Two equal weight monkeys each hang from the ends of a rope passing over a weightless, frictionless pulley. If one accelerates up the rope, what happens to the other?

Answer 1

THe monkey is putting a tension on the rope of mg+ma. THe opposite monkey is putting a tension of -mg. The resultant force on the rope is ma.

F=ma
ma= (2m)A
A= 1/2 a

Therefore, both monkeys accelerate upwards at the same rate, which is half the rate as the acceleration (relative to the rope) as the climbing monkey. In short, the other monkey goes up.