Question

A simple pendulum is hanging from a support and is at rest as seen from the lab where the latitude is 60⁰. Assume the earth to be a uniform sphere. The angle between the string and the true vertical at that place is

(a) less than 2⁰

(b) between 2⁰ and 4⁰

(c) between 4⁰ and 6⁰

(d) more than 6⁰?

Answer 1

$T=m\omega^2 l=m \omega^2Rcos \varphi,$

$P=mg=m(\frac{GM}{R^2}-\omega^2 R),$

$F=\frac{GmM}{R^2},$

$\varphi=60°,$

$R=6.37\cdot 10^6~m,$

$\omega=\frac{2\pi}{t}=\frac{2\pi}{86164},$

$T^2=P^2+F^2-2PTcos \beta,\implies$

$cos\beta=\frac{P^2+F^2-T^2}{2PF}=\frac{(\frac{GM}{R^2}-\omega^2 R)^2+(\frac{GM}{R^2})^2-(\omega^2 R cos\varphi)^2}{2(\frac{GM}{R^2}-\omega^2 R)\frac{GM}{R^2}}\approx 1,$

$\beta\approx 0°<2°,$

answer (a).