Video Transcript
A satellite is held in an orbit that is 310 kilometers above the surface of the Earth by gravitational force of magnitude 14,637 newtons. Given that the mass of the Earth is six times 10 to the power of 24 kilograms, its radius is 6,360 kilometers, and the universal gravitational constant is 6.67 times 10 to the power of negative 11 newton meters squared per kilogram squared, determine the mass of the satellite.
Well, in this problem, we can see that we’re told what the universal gravitational constant is. We also know about the gravitational force. So therefore, what we’re going to do is use Newton’s law of universal gravitation to solve the problem. That tells us that 𝐹 sub 𝐺 is equal to capital 𝐺 multiplied by 𝑚 sub one multiplied by 𝑚 sub two over 𝑟 squared. And this is where 𝐹 sub 𝐺 is gravitational force. Big 𝐺 is the universal gravitational constant. Then we’ve got our masses 𝑚 sub one and 𝑚 sub two. And then the separation between our bodies or distance between them is 𝑟.
Well, whenever we’re looking at a problem like this, the first thing we want to do is write down the information we’ve been given. Well, we know that the gravitational force is 14,637. Well, our 𝑚 sub one, which we’re gonna call 𝑚 sub 𝐸 because it’s the mass of the Earth we’re looking at, is equal to six times 10 to power of 24 kilograms. Our 𝑚 sub 𝑠, which is the mass of the satellite, we don’t know because that’s what we’re trying to find.
Then next, what we have is the universal gravitational constant, which is 6.67 times 10 to the power of negative 11 newton meters squared per kilogram squared. Well then, our separation or distance between the center of Earth and the satellite is gonna be 6,360 plus 310 because that’s the radius of the Earth added to the distance above the Earth that the satellite orbits, which is gonna give us 6,670 kilometers.
However, we aren’t quite finished here with 𝑟 because we notice that this is in kilometers. However, if we check out the units, then we can see that in fact when we’re looking at the universal gravitational constant, we’re dealing with meters because it’s newton meters squared per kilogram squared. So therefore, we need this to be in consistent units throughout. So we need to change our 6,670 kilometers into meters. And we do that by multiplying by 1,000. And when we do that, what we’re gonna get is 6,670,000 meters.
Okay, great, so now what we need to do is look at our formula because what we’re trying to do is work out the mass of the satellite. Now, before we substitute in our values into the formula, what we can do is rearrange to make 𝑚 sub 𝑠 the subject of the formula. And this is because this is what we’re trying to find.
So first of all, we multiply by 𝑟 squared. So then what we get is 𝐹 sub 𝐺 𝑟 squared equals 𝐺 𝑚 sub 𝐸 𝑚 sub 𝑠. So then, next, what we do is divide by 𝐺 𝑚 sub 𝐸. So therefore, what we’re gonna get is 𝑚 sub 𝑠 is equal to 𝐹 sub 𝐺 𝑟 squared over 𝐺 𝑚 sub 𝐸. So now we’ve got the mass of the satellite as the subject of our formula. What we can do is substitute our values in.
So now if we substitute in our values, what we get is 𝑚 sub 𝑠, so the mass of the satellite, is equal to 14,637 multiplied by 6,670,000 squared all over 6.67 times 10 to the power of negative 11 multiplied by six times 10 to the power of 24. And when we calculate this, we’re gonna get 1,627.1465. So therefore, if we round to the nearest kilogram, we can say the mass of the satellite is 1,627 kilograms. Or if we put it in standard form, it’s gonna be 1.627 times 10 to the power of three kilograms.
