Workers who wash windows or paint the outside of buildings use an interesting contraption known as a painter's lift. This consists of a harness that the worker wears suspended by a rope. The rope runs through a pulley mounted on the roof of the building and back down to hang beside the worker. The worker simply pulls down on the hanging rope to raise herself up, and releases it to lower herself down (tieing the hanging rope to her harness keeps her at a constant height). What's neat is that the configuration also makes it easier for the worker to move up and down than if she was just hanging by a single rope. Let F1 be the force the worker exerts on the hanging rope in the painter's lift configuration to move upward at a constant speed. Let F2 be the force the worker would need to exert on a single rope to move upward at a constant speed. What is F1/F2? You can assume the rope itself doesn't have any significant mass.

Question

Workers who wash windows or paint the outside of buildings use an interesting contraption known as a painter's lift. This consists of a harness that the worker wears suspended by a rope. The rope runs through a pulley mounted on the roof of the building and back down to hang beside the worker. The worker simply pulls down on the hanging rope to raise herself up, and releases it to lower herself down (tieing the hanging rope to her harness keeps her at a constant height). What's neat is that the configuration also makes it easier for the worker to move up and down than if she was just hanging by a single rope. Let F1 be the force the worker exerts on the hanging rope in the painter's lift configuration to move upward at a constant speed. Let F2 be the force the worker would need to exert on a single rope to move upward at a constant speed. What is F1/F2? You can assume the rope itself doesn't have any significant mass.

Answer 1

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Answer 2

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-Calvin Lin
Brilliant Maths Challenge Master

Answer 3

To calculate the ratio of F1 to F2, we need to understand the forces involved in both the painter's lift configuration and the single rope scenario.

Let's start with the single rope scenario. In this case, if the worker wants to move upward at a constant speed, she needs to exert a force equal to her body weight. This is because forces must be balanced for an object to remain at a constant speed. So F2 is equal to the worker's body weight.

Now, let's consider the painter's lift configuration. The worker exerts a force, F1, on the hanging rope to move upward at a constant speed. However, since the rope goes through a pulley, the force is redirected to support the worker's weight vertically. By pulling down on the hanging rope, she can lift herself up. Therefore, the force she needs to exert is equal to her body weight.

Since both F1 and F2 are equal to the worker's body weight, F1/F2 is equal to 1. In other words, the ratio of the forces in the painter's lift configuration to the single rope scenario is 1.

So, F1/F2 = 1.