Question

If all objects on the equator of the earth feel weightless then the duration of the day will nearly become

• A. $6.2hr$
• B. $4.4hr$
• C. $2.2hr$
• D. $1.412hr$

Answer 1

Answer: (D)

$1.412hr$

We know that acceleration due to gravity at equator is,

$g_e=g-R{\omega }^2$

where, ${\omega }_0$ is the angular velocity of the earth.

Angular velocity of Earth$=\frac{2\pi }{T}=\frac{2\pi }{86400}$

If the acceleration due to gravity at the equator becomes zero, the body at equator experiences weightlessness.

Let $\omega$ be the angular velocity of rotation at which the acceleration due to gravity at the equator becomes zero.

That is,

$g_e=g-R{\omega }^2=0$

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

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

Now, $\frac{\omega }{{\omega }_0}=\frac{1}{{\omega }_0}\sqrt{\frac{g}{R}}=\frac{86400}{2\pi }\sqrt{\frac{9.8}{6.4\times {10}^6}}=17$

$\Rightarrow \omega =17{\omega }_0$

Thus, the earth should rotate $17$ times faster than the present rate.

Rotation speed increases by a factor of $17.$

$\therefore$ the duration of day decreases by factor $17.$

New duration of day, $T=\frac{24}{17}=1.412$hr