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?

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.

When one of the monkeys accelerates up the rope, the other monkey will also accelerate, but in the opposite direction. This is governed by Newton's third law of motion, which states that for every action, there is an equal and opposite reaction.

To understand why this happens, let's break down the scenario. The monkeys are hanging from the ends of a rope, which is passing over a weightless, frictionless pulley. Since the monkeys are of equal weight, the forces acting on them are also equal. The force of gravity is pulling each monkey downwards with the same magnitude.

When one monkey starts accelerating up the rope, it exerts an upward force on its side of the rope. According to Newton's third law, the rope exerts an equal and opposite force on the other monkey. This force causes the other monkey to accelerate in the opposite direction, downwards.

In other words, the monkey going up pulls the rope, which pulls the other monkey down. Both monkeys experience an acceleration, but in opposite directions.