Chapter 6.6, Problem 25ES

It follows from Newton’s Law of Universal Gravitation that he gravitational force exerted by the Earth on an object above the Earth’s surface varies inversely as the square of its distance from the Earth’s center. Thus, an object’s weight $\omega \left(x\right)$ is related to its distance $x$ from the Earth’s center by a formula of the form

$\omega \left(x\right)=\frac{k}{x^2}$

where $k$ is a constant of proportionality that depends on the mass of the object.

(a) Use this fact and the assumption that the Earth is a sphere of radius 4000 mi to obtain the formula for $\omega \left(x\right)$ in Example $4$ .

(b) Find a formula for the weight $\omega \left(x\right)$ of a satellite that is x mi from the Earth’s surface if its weight on Earth is 60001b.

(c) How much work is required to lift the satellite from the surface of the Earth to an orbital position that is $1000\text{ mi}$ high?