Video Transcript
An object orbits a planet at a distance 𝑟 from the planet’s center of mass. The planet has a mass 𝑀. If the object is moved to an orbit three 𝑟 from the planet’s center of mass, by what factor does the local acceleration due to gravity at its orbital radius change?
So let’s start with a diagram. Here’s an object in orbit around the planet. And we know that the distance between the object and the planet’s center of mass is some distance 𝑟. We then move this object to a new orbit, at which point its distance from the planet’s center of mass is three 𝑟. We need to calculate the factor by which the local acceleration due to gravity changes. So let’s recall the equation for the local acceleration due to gravity. That is, 𝑎 equals 𝐺𝑀 over 𝑟 squared, where 𝑎 is the acceleration due to gravity, 𝐺 is the universal gravitational constant, 𝑀 is the mass of the planet, and 𝑟 is the orbital radius.
Now, in this situation, the mass of the planet is not changing. The universal gravitational constant 𝐺 never changes. So the only quantity that’s changing is the orbital radius 𝑟. And 𝑟 is in the denominator of this fraction, which means that if 𝑟 increases, 𝑎 will decrease. So we know that the acceleration due to gravity once the orbital radius is increased will be lower. And we just have to work out by how much.
So let’s start with the acceleration due to gravity at the original orbital radius, which we’ll call 𝑎 sub one. And this is equal to 𝐺𝑀 over 𝑟 squared. The local acceleration due to gravity after the object has moved, which we’ll call 𝑎 sub two, is equal to 𝐺𝑀 over three 𝑟 squared. Now, here it’s useful to recall that any time we have two quantities multiplied together, say 𝑥 times 𝑦, all squared, this is equivalent to 𝑥 squared times 𝑦 squared. So here we can say that 𝑎 sub two is equal to 𝐺𝑀 divided by three squared times 𝑟 squared. And three squared is equal to nine. So we have 𝑎 sub two is equal to 𝐺𝑀 over nine 𝑟 squared.
Now, it’s useful to note here that 𝐺𝑀 over 𝑟 squared is the same as 𝑎 sub one. So we can separate the one over nine and write that 𝑎 sub two is equal to one divided by nine times 𝐺𝑀 over 𝑟 squared. And 𝐺𝑀 over 𝑟 squared is equal to 𝑎 one. So we found that the local acceleration due to gravity at this point after the object has been moved is one-ninth of the value that it had before the object was moved. So the factor by which the local acceleration due to gravity at the orbital radius of the object has changed is one-ninth.
