91Ó°ÊÓ

An object faces the condition of weightlessness when its centripetal acceleration equals the acceleration due to gravity.

The velocity of rotation can be calculated as,

\(v = \frac{{2\pi {R_{{\rm{Earth}}}}}}{T}\)

Here, \({R_{Earth}}\) is the radius of the Earth and its value is \(6.38 \times {10^6}\;{\rm{m}}\) and \(T\) is the time period.

The acceleration due to gravity is given by,

\(\begin{aligned}g &= \frac{{{v^2}}}{{{R_{{\rm{Earth}}}}}}\\g &= \frac{{4{\pi ^2}{R_{{\rm{Earth}}}}}}{{{T^2}}}\\T &= 2\pi \sqrt {\frac{{{R_{{\rm{Earth}}}}}}{g}} \end{aligned}\)

Substitute the values in the above equation,

\(\begin{aligned}T &= 2\pi \sqrt {\frac{{6.38 \times {{10}^6}\;{\rm{m}}}}{{9.8\;{\rm{m/}}{{\rm{s}}^2}}}} \\ &= 5.07 \times {10^3}\;{\rm{s}}\left( {\frac{{1\;{\rm{min}}}}{{60\;{\rm{s}}}}} \right)\\ &= 84.5\;{\rm{min}}\end{aligned}\)

Thus, the length of the day is \(84.5\;min\).