#pulleys

I don’t remember elementary school, but I do remember being sick or something the day my high school physics class started learning about tension, so I never really understood pulleys very well until now (hopefully).

A pulley is a simple machine that can change the direction of a force. If you don’t want to pull up to lift a box, you can use a pulley to change the direction and pull down instead, using the same amount of force. But that doesn’t really work to lift 500kg of tree, the way Senku does here in Ch 16.

As you can see in the first image, Senku uses 3 pulleys in order to lift a tree weighing about 5000 Newtons (500kg of tree mass * 10m/s/s of acceleration due to gravity), using his own weight of about 600N (from 60kg).

Senku’s pulley system uses 3 moveable pulleys, plus one fixed “pulley” from the loop around the tree that he pulled down on (right above the little sack in his diagram). From left to right, each moveable pulley cuts the force that the next pulley has to carry in half. Why is that?

Here’s a rough diagram of the forces at play (making various assumptions like ignoring the weights of the pulleys themselves and ignoring that Senku’s weight here isn’t enough to keep the system at rest, plus ignoring friction that Senku reduced by lathering the rope with soap). Hopefully you can read the numbers, but essentially the two sides of the rope holding one pulley up are under equal amounts of tension, the sum of which being equal to the tension pulling the pulley down. This means that the force on one side of the rope pulling up is equal to half the force pulling down.

For example, taking the lowest pulley that is attached to the tree:

The weight of the tree is exerting 5000N of force downward on the pulley. To keep the tree and pulley suspended at rest, the two sides pulling upward must exert a force so that the net force is 0, meaning that the total sum of the upward forces (opposite in direction to the weight of the tree pulling down) must equal 5000N. Therefore, each side of the rope is pulling up with 2500N of force.

(This might be a little different in Senku’s actual execution since it looks like some pulleys are being pulled at an angle, but it probably balances out enough that I will ignore this detail as well. Yay physics!)

And now the right side of the rope is pulling down on the middle pulley with 2500N of force, as opposed to the original 5000N. The same thing happens again, where the two sides of the rope holding up this middle pulley are pulling up with 1250N of force each, cutting that 2500N in half. And then the last pulley cuts that in half again, to 625N of force on each side. Then Senku just needs to use 625N of force to pull the rope and lift the tree! Since he was only estimating the weight of the tree, the weight could have been less, so that his 60kg body mass would be enough to move everything.

To summarize, since each pulley cuts the force in half, Senku needs the number of pulleys that will cut 5000N in half enough times to reach 600N. That’s the 2^3 that he mentions: 5000N / (2^3) = 625N which is close enough to 600N (or going the other way around, each moveable pulley from right to left doubles the force of his weight: 600N * 2^3 is close enough to 5000N), meaning that he needs to use 3 pulleys. And thus Kohaku is rescued!

But why does Senku say that the tree is 500kg after first saying the whole thing weighs a metric ton (1000kg)? Answer: It’s a lever! I’ll cover that in the next post, so stay tuned!

If you have any questions on or corrections to this post, or have suggestions for me to research and explain, send me an ask!

Sources