Tip vs Slide: Block on an Incline

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 — this is the same "toppling block" problem used to model chimneys, cranes and furniture tip-overs.

Try it: a short, wide block (large b/h) resists tipping but still slides on a low-friction ramp; a tall, narrow block (small b/h) topples at a shallow angle even with plenty of grip.

Block (center of mass marked) Ramp surface

⚠ Couldn't load the 3D engineThree.js failed to load from the CDN. Check your connection and reload.