Question

The rotation of earth having radius $R$ about its axis speed upto a value such that a man at latitude angle ${60}^{\circ }$ feels weighthless.The duration of the day in such case will be:

• A. $\pi \sqrt{\frac{R}{g}}$
• B. $8\pi \sqrt{\frac{g}{R}}$
• C. $8\pi \sqrt{\frac{R}{g}}$
• D. $4\pi \sqrt{\frac{g}{R}}$

Answer 1

The correct option is A $\pi \sqrt{\frac{R}{g}}$

We know that,
$g^{\prime }=g-{\omega }^{2}R{\cos }^2\varphi$

Here, $\varphi ={60}^{\circ }\text{and}{g}^{{}^{\prime }}=0$ for weightless.
$\therefore 0=g-{\omega }^{2}R\times {\cos }^2{60}^{\circ }$

$\Rightarrow 0=g-\frac{{\omega }^{2}R}{4}$

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

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

Now, time period, $T=\frac{2\pi }{\omega }$
$\Rightarrow T=\frac{2\pi }{2\sqrt{\frac{g}{R}}}=\pi \sqrt{\frac{R}{g}}$