Problem — Monkey and weight on a pulley

Problem

A monkey climbs a rope running over a fixed pulley, at the other end of which hangs a weight exactly equal to the monkey’s mass. When the monkey climbs $2\;\mathrm{m}$, by how much does the weight rise?

Solution

Symmetry of the system. The rope is inextensible and the pulley frictionless, so the tension $T$ is the same throughout the rope and acts equally on the monkey and on the counterweight. The masses are equal ($m_\text{scimmia} = m_\text{peso}$), so the two ends of the rope are subject to identical forces: the system is perfectly balanced.

Reasoning about acceleration. For the monkey: $T - m g = m\,a_\text{s}$. For the weight: $T - m g = m\,a_\text{p}$. The two equations are identical, so at every instant $a_\text{s} = a_\text{p}$: the monkey and the weight always have the same acceleration and the same vertical speed. Whatever effort the monkey makes to climb, it pulls the rope and raises the counterweight in exactly the same way.

Conclusion. If the monkey climbs $2\;\mathrm{m}$ relative to the ground, the weight also rises by $2\;\mathrm{m}$: their distance from the pulley decreases by the same amount on both sides. (Consistent with conservation of energy: both gain the same potential energy, supplied by the monkey’s muscular work.) $\ev{\text{the weight rises by } 2\;\mathrm{m}}$

Collegamenti

Argomenti: Lavoro ed energia Concetti: Conservazione dell’energia meccanica Competenze: Analisi di casi limite e fantafisica Oggetti: Puleggia