Video Transcript
A satellite of mass 2,415 kilograms is orbiting the Earth 540 kilometers above its surface. Given that the universal gravitational constant is 6.67 times 10 to the power of negative 11 newton meters squared per kilogram squared and the Earth’s mass and radius are six times 10 to the power of 24 kilograms and 6,360 kilometers, determine the gravitational force exerted by the Earth on the satellite.
So the first thing I’ve done is drawn a sketch of our scenario just so we can see what’s happening. So we’ve got our satellite. And we can see that it’s 540 kilometers above the surface of the Earth. Now we can also see the radius of the Earth as 6,360 kilometers. But why is this useful? Well, it’s gonna be useful because what we’re going to use to solve this problem is Newton’s law of universal gravitation. And what we know is that 𝐹 sub 𝑔 is equal to big 𝐺 multiplied by 𝑚 sub one multiplied by 𝑚 sub two over 𝑟 squared, where 𝐹 sub 𝑔 is the gravitational force, capital 𝐺 or big 𝐺 is the universal gravitational constant, then our 𝑚 sub one and 𝑚 sub two are the masses of our bodies, and then our 𝑟 is our separation or distance between them.
So it’s gonna be this 𝑟 value that our diagram is gonna help us to find out. Okay, so the next thing we want to do is, in fact, write out which of the values we’ve been given in our question. Well, we don’t know 𝐹 sub 𝑔 because we’re trying to find the gravitational force. Well, we know that the mass of the satellite is 2,415 kilograms. The mass of Earth is six times 10 to the power of 24 kilograms. The universal gravitational constant is 6.67 times 10 to the power of negative 11 newton meters squared per kilogram squared. So then, from our diagram, we can see that our 𝑟 value, so the distance between the center of the Earth and our satellite, is gonna be 540 plus 6,360, which equals 6,900 kilometers.
Well, we might just think at this point, “Okay, great, all we need to do is substitute our values in and we can find our gravitational force.” However, there’s a common mistake and it’s one that we’ve got to watch out for here. We’ve seen that our 𝑟 value, so our distance, is in kilometers, so we’ve got 6,900 kilometers. However, just on inspection, we can already see that if we look at 𝐺, which is the universal gravitational constant, we can see that, in there, the distance element of the units is actually measured in meters. And actually, that’s what we’re going to be using because, again, if we’re thinking about newtons, it’s meters that we’re concerned with. So what we want to do is convert 6,900 kilometers into meters. And when we do that, what we get is 6,900,000 meters.
Okay, great, so now we’ve got everything we need. So now let’s substitute into our formula to find out what our gravitational force is going to be. So when we do that, we get the gravitational force is equal to 6.67 times 10 to the power of negative 11 multiplied by 2,415 multiplied by six times 10 to the power of 24 all over 6,900,000 squared. So then, if we put this in the calculator, what we get out is that the 𝐹 sub 𝑔, so our gravitational force, is equal to 20,300. So therefore, we can say given the information that we have in the question, the gravitational force exerted by the Earth on the satellite is 20,300 newtons.
