Setup: A monkey of mass m climbs a rope hanging from the ceiling.

Case 1: Monkey climbs with constant velocity v: T = mg (no net force needed)

Case 2: Monkey accelerates upward at a: T - mg = ma => T = m(g + a)

Case 3: Monkey decelerates while climbing (a downward): T = m(g - a)

Case 4: Rope breaks at tension $T_{max}$: Maximum upward acceleration: $a_{max}$ = $\frac{T_max}{m}$ - g

Two monkeys on a rope over a pulley: Treat like Atwood machine if masses differ. If the lighter monkey climbs, analyze using relative acceleration.