A ladder against a wall. How far up can he climb?

The ladder holds on its own. It holds with a man on the bottom rung. Yet somewhere up the ladder it will slip. Where, and why there?

scroll ↓

The smooth wall pushes straight out: $S$. The rough ground pushes up with $R$ and grips with friction $F$, towards the wall. Resolving:

As he climbs, his weight turns the ladder about its foot more and more. Only the wall's push can balance that turning, so $S$ grows, and the foot needs more friction to match it. $R$ stays at $100g$, so friction can be at most $0.4 \times 100g = 40g$.

At 2.71 m up the ladder, the foot needs all the friction the ground can give, $40g$. The gauge is full: the ladder is on the point of slipping.

Any higher and the foot needs more than $\mu R$: it slides out and the ladder comes down the wall. A rougher floor, or a steeper ladder, would let him climb further.

Resolve, take moments, respect the limit.

Particles pushed up rough slopes, rods held on rough ground, ladders on their own, connected particles over pulleys, strings that go slack, and twelve graded exam questions with full worked solutions are waiting.

Read the full chapter › Jump to the questions ›