Tip vs Slide: Block on an Incline (2D)

Block Geometry

Base width b 1.00 m

Height h 1.40 m

Static friction μ 0.45

Ramp

Incline angle θ 10.0°

Live State

Resting — increase θ to test.

Critical slide θ_s

24.2°

Critical tip θ_t

35.5°

Current θ

10.0°

Predicted mode

Tips first

How it works

A rigid block of base width b and height h rests on a ramp tilted to angle θ. As θ rises from zero, one of two failure modes wins depending only on the geometry and the static friction coefficient μ:

Sliding starts when:  tan θ > μ           →  θ_s = atan(μ)
Tipping starts when:  tan θ > b / h       →  θ_t = atan(b / h)

Whichever critical angle is smaller happens first as θ increases — this simulator computes both live and drives the matching real dynamics:

Sliding — once tan θ > μ, the block accelerates down the slope with a = g(sin θ − μ cos θ), no rotation.
Tipping — once tan θ > b/h, the block rotates about its downhill bottom edge. The torque about that pivot is τ = −g·r_x(φ) per unit mass, where r(φ) is the pivot→center-of-mass vector after rotating by φ, and the moment of inertia per unit mass about the pivot is I/m = (b²+h²)/12 + d², with d the pivot-to-CG distance (parallel-axis theorem). Angular acceleration α = τ/(I/m) is integrated frame by frame — verified independently here by checking that (1/2)(I/m)ω² + g·y_cg stays constant during the fall, exactly as energy conservation for a frictionless pivot requires.

This is the flat 2D side view of the same rigid-body model as the 3D version — same equations, same numbers, just without the orbiting camera. Drag to pan, scroll or pinch to zoom.

Block (center of mass marked) Ramp surface

drag to pan · scroll to zoom

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Tip vs Slide: Block on an Incline — the 2D Companion