Chapter 6, Problem 19P

(III) (a) Use the binomial expansion

${(1\pm x)}^n=1\pm nx+\frac{n(n-1)}{2}{x}^2\pm \cdots$

to show that the value of g is altered by approximately

$\Delta g\approx -2g_{r_{\text{E}}}^{\Delta r}$

at a height Δr above the Earth’s surface, where r_E is the radius of the Earth, as long as Δr << r_E. (b) What is the meaning of the minus sign in this relation? (c) Use this result to compute the effective value of g at 125 km above the Earth’s surface. Compare to a direct use of Eq. 6–1.