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Question Video: Finding the Force of Gravity between Two Planets Using Newton’s Law of Universal Gravitation Mathematics

Two planets are separated by a distance of 3 × 10⁶ km. The mass of the first is 9.9 × 10²² metric tons, and that of the other is 10²⁷ metric tons. Given that the universal gravitational constant is 6.67 × 10⁻¹¹ N⋅m²/kg², find the force of gravity between them.

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Video Transcript

Two planets are separated by a distance of three times 10 to the power of six kilometers. The mass of the first is 9.9 times 10 to the power of 22 metric tons, and that of the other is 10 to the power of 27 metric tons. Given that the universal gravitational constant is 6.67 times 10 to the power of negative 11 newton meters squared per kilogram squared, find the force of gravity between them.

So in this problem, what we can see is that we’re using Newton’s law of universal gravitation. And we know that because of some of the things that we’ve been given, because we’ve been given the universal gravitational constant. And what we’re trying to find is the force of gravity. And we’re looking at two bodies, in this case two planets.

So what the law tells us is that 𝐹 sub G is equal to 𝐺 multiplied by 𝑚 sub one multiplied by 𝑚 sub two over 𝑟 squared, where 𝐹 sub G is the gravitational force. 𝐺 or capital 𝐺 is the universal gravitational constant. And we use this to make sure that all of our units are lined at the end if we’re using this formula. Then we’ve got 𝑚 sub one and 𝑚 sub two, which are our masses. And then we’ve got 𝑟, which is our separation between our bodies, in this case between our planets.

So the first thing we need to do when solving a problem like this is look at the information we’ve been given. Well, if we’re looking at our term, so 𝐹 sub G starting off, which is our gravitational force, well, we don’t know what this is cause in fact this is what we’re trying to find out in this question. However, we do know big 𝐺, our universal gravitational constant, cause this is equal to 6.67 times 10 to the power of negative 11 newton meters squared per kilogram squared. Then we have the mass of our first planet, which is 𝑚 sub one, which is gonna be equal to 9.9 times 10 to the power of 22 metric tons.

However, if we think about what units we want to be using, well, we want to be using kilograms. And actually, we can see this in our universal gravitational constant. So what can we do to make our metric tons into kilograms? So how do we convert?

Well, what we do is we multiply it by 1000. So therefore, we can say that 𝑚 sub one, so the mass of the first planet, is 9.9 times 10 to the power of 25 kilograms. Well then, for the mass of our second planet, which we’re gonna call 𝑚 sub two, well, this is equal to 10 to the power of 27 metric tons. Again, we need to multiply by 1000, which is gonna give us 10 to the power of 30 kilograms.

Okay, great, so we’ve now found 𝑚 sub one and 𝑚 sub two. So now let’s have a look at the separation. Well, for our separation, we can see that we’ve got three times 10 to the power of six kilometers. Once again, we must remember here to stop and have a look at what units we’ve got. So we’ve got kilometers. However, again, we can use the universal gravitational constants units just to help us see what we need. We can see that we want to work in meters. So therefore, again, we’re gonna have to multiply this by 1000 to convert it into meters. And when we do that, we get three times 10 to the power of nine meters.

Okay, great, so we’ve got everything we need. So now what we need to do is substitute in our values into our formula to find our gravitational force or the force of gravity between the two planets. And when we do that, what we’re gonna get is 𝐹 sub G, so our gravitational force, is gonna be equal to 6.67 times 10 to the power of negative 11 multiplied by 9.9 times 10 to the power of 25 multiplied by 10 to the power of 30. And then this is all over three times 10 to the power of nine all squared, which is gonna give us 7.337 times 10 to the power of 26.

So therefore, we can say that the force of gravity between our two planets is 7.337 times 10 to the power of 26 newtons.

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