At what distance above the surface of the earth is the accelerati...

Question

At what distance above the surface of the earth is the acceleration due to the earth’s gravity 0.980 m/s^2 if the acceleration due to gravity at the surface has magnitude 9.80 m/s^2 ?

Accepted Answer

everyone welcome back in this problem, we're told that the acceleration due to Earth's gravity at a given point in Earth's atmosphere is 8.7 m per second squared OK? Instead of 9.8 m per second squared when we're at the Earth's surface and were asked to find the altitude of the point above the Earth's surface. Okay, now let's recall that the gravitational acceleration okay. At altitude can be found by the gravitational acceleration of Earth times the radius of the earth divided by the radius of the Earth plus H where h is the height or the altitude above the Earth's surface. So that means that H is the quantity that we're going to be looking for. Okay. Alright. So this equation allows us to compare the gravitational acceleration at some altitude versus on Earth's surface. Now we're told that the gravitational acceleration at The atmosphere point is 8.7. Hey, meters per second squared. And on the earth's surface it's 9.8 m/s squared. Do not be. It's gonna is a little bit smaller. It's easier for us to keep writing here. Okay, so now we have the radius of the Earth. Okay, The radius of the Earth 6.37 times to the six m. Okay, And you can use your textbook or table that your professor provided um for these values the radius of the Earth again. 6.37 times 10 to the six m plus H. R. Altitude that we're trying to find. All squared. Okay, so now we have this equation, we know all of the values except for the one we're looking for. All we need to do is go ahead and solve for H. Alright, so we get 8.7 m/s squared on the left, 9.8 m/s squared on the right. Then we have times 6. times 10 to the six meters all squared, decided by 6.37 times 10 to the six m plus H. All squared. Okay. Alright, so let's multiply both sides by this big denominator here. Okay. And then we're gonna divide by 8.7. So we're gonna get the term with h by itself on the left hand side. So we're gonna have 6.37 times 10 to the six m plus H or altitude squared is going to equal 9.8 divided by 8.7 K. And the unit meter per second squared is going to cancel when we do that division And then we have times 6.37 times 10 to the six m squared. Okay, All right now, what we want to do here, we have square on both sides. Let's go ahead and take the square root of both sides. So let's give ourselves some more room. Taking the square root on both sides. We're just going to be left with 6.37 times 10 to the six m plus H. Okay. On the left hand side, we don't need the bracket anymore. And on the right hand side we're gonna have this square root Of 9.8 divided by 8.7 times 6.37 times 10 to the six m. Alright, so again we're solving H. We want to get H by itself. We can move this term 6.37 times 10 to the six m to the right hand side. We'll have H is equal to the square root of 9.8, divided by 8.7 Times 6.37 times 10 to the six m -6.37 times 10 to the six m. You can go ahead and factor. So you're gonna have 6.37 times 10 to the six m times from the first term, we get the square root of 9.8 divided by 8.7 and a -1. Alright. And if we work this out on our calculator we're going to get 382,200 m approximately. Okay, Um that's using some approximation of the square root. So just check with your professor or your textbook on how many significant digits they want you to carry in your intermediate steps. Okay? You get 382,200 m. And if we write this, if we're looking at the answers We need to write this as 10 to the six. Okay, so we can go ahead and write this 0. times 10 to the six m. Okay, so that is our height h. That is the altitude and that is going to correspond with answer. Hey, thanks everyone for watching. See you in the next video.

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