Inclined Plane
Solve inclined-plane acceleration, parallel force, normal force, or the minimum friction coefficient. Includes a = g sinθ, friction cases, and clear step-by-step working.
Key formula
a = g sin(theta), N = mg cos(theta), mu_min = tan(theta)
How do you solve inclined plane problems?
Resolve weight: F∥ = mg sinθ down the plane and N = mg cosθ into the plane. Frictionless acceleration is a = g sinθ. With kinetic friction sliding down, a = g(sinθ − μ cosθ). To stay at rest, you need μ_s ≥ tanθ.
About this tool
An inclined plane splits weight into a down-ramp component and a into-the-surface component. This page keeps a = g sinθ, N = mg cosθ, and μ_min = tanθ visible so ramp homework stays consistent on units and free-body signs.
How to use this calculator
- Choose the variable you want to solve for and confirm which values are already known.
- Enter inputs in consistent units before calculating so the formula is applied correctly.
- Check whether the result is physically reasonable by comparing it with the size of the inputs.
- Use the step-by-step breakdown to see how the equation was rearranged and evaluated.
Worked examples
Example 1: frictionless acceleration at 30°
With g = 9.8 m/s² and θ = 30°, a = 9.8 sin(30°) = 9.8 × 0.5 = 4.9 m/s² down the plane. Mass cancels for acceleration on a frictionless ramp.
Example 2: parallel force for 2.0 kg at 30°
F∥ = mg sinθ = (2.0)(9.8)(0.5) = 9.8 N down the plane. Check: a = 4.9 m/s² and F = ma = 9.8 N.
Example 3: minimum μ to hold at 30°
μ_s,min = tan(30°) ≈ 0.577. Any smaller static coefficient and the block starts to slide.
Frequently asked questions
What is the acceleration on a frictionless incline?
a = g sinθ down the plane, where θ is the angle above the horizontal. Mass does not appear in the acceleration formula when friction is zero.
How do I find normal force on an incline?
For a block resting or sliding on the ramp with no other forces perpendicular to the surface, N = mg cosθ. On a horizontal surface θ = 0° and N = mg.
What friction coefficient keeps a block from sliding?
The minimum static coefficient is μ_s,min = tanθ. That comes from setting mg sinθ = μ_s N = μ_s mg cosθ.
How does kinetic friction change acceleration?
Sliding down: a = g(sinθ − μ_k cosθ). If μ_k cosθ > sinθ, the net acceleration is up the plane (or the block decelerates while moving down).
Should the angle be in degrees or radians?
Enter degrees in this calculator; the working converts to radians inside the sine and cosine. Never mix calculator degree/radian modes when checking by hand.