If the radii of two planets are 𝑅 sub one equals 1,552 kilometers and 𝑅 sub two equals 6,208 kilometers and the ratio between their gravitational accelerations 𝑔 sub one to 𝑔 sub two is equal to one to two, find the ratio between their masses 𝑚 sub one to 𝑚 sub two.
Well, to help us solve this problem, what we have is a formula for our acceleration due to gravity. And that is that 𝑔 is equal to capital 𝐺𝑚 over 𝑟 squared, where 𝑔 is the acceleration due to gravity, capital 𝐺 is the universal gravitational constant, 𝑚 is the mass, and 𝑟 is the radius. And this is actually derived by combining Newton’s law of universal gravitation with Newton’s second law.
Okay, great, so we have this. But how’s it gonna help us? Well, what we can do is actually work out the acceleration due to gravity of both of our planets. So therefore, we can say that the gravitational acceleration of the first planet is gonna be equal to capital 𝐺 multiplied by 𝑚 sub one over 𝑟 sub one squared, which is gonna give us 𝑔 sub one is equal to capital 𝐺 multiplied by 𝑚 sub one over 1,552,000 squared. And the reason we have this is because if we look at the radius of the first planet, it’s 1,552 kilometers. However, whenever we’re looking to work acceleration due to gravity, the units we want our radius to be in are in fact meters. So what we’ve done is we multiplied this by 1,000 to give us 1,552,000 meters.
Now, if we repeat the same for the second planet, we’ll have 𝑔 sub two is equal to capital 𝐺 multiplied by 𝑚 sub two over 𝑟 sub two all squared. So then what we’re gonna get is 𝑔 sub two is equal to capital 𝐺 multiplied by 𝑚 sub two over 6,208,000 squared. And once again, what we’ve done is converted our kilometers to meters, so we’ve done 6,208 kilometers multiplied by 1,000, which gives us 6,208,000 meters.
Well, next, what we’re going to do is take a look at another bit of information we’ve been given in the question. And that is that the ratio between the gravitational accelerations is one to two. So therefore, what this means is the acceleration due to gravity on the second planet, so 𝑔 sub two, is twice that of the acceleration due to gravity on the first planet, 𝑔 sub one. So therefore, what we can do is form an equation. And we can do that by multiplying our expression for 𝑔 sub one by two. Because what we get is that two multiplied by capital 𝐺𝑚 sub one over 1,552,000 squared is gonna be equal to, then our expression for 𝑔 sub two, which is capital 𝐺𝑚 sub two over 6,208,000 squared.
Well, what we can do is divide through by capital 𝐺, our universal gravitational constant, ’cause this is unchanged on each side of our equation. And then we can change the subject to make 𝑚 sub two the subject. So we have 𝑚 sub two in terms of 𝑚 sub one. And to do that, what we’ve done is multiplied through by 6,208,000 squared. So what we have is two multiplied by 6,208,000 squared over 1,552,000 squared 𝑚 sub one is equal to 𝑚 sub two. So calculating this, we get 32𝑚 sub one is equal to 𝑚 sub two. So 𝑚 sub two is 32 times bigger than 𝑚 sub one. So therefore, what we can say is that the ratio between the masses of our two planets, so 𝑚 sub one to 𝑚 sub two, is going to be one to 32.