# Acceleration due to gravity above Earth

"How far above the Earth's surface will the acceleration due to gravity be a quarter of what it is on the surface?"

I tried to use the g' = (G * m) / r^2 equation (with the mass of the earth as "m") because it was used with a somewhat similar problem in my book about the value of g on the top of Mount Everest, but I came up with a ridiculous number.

Can you tell me how to do this problem correctly?

## Solution This solution is **FREE** courtesy of BrainMass!

Hi,

<br>To begin with, I'd like to say you were on the right track when you used g=G*m/r^2 but I guess something went wrong in your calculation. I'm going to suggest 2 approaches: one of them quick and intuitive, the other a more brute force method that will succeed if the first way doesn't occur to you.

<br>

<br>Method 1

<br>since G and m are constants regardless of your height above the earth, the only variable here is "r", ie distance to the centre of the earth. Thus g is proportional to 1/r^2 => to reduce g by a factor of 4, you have to double r. i.e. your distance to the centre of the earth has to be twice the distance from the surface of the earth to the centre. Which means your height above the surface of the earth must be 1 earth radius.

<br>

<br>Method 2

<br>If you'd prefer to use the equation for gravitational force, that works too, it just takes a bit more time. From F = ma, you get

<br>

<br>Gm1m2/r^2 = m2a,

<br>i.e. a = Gm1/r^2 where a is the acceleration due to gravity (which on earth we call g), m1 is the mass of the earth, G is the universal constant and r is the distance from the centre of the earth. (This is the equation you've tried to use).

<br>

<br>We're trying to compare surface g with the g' you would get if you were a height h above the surface, and you want g' to to be equal to g/4.

<br>

<br>Here are the 2 equations for acceleration at the 2 different points:

<br>

<br>g = GM/r^2 (at surface, where r is radius of earth)

<br>g' = GM/(r+h)^2 (at height h above the earth, so that distance to

<br> centre of earth is (r+h)

<br>letting g' = g/4, you get

<br>

<br>(1/4)*GM/r^2 = GM/(r+h)^2

<br>

<br>G and M cancel from both sides so you get

<br>

<br>1/(4r^2) = 1/(r+h)^2

<br>or (r+h)^2 = 4r^2

<br>Now just take square root of both sides:

<br>

<br>r+h = 2r

<br>h = r

<br>

<br>=> your height above the earth is one earth radius.

<br>