Answer to Question #103925 in Calculus for Martin

Question

The acceleration due to gravity, g, is given by
g=GMr2,
where M is the mass of the Earth, r is the distance from the center of the Earth, and G is the uniform gravitational constant.

(a) Suppose that we increase from our distance from the center of the Earth by a distance Δr=x. Use a linear approximation to find an approximation to the resulting change in g, as a fraction of the original acceleration:
Δg≈ g×

(Your answer will be a function of x and r.)

(b) Is this change positive or negative?
Δg is
(Think about what this tells you about the acceleration due to gravity.)

(c) What is the percentage change in g when moving from sea level to the top of Mount Elbert (a mountain over 14,000 feet tall in Colorado; in km, its height is 4.4 km; assume the radius of the Earth is 6400 km)?
percent change =

Answer 1

g = GM/r²

a) Δg ≅ dg/dr Δr

dg/dr = -2GM/r³ = -2g/r

Δg ≅ -2g/r Δr = -2g(Δr/r)

b) r is measured from the center of Earth. As r increases, g decreases; hence, the minus sign shows the change in g with a change in r.

c) Rearrange the result in part a to get Δg/g ≅ -2(Δr/r) =-2(4.4/ 6400) = -0.001375 or -0.1375% change in g.