How do i find gravity given a radius and an altitude?

Question

Using g = m/r^2?
What is the acceleration due to gravity at an altitude of 1.00 x 10^6 above the earth's surface, given that the radius of the earth is 6.38 x 10^6 m?
How do i go about solving that?

Answer 1

Let's try this:
First: Fg = GMm/r^2
So, if you are only changing r then write Fg1 = GMm/r1^2.
Now write Fg1 / Fg2 = GMm/r1^2 divided by GMm/r2^2
Canceling gives Fg1 / Fg2 = r2^2/r1^2

Um ... you can do the same thing with g, since g = Fg/m
.: g1/g2 = r2^2/r1^2 (you can work it out using the same procedure as in the previous paragraph. Maybe I should just have started with this, but I am figuring it out as I type).

Since you know g1 = 9.81/m/s/s and r1=6.38E6
and r2=1.00E6 + r1 (you MUST add the earth's radius to the altitude. ALWAYS do this anytime you see the code word ALTITUDE), you know all of the variables in the equation except for g2 ---- which is what you are trying to find.

Answer 2

First, the equation is g = Gm/r^2, you left out the constant G. m is Earth's mass and r is the distance from Earth's center of mass to wherever you are measuring g.

Then, r = R + h; where R is the Earth's radius and h is the height above the surface. Thus r is the distance from Earth's center of mass. Thus g = Gm/(R + h)^2; plug and chug as my colleague Bekki likes to say.

There is one other way; assume you know that g0 = 9.81 m/sec^2 at r = R on Earth's surface. Then g = g0 (R/(R + h))^2 would be true. That is, the g's are inversely proportional to the square of the distances from the center of mass. Since R and h are given, you can solve for g. This second way eliminates having to know what G and Earth's mass are.