Gravitation Reasonings

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.