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 Incoming links: Frictionless sliding ladder

Treat the ladder as a uniform slender rigid body. While both contacts remain imposed, its centre of mass is

(l sin θ, l cos θ)

, with speed

l ∣ \dot{θ} ∣

. Its central moment of inertia is

m (2 l)^{2} /12 = m l^{2} /3

. Adding translational and rotational kinetic energy gives

K = \frac{1}{2} m l^{2} \dot{θ}^{2} + \frac{1}{2} \frac{m l ^{2}}{3} \dot{θ}^{2} = \frac{2}{3} m l^{2} \dot{θ}^{2} .

L = \frac{2}{3} m l^{2} \dot{θ}^{2} - m g l cos θ, p_{θ} = \frac{4}{3} m l^{2} \dot{θ}, \ddot{θ} = \frac{3 g}{4 l} sin θ .

The positive sign is appropriate: gravity increases the angle from the vertical. These are the constrained equations of a frictionless sliding ladder. Real wall contact is unilateral: a wall cannot pull the ladder. Once the computed normal force would become negative, the two-contact model ceases to apply; the question explicitly assumes that contact persists.