Physics of Fitness Fridays - The Pull-Up

Today let’s look at the humble pull-up. Specifically, how much energy is exerted in a pull-up? Can physics help us make a pull-up easier?

Today I’m going to define a pull-up by your change in potential energy. Potential energy is related to the height of your center of mass (COM) above some reference point—here, I use the ground. Potential energy (PE) is defined as: PE = mgh. Here, m is mass, g is the gravitational constant, and h is your height above the ground.

When you perform a pull-up, your height above the ground increases from h to h’. So your potential energy increases by the change in height, or change(PE) = mg(h’-h). Your body must exert at least this much energy to pull yourself up. (In reality, your body is rather inefficient, so you will exert a LOT more energy than this to pull-up, but let’s keep things simple.)

Looking at the equation [change(PE)] = mg(h’-h), one thing is pretty obvious: the higher your mass, the more energy is exerted. So I wouldn’t recommend doing pull-ups after a large meal, or, say, wearing ankle weights, unless you really want a challenge.

We know that g is a constant, but what about (h’-h)? How much distance does your body really travel? If you look at the beginning and end positions, notice that your forearms stay relatively vertical, while your upper arm kind of “swings” around your elbow joint. That means that the total distance traveled is about twice the length* of your upper arm (L). So if you have long arms, you’ll have to work harder than someone who is the same length but has shorter arms. Not fair! You say. I say, hey, it will make you that much stronger!

*Yes, it’s a bit of an approximation. We could always measure the distance traveled by the top of your head, etc, but it does depend on arm length and I wanted to make that point here.

So what happens if you kick your heels back to do your pull-up? Well, it will shift your COM up slightly, but you’ll still need to move that COM up a distance of ~2L, so that doesn’t really help much.

Unless…

…Let’s compare two pull-ups: one with your feet kciked back behind you, one with your legs out in front of you. The total distance traveled in each pull-up is the same. But note how your COM doesn’t go quite as far up with your legs out as it does with your feet kicked back. That means that if we start with our feet behind us but end with our feet in front, the distance our COM travels decreases from what it would be if we kept our feet in one place. So that means we can make the pull-up a little easier by kicking our feet out. No, you don’t have to do this with any momentum, just let your feet drop as you get to the top. Try it! Do you notice any difference?

So by manipulating our center of mass, we can “cheat” on our pull-ups and make them a little easier. But why would I encourage you to cheat? Do your full pull-up and be proud!