# What will be the time duration of one day if the earth rotates so fast that bodies just float at the equator?

## OR: At what condition does a body becomes weightless at the equator?

The value of acceleration due to gravity considering the rotation of the earth is, $g’=g-\omega^2R\cos^2\theta$ At equator, $\theta=0°$, $\therefore g’=g-\omega^2R$

In order that the bodies start to float or a body becomes weightless, $g’=0.$

$\text{i.e.}\;g-\omega^2R=0$ $g=\omega^2R$ $\frac{g}{R}=\omega^2$ $\sqrt{\frac{g}{R}}=\omega=\frac{2π}{T}$ $\therefore T=2π\sqrt{\frac{R}{g}}=2π\sqrt{\frac{6400000}{9.8}}$ $=5077.6\;\text{sec}=1.41\;\text{hrs}$

Therefore, when earth rotates in such a way that the time period of rotation is 1.41 hours, then the bodies will start to float at the equator or the bodies will become weightless.