LADDER
Demand grows rung by rung; supply never moves — the slip point, computed first.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
A ladder against a slick wall, a rough floor, and someone about to climb it. Every window washer performs this calculation by instinct; here it is performed by torque, with a verdict and — when the verdict is bad — an exact address for the disaster.
Forces
Four forces, one weakness. The floor pushes up and grips sideways; the frictionless wall can only push, never hold. So the entire fate of the ladder rests on one number: whether the floor’s grip can match the wall’s push.
Solve
Torque about the foot kills two unknowns at once and hands over the demand: f = g(M/2 + ms)/tanθ. Demand grows as the climber rises; supply — μ times the total weight — never moves. Set them equal and solve for s: the slip point, computed before anyone steps on a rung.
Climb
The climb, quasi-static, with the demand bar rising rung by rung against a supply bar that cannot answer back. Either the climber arrives with margin — or the bars meet at exactly the predicted s, and the floor lets go. The slide itself is shown presentationally; statics has nothing to say past the slip.
Audit
The books: torques vanish about the top as well as the foot — the balance is real, not a pivot artifact. And g cancels from the verdict entirely: the same ladder at the same angle holds or slips identically on the Moon. Grip and geometry decide; gravity only raises the stakes.