Why did an older version sometimes cite arccos(√(2/3))?

That value appears in some textbook special cases but does not generally equal the separation angle for arbitrary release θ₀; the sin θ = (2/3) sin θ₀ condition matches the standard energy-integral derivation for the smooth smooth case.

What if the floor or wall has friction?

The ODE and separation criterion change; this simulator keeps both contacts frictionless.

Sliding ladder

A uniform ladder leans on a smooth wall and smooth floor with no friction. With θ measured from the horizontal, Lagrangian mechanics gives θ̈ = −(3g/2L) cos θ for the angle while both contacts persist. Released from rest, the horizontal reaction on the wall vanishes when sin θ = (2/3) sin θ₀, where θ₀ is the release angle to the horizontal — not a universal constant independent of θ₀. After separation the real ladder would rotate differently; this page freezes the pose at the critical angle as a teaching boundary.

Who it's for: Intermediate mechanics and AP physics students working constraint forces.

Key terms

Sliding ladder
Constraint force
Lagrangian
Wall reaction
Critical angle

How it works

Smooth wall and floor: the ladder’s angle θ (from horizontal) obeys θ̈ = −(3g/2L) cos θ with I = mL²/12. Released from rest, the wall normal vanishes when sin θ = (2/3) sin θ₀ — after that this demo freezes the pose.

Frequently asked questions

Why did an older version sometimes cite arccos(√(2/3))?: That value appears in some textbook special cases but does not generally equal the separation angle for arbitrary release θ₀; the sin θ = (2/3) sin θ₀ condition matches the standard energy-integral derivation for the smooth smooth case.
What if the floor or wall has friction?: The ODE and separation criterion change; this simulator keeps both contacts frictionless.