 m at 45° and μ = 0.2 also gives t and vₖ.
Real-life examples
Example 2
α = 30°, μ = 0.2: a ≈ 3.21 m/s².
Example 3
α = 10°, μ = 0.3 → a ≤ 0.
Example 4
α = 45°, μ = 0.2, L = 12 m.
Example 5
α = 20°, μ = 0.35.
Example 6
α = 15°, μ = 0.7: often a ≤ 0.
Example 7
α = 25°, μ = 0.15, L = 20 m.
Example 8
α = 30°, μ = 0.1, g = 1.62 m/s².
Example 9
α = 35°, μ = 0.25, L = 50 m.
Ways to use this calculator
- You compare 30° with no friction and with μ = 0.2.
- You check whether 10° and μ = 0.3 even starts to slide.
Frequently asked questions
How much a at 30° and μ = 0.2?
Acceleration is about 3.21 m/s². You compute 9.81 × (0.5 − 0.2 × 0.866). With no friction at 30° it would be about 4.9 m/s².
When does the block stay put?
When μ is at least tan α, then a is not positive. Example: 10° and μ = 0.3, because tan 10° is about 0.18.
Which units do I type?
Angle α [°], coefficient μ with no unit, optional L [m] and g [m/s²]. Blank g is 9.81. Result a [m/s²].
Do I have to type L?
No. Without L you only get a. With L you also get t [s] and vₖ [m/s], if a is positive and you start from rest.
How much at 45°, μ = 0.2 and L = 12 m?
You get a about 5.55 m/s², t about 2.08 s and vₖ about 11.5 m/s. Start from rest, g = 9.81.
Is the angle in radians?
No. The field wants degrees. Typed 30 is thirty degrees, not 30 rad.
How is this different from friction force?
Here you get slide a from α and μ. There T = μ N, with no time and no vₖ. Different fields, the same μ.
Does mass enter a?
In this model a does not depend on m. A heavier block slides at the same a at the same α and μ.
What does μ = 0 do?
It returns to the smooth incline: a = g sin α. At 30° that is about 4.9 m/s², the pair of the no-friction page.
Does a comma in 0.2 work?
Yes. 0.2 and 0,2 are the same μ. A comma and a period mean the same value.
Knowledge sources
The formula is the school one. Units follow SI; NIST SP 330 and BIPM define the measures, not your result.
Page updated in 2026.