The acceleration due to gravity, g, is given by g=(GM/r^2)

Question

The acceleration due to gravity, g, is given by g=(GM/r^2), where M is the math of the earth, r is the distance from the center of the earth, and G is the universal gravitational constant. 

(a) show that when r changes by Δr, the change in the acceleration due to gravity, Δg, is given by Δg≈-2g(Δr/r).

(b) What is the significance  the negative sign?

(c) What is the percent change in g when moving from sea level to the top of Pike's peak (4315 km)? Assume the radius of the earth is 6400 km. 

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(4.315 / 6400) = -0.00135 or -0.135% change in g.

 

Pike's Peak is 4.315 km above sea level.