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Question Video: Finding the Acceleration Due to Gravity between a Planet and the Earth Mathematics

A planet has a mass of 8.4 × 10²⁴ kg and a radius of 5,723 km. Given that the mass of Earth is 5.97 × 10²⁴ kg, its radius is 6,340 km, and the acceleration due to gravity at its surface is 9.8 m/s², find the acceleration due to gravity 𝑔 at the surface of the other planet, approximating your answer to the nearest two decimal places.

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Video Transcript

A planet has a mass of 8.4 times 10 to the power 24 kilograms and a radius of 5,723 kilometers. Given that the mass of Earth is 5.97 times 10 to the power of 24 kilograms, its radius is 6,340 kilometers, and the acceleration due to gravity at its surface is 9.8 meters per second squared, find the acceleration due to gravity 𝑔 at the surface of the other planet, approximating your answer to the nearest two decimal places.

Now to solve this problem, what we’re gonna do is recall a couple of formula that we have. The first one is Newton’s universal gravitation. And what this is is that 𝐹 sub 𝐺 is equal to 𝐺 multiplied by 𝑚 sub one 𝑚 sub two over 𝑟 squared. And this is where 𝐹 sub 𝐺 is the gravitational force, capital 𝐺 or big 𝐺 is the universal gravitational constant or the constant of universal gravitation. Then we’ve got 𝑚 sub one and 𝑚 sub two, our masses, and then 𝑟 is our separation. We look at this and think, “Well, how is this gonna be useful?” because what we’re trying to find in this question is acceleration due to gravity.

Then what we’re gonna do now is recall the other formula or equation that we mentioned. Well, that is that 𝑔 is equal to capital 𝐺𝑚 over 𝑟 squared. Then we got little 𝑔, which is acceleration due to gravity. And then we got capital 𝐺 again, which we’ve looked at already and 𝑚, and this 𝑚 is going to be the mass of the body or planet we’re looking at, and then capital 𝑅 just to distinguish the difference here. And that is because we’re looking at the radius of the planet, not the separation. Okay and we actually find this second formula from using the first formula and actually combining it with Newton’s second law, which is 𝐹 equals 𝑚𝑎.

Now we might think, “Well, where do we go from here?” Well, what we’re going to do is work out what capital 𝐺 is, so the universal gravitational constant, because you might think, “Well, we already know what this is.” However, we haven’t been told this in the question. So what we’re gonna do is calculate it. And we can do that using the information we know about planet Earth. Well, what we know about Earth is that the acceleration due to gravity is 9.8 meters per second squared. We’ve got the mass of Earth is 5.97 times 10 to the power of 24 kilograms. And then we’ve got the radius of Earth is 6,340 kilometers. However, what we want is in fact the radius in meters because when we’re looking to work out what our universal gravitational constant is, due to the units that that’s going to be in, we want to be working in meters.

So therefore, we multiply 6,340 by 1000, and this is gonna give us 6,340,000 meters. So therefore, we can say that 9.8 is gonna be equal to 𝐺 multiplied by 5.97 times 10 to the power of 24 over 6,340,000 squared. So then, if we rearrange this, we get 𝐺 is equal to 9.8 multiplied by 6,340,000 squared over 5.97 times 10 to the power of 24. So then what we get is capital 𝐺 is equal to 6.598 continued multiplied by 10 to the power of negative 11. We haven’t rounded at this point because we want to maintain accuracy. So we will keep this for the second part of the question. So now what we can do is have a look at the planet to work out the acceleration due to gravity on the surface of that planet.

So now we’ve cleared some room. So what we’ve got is the acceleration due to gravity of the planet we don’t know cause that’s what we’re trying to find. The mass of the planet is 8.4 times 10 to the power of 24 kilograms. Then we have the radius of the planet which is 5,723 kilometers. And again, we want to convert this into meters. So we multiply by 1000 which is gonna be equal to 5,723,000 meters. And then finally, we’ve got 𝐺 which is our final piece of the puzzle, which we calculated earlier. Okay, so now we can just substitute this into our formula to find our acceleration due to gravity on the planet.

So when substituted into the formula, we get the acceleration due to gravity of the planet is equal to 6.598 continued multiplied by 10 to the power of negative 11 multiplied by 8.4 times 10 to the power of 24 all over 5,723,000 squared. which is gonna give us 16.9224 continued. Then checking the question, we can see that we want the answer to two decimal places. So therefore, we can say that the acceleration due to gravity at the surface of the other planet is 16.92 meters per second squared, and that’s to the nearest two decimal places.

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