Video Transcript
Given that the force of gravity acting between the Sun and a planet is 4.37 times 10 to the power of 21 newtons, where the mass of that planet is 2.9 times 10 to the power of 24 kilograms, and that of the Sun is 1.9 times 10 to the power of 30 kilograms, find the distance between them. Take the universal gravitational constant 𝐺 is equal to 6.67 times 10 to the power of negative 11 newton meters squared per kilogram squared.
So in order to solve this problem, what we’re going to be doing is using Newton’s law of universal gravitation. And what Newton’s law of universal gravitation does is it allows us to actually look at the attraction between two bodies’ mass. And in fact, it is universal because it doesn’t matter on the size or mass of those two bodies. So what we’ve got is that 𝐹 sub 𝐺 is equal to 𝐺 multiplied by 𝑚 sub one 𝑚 sub two over 𝑟 squared. This is where 𝐹 sub 𝐺 is our gravitational force, capital 𝐺 is the universal gravitational constant, 𝑚 sub one and 𝑚 sub two are the masses of the bodies, then 𝑟 is the separation of the bodies or, in fact, the distance between their centers.
So the first thing we can do is write down what information we’ve been given. So we know that 𝐹 sub 𝐺, our gravitational force, is equal to 4.37 times 10 to the power of 21. And the units of this would be newtons. And then what we have are the masses of the planet and the Sun. And these are 2.9 times 10 to the power of 24 kilograms and 1.9 times 10 to the power of 30 kilograms, respectively. And these are what we’re gonna use, so we’re gonna have 𝑚 sub 𝑝 and 𝑚 sub 𝑠 instead of our 𝑚 sub one 𝑚 sub two. Then, we have our universal gravitational constant 𝐺, which is equal to 6.67 times 10 to the power of negative 11 newton meters squared per kilogram squared. Then finally, we have 𝑟, which is our distance between them, which is what we’re trying to find, so we don’t know this. Okay, great!
So now what we need to do is substitute them into the formula to find 𝑟. But before we do that, we can quickly check our units to make sure that we’re working in the correct ones for this problem. We’ve got newtons and kilograms, so that’s what we want. So great. So now let’s move on and substitute our values in. So when we do that, we get 4.37 times 10 to the power of 21 equals 6.67 times 10 to the power of negative 11 multiplied by 2.9 times 10 to the power of 24 multiplied by 1.9 times 10 to the power of 30 all over 𝑟 squared.
So therefore, we can say that 𝑟 squared is equal to 6.67 times 10 to the power of negative 11 multiplied by 2.9 times 10 to the power of 24 multiplied by 1.9 times 10 to the power of 30 all over 4.37 times 10 to the power of 21. So therefore, we can say that 𝑟 squared is equal to 8.41 times 10 to the power of 22. So now all we need to do is take the square root of both sides to find out what 𝑟 is. So we can say that 𝑟 is equal to 2.9 times 10 to the power of 11. And we’re only interested in the positive result, and that’s because we’re just looking at a length. So therefore, we can say that the distance between the Sun and the planet is 2.9 times 10 to the power of 11 meters.
