The radius of the earth is R and g is the acceleration due to gravity on its surface. What should be the angular speed of the earth so that bodies lying on the · equator may appear weightless? 

At the equator, the value of g is
$g^{\text{'}}=g-R{\omega }^2$
where $\omega$ is the angular speed of the earth. For bodies to appear weightless at the equator, $g^{\text{'}}=0$, i.e.
$g-R{\omega }^2=0$
which gives $\omega =\sqrt{\frac{g}{R}}$. Hence the correct choice is (1)