Question Video: Finding the Ratio of Acceleration Due to Gravity between Two Planets Using Newton’s Law of Universal Gravitation

In a star system, there are two planets. The first has a mass of 1.2 × 10²² kg, a radius of 4,000 km, and the acceleration due to gravity at its surface is 𝑔₁, whereas the second planet has a mass of 6 × 10²¹ kg, a radius of 8,000 km, and the acceleration due to gravity on its surface is 𝑔₂. Find 𝑔₁ : 𝑔₂, given that the universal gravitational constant is 6.67 × 10⁻¹¹ N ⋅ m²/kg².

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Video Transcript

In a star system, there are two planets. The first has a mass of 1.2 times 10 to the power of 22 kilograms, a radius of 4,000 kilometers, and the acceleration due to gravity at its surface is 𝑔 sub one, whereas the second planet has a mass of six times 10 to the power of 21 kilograms, a radius of 8,000 kilometers, and the acceleration due to gravity on its surface is 𝑔 sub two. Find the ratio 𝑔 sub one to 𝑔 sub two, given that the universal gravitational constant is 6.67 times 10 to the power of negative 11 newton meters squared per kilogram squared.

So in this problem, what we’re looking at is Newton’s law of universal gravitation. So therefore, we’d think, well, let’s use the formula, which is that gravitational force, 𝐹 sub 𝐺, is equal to capital 𝐺, which is the universal gravitational constant, multiplied by 𝑚 sub one multiplied by 𝑚 sub two over 𝑟 squared, where 𝑚 sub one and 𝑚 sub two are the masses of two bodies and 𝑟 is their separation. However, this isn’t particularly useful in this problem because in this problem we’re actually looking at the acceleration due to gravity on each of the planets’ surfaces. And we’re actually not looking at the gravitational force between them. So what are we going to do?

Well, what we’re gonna do is use a formula which is derived from the universal law of gravitation and Newton’s second law. And this is 𝑔 is equal to capital 𝐺𝑚 over 𝑟 squared. And this is where small 𝑔 is the acceleration due to gravity. Then we’ve got our 𝑟 is the radius. In this case, it’s gonna be the radius of the planet that we’re looking at. Then we’ve got big or capital 𝐺, which is our universal gravitational constant, and then 𝑚 is the mass of the planet. Now we have this, what we can do is use it to find out the acceleration due to gravity on the surface of each of our planets.

Well, if we start with our first planet, and we’re gonna call the mass of this first planet 𝑚 sub one, this is equal to 1.2 times 10 to the power of 22 kilograms. And then we have 𝑟 sub one, which is our radius, and this is gonna be 4,000 kilometers. However, if we now take a look at our universal gravitational constant, we can see that its units are in newton meters squared per kilogram squared. So therefore, we want to be working in meters, so we’re gonna convert 4,000 kilometers to meters. And to do that, we’ll just multiply by 1,000, which means that our radius of the first planet is gonna be four million meters.

Okay, great, so now we have everything we need to work out the acceleration due to gravity on the surface of our first planet. So therefore, we can say that 𝑔 sub one, so the acceleration due to gravity on the first planet, is gonna be equal to 6.67 times 10 to the power of negative 11 multiplied by 1.2 times 10 to the power of 22 over four million squared. And this is gonna give us an acceleration due to gravity on the first planet of 0.050025 meters per second squared. Okay, now what we can do is move on to the second planet, and we can clear some space to help with our working.

Well, for our second planet, we know that the mass is six times 10 to the power of 21 kilograms and the radius is equal to 8,000 kilometers. But once again, we want to be working in meters. So what we’re going to do is multiply this by 1,000, which gonna be equal to eight million meters. So therefore, we now have all the information we need to work out the acceleration due to gravity on the surface of the second planet. So what we can say is that 𝑔 sub two, so the acceleration due to gravity on the second planet, is gonna be equal to 6.67 times 10 to the power of negative 11 times six times 10 to the power of 21 all over eight million squared, which is gonna give us an acceleration due to gravity of 6.253125 times 10 to the power of negative three meters per second squared.

Okay, great, so we now have the accelerations due to gravity on both surfaces. But this isn’t the answer because what we want to do is find the ratio between them, which we’ll have a look at now. So this is gonna give us a ratio of 0.050025 to 6.253125 times 10 to the power of negative three. Well, this isn’t very neat. So what we can do is actually simplify this. And what we can try is dividing through by the smaller amount, so the value on the right-hand side. So when we do divide through by 6.253125 times 10 to the power of negative three, we get the ratio eight to one. So therefore, we can say that the ratio of the first planet to the second planet when we look at their acceleration due to gravity at the surfaces is eight to one.

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