A body stretches a spring by a particular length at the earth's surface at the equator. At what appr

Question

A body stretches a spring by a particular length at the earth's surface at the equator. At what approximate height above the south pole will it stretch the same spring by the same length? Assume the earth to be spherical. (Given $R = 6400~km$, $g = 9.8~m~s^{-2}$, $\omega = 7.3 	imes 10^{-5}~rad~s^{-1}$)

Professor PC Thomas Classes · Answer

For the stretch to be equal, effective gravity must be equal. At equator, $g_{eq} = g - R\omega^2$. At height $h$ above the pole, $g_h = g(1 - \frac{2h}{R})$. Equating them gives $R\omega^2 = \frac{2gh}{R} \implies h = \frac{R^2\omega^2}{2g}$. Substituting the values yields $h \approx 11~km$.