Question:
A body stretches a spring by a particular length at the earth's surface at equator. At what height above the south pole will it stretch the same spring by the same length? Assume the earth to be spherical.
Solution:
At equator, let stretch in spring be $x$
Spring force $={ }^{m} g_{e}$
$k x=m\left(g-R \omega^{2}\right)-(1)$
Now, at height $h$ above surface, stretch in spring is $x$
Sprina force $=m g_{h}$
$k x=m g\left(1-\frac{2 h}{R}\right)-(2)$
From (1) and (2)
$m\left(g-R \omega^{2}\right)=m g\left(1-\frac{2 h}{R}\right)$
$R \omega^{2}=\frac{2 h}{R} g$
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