# The Physics Philes, lesson 72: You’ve Got Potential

Over the past few weeks we’ve been learning about this super-awesome thing called gravity. In a way, we already know a lot about it because we know about weight and we know about gravitational potential energy. Last week we went a little deeper into the concept of weight and explored how gravity relates. This week we’ll dive into gravitational potential energy.

But Mindy! you might say. We’ve already talked about gravitational potential energy. That’s true, but what we talked about was actually a special case of a much more general formula. The equation we discussed, in which gravitational potential energy is equal to the product of the mass of the body, acceleration due to gravity, and the height above the ground, is actually just a special case of a general formula.

While on Earth, the gravitational potential energy formula we learned way back in lesson 15 works great. But Earth isn’t the only place in the universe! In fact, this equation cannot be valid for every arbitrary height because the acceleration due to gravity decreases the further you get from Earth. Luckily, we can use the law of universal gravitation to find gravitational potential energy anywhere.

If we use the law of universal gravitation to find gravitational potential energy on Earth, the equation is:

Remember that G stands for the force of gravity, m is the mass of a body, M is the mass of the Earth, and r is the radius of the Earth.

I’d love to tell you how very smart people got to this equation, but it has to do with integral and antiderivatives and I’m simply not equipped to explain it to you. I might be able to understand what is going on in a month or so, but right now I just have to trust that it’s true. If I discover that it’s wrong I’ll be sure to let you know after I claim my Nobel.

What I do know, however, is that as the gravitational potential energy gets closer and closer to zero, the distance r gets bigger and bigger and bigger. Where we measure gravitational potential energy as zero is largely arbitrary because it’s the difference in potential energy that is significant. When we talk about astronomical distances it’s just easier for U to equal zero at an infinite distance.

Remember, like, five seconds ago when I wrote that the gravitational potential energy equation we learned about in lesson 15 is just a special case of the general equation? You wanna see how that happens? You do? What luck! I can show you.

First, let’s look at this equation:

The first part of this equation is the equation for some height above the Earth’s surface, which is denoted by h. We’re subtracting that value from the value of the gravitational potential energy at the Earth’s surface.

Alrighty. Let’s rewrite that equation so we can really see what we’re dealing with.

Do you see what I did to the first expression? I just separated it out. Now let’s put it all back together.

Simplify!

Mmm…there’s something familiar about this. Let’s do a little more rearranging.

Yes! There it is! Do you remember what that middle term is? It looks an awful lot like the value for g that we found a couple of weeks ago. Suddenly, this equation looks a lot simpler.

Whoa! Gravitational potential energy equals mass times the acceleration due to gravity times the height. Pretty cool.

That’s all for today. Next week we’ll get into some cool stuff about how gravity effects the motion of satellites. You’re not going to want to miss it.

Featured image credit: Ryan Dickey via Flickr