Chapter 11.R, Problem 60E

The force due to gravity on an object with mass m at a height h above the surface of the earth is

$F=\frac{mg{R}^2}{{\left(R+h\right)}^2}$

where R is the radius of the earth and g is the acceleration due to gravity for an object on the surface of the earth.

(a) Express F as a series in powers of h/R.

(b) Observe that if we approximate F by the first term in the series, we get the expression $F\approx mg$ that is usually used when h is much smaller than R. Use the Alternating Series Estimation Theorem to estimate the range of values of h for which the approximation $F\approx mg$ is accurate to within one percent. (Use R = 6400 km.)