Physics earth radii question

Question

How much would a 100-kg object weigh at two earth’s radii above the earth’s surface?

I’m thinking it would mass times gravity, and then divide by 4 because the square of 1/2 is 1/4 for two earth radii. Am I thinking in the right direction or am I wrong?

Answer 1

My students in AP physics 1 and C courses do this problem like this:

On the surface of the Earth the weight of the body W₀ = mg₀,

here m – mass of the body,

 g₀ = GM / R² = 9.8 m/s² - acceleration due to gravity on the Earth’s surface,

where G – gravitational constant,

M – mass of the Earth.

R - radius of the planet.

If the body at a distance h above the earth surface, then its weight W = mg,

where g = GM / (R + h)².

In our case h = 2R, hence the formula for g looks like this:

                                           g = GM / (3R)² = GM / 9R²

If we divide W by W₀ after cancelations of mass, we get

                                                W / W₀ = g / g₀     

or

                                                 W / W₀ = R² / 9R² = 1 / 9

Hence, we see that weight W on this distance from the center of the Earth is 9 times smaller than on Earth's surface. Knowing that W₀ = mg₀, we calculate it. W₀ = 980 N.

Divide this number by 9 to get the weight of the body on a distance equal to 2R from the Earth’s surface. Final answer is 109 N

                                                      

Answer 2

Fg=mg=GMm/r^2 = 100kg (9.8 m/s^2)= 980 Newtons (r being the distance from the surface of the earth to its center)

If we put this mass at 2 Earth Radii above the surface that means that the object would be 3 radii from the earths center.

We take the same equation but put 3r on the bottom and square it:

Fg=GMm/(3r)^2=GMm/9r^2 = 1/9 (GMm/r^2)= 1/9 (980N) = 109 Newtons