Slope diagram
Weight (mg)
Normal (N)
Friction (f)
Net force
| Gravity along slope (mg·sinθ) | |
| Gravity into slope (mg·cosθ) | |
| Normal force (N = mg·cosθ) | |
| Max static friction (μₛN) | |
| Friction force applied | |
| Net force along slope | |
| State | - |
Decomposing gravity on a slope
F∥ = mg·sin θ F⊥ = mg·cos θ
On an incline, weight (mg) is split into two perpendicular components: one pulling the object down the slope (F∥), and one pressing it into the surface (F⊥). As θ increases, more weight acts along the slope and less presses into it.
Normal force
N = mg·cos θ
The surface pushes back perpendicular to itself with force N, balancing the perpendicular component of weight. N shrinks as the incline steepens — at 90°, N drops to zero.
Static vs. kinetic friction
f₍s,max₎ = μₛN f₍k₎ = μₖN
While stationary, friction opposes the tendency to slide and can take any value up to μₛN. Once motion begins, friction drops to the (usually smaller) kinetic value μₖN.
Net force and acceleration
F₍net₎ = mg·sin θ − f a = F₍net₎ / m
If mg·sinθ exceeds the maximum static friction, the object accelerates down the slope using kinetic friction. If not, the object stays at rest and friction exactly cancels the downslope pull.
Critical angle
θ₍critical₎ = arctan(μₛ)
This is the steepest angle at which an object can still rest without sliding. Beyond it, gravity along the slope always wins over the maximum friction available.
Mass cancels out (for sliding threshold)Whether an object slides depends only on θ and μₛ, not mass — heavier objects have more gravity pulling them down, but also more normal force creating friction, and the two scale together.
Steeper isn't always faster… at firstNear 0°, nothing moves. Past the critical angle, acceleration increases smoothly with θ, approaching g as the incline approaches 90°.
Static friction has a max, not a fixed valueBelow the critical angle, friction force equals mg·sinθ exactly (just enough to hold still) — not μₛN, which is only the ceiling.
Kinetic friction is usually weakerμₖ < μₛ in most materials, which is why once something starts sliding it tends to keep accelerating rather than re-settling.
01
Set the incline angleDrag the θ slider from 0° (flat) to 80° (steep). Watch the slope in the diagram rotate and the weight vector's components shift.
02
Adjust the massChange the mass slider to see how gravity, normal force, and friction all scale up together — notice the State (held / sliding) doesn't change just because mass changes.
03
Pick a surface or set coefficients manuallyUse a preset button (ice, wood, rubber, rock) to instantly load realistic μₛ and μₖ values, or drag the sliders directly.
04
Watch the state changeAs you raise θ past the critical angle (arctan μₛ), the State row flips from "Held by friction" to "Sliding," and the net force arrow appears.
05
Copy individual valuesHover over any stat card or table row — a copy icon appears next to the number. Click it to copy just that value to your clipboard.
06
Download an HTML reportClick 🖨️ HTML Report to save a self-contained printable report with the current diagram snapshot, all inputs, and all outputs.
Experiments to try
Find the critical angle: With μₛ = 0.40, slowly increase θ from 0°. The object starts sliding right around 21.8° (arctan 0.40).
Mass independence: Set θ to 30°, μₛ to 0.40. Change mass from 5 kg to 50 kg. The acceleration stays the same — only the force magnitudes scale.
Static "stickiness": Set θ just below the critical angle. Friction force shown equals mg·sinθ exactly, not μₛN — it only borrows as much as needed to stay still.
Ice vs. rock: Keep θ fixed at 25° and switch between Ice and Rough rock presets.
