The rotation of the earth having R radius about its axis speeds up to a value such that a man at latitude angle 60° feels weightlessness. The duration of the day in such a case is

$4 \pi \sqrt{\frac{R}{g}}$

$0=g-R \omega^2 \cos ^2 60^{\circ}$

$\Rightarrow \frac{R^2 \omega^2}{4}=g$

$\Rightarrow \omega=2 \sqrt{\frac{g}{R}}$

$\Rightarrow T=4 \pi \sqrt{\frac{R}{g}}$