Answer to Question #105124 in Classical Mechanics for Gideon

Question

1. In most calculations, we use W ≈ mg. But we know that, for large changes in altitude, we need to use W∝1/r^2. How far above the Earth's surface can we use W = mg before our systematic error reaches 1%? Use only the information given in this question, and the radius of the Earth re=6400 km. Do not explicitly use G or the mass of the Earth, and do the calculation for the pole so that we don't worry about the effect of centripetal acceleration.

Hint: Does g become larger or smaller with altitude?

Altitude = ___ km (to one significant figure)

Answer 1

The law of universal gravitation and the force of gravity at earth's surface and at some height are equal:

$(1)\space\space \space \space G\frac{Mm}{r^2}=mg,\\ \space\\ (2)\space\space \space \space G\frac{Mm}{(r+h)^2}=mg_h.$

Divide the second by the first:

$\frac{g_h}{g}=\frac{r^2}{(r+h)^2}.$

The systematic error between the two values is

$\epsilon=\bigg(1-\frac{g_h}{g}\bigg)\cdot100\%=\\ \space\\ =\bigg(1-\frac{r^2}{(r+h)^2}\bigg)\cdot100\%=\frac{h^2+2rh}{(r+h)^2}\cdot100\%.$

Since the systematic error must be 1%, the last equation gives us the following value of height:

$1\%=\frac{h^2+2\cdot6400h}{(6400+h)^2}\cdot100\%,\\ \space\\ h=32\text{ km}.$