The radius of the earth is $6400\, km$ and $g = 10\,m/{\sec ^2}$. In order that a body of $5 \,kg$ weighs zero at the equator, the angular speed of the earth is

If the acceleration due to gravity experienced by a point mass at a height $h$ above the surface of earth is same as that of the acceleration due to gravity at a depth a depth $\alpha$ h $\left(h \ll R_{e}\right)$ from the earth surface. The value of $\alpha$ will be$….$(use $R _{ e }=6400\,km$ )

A ball is launched from the top of Mt. Everest which is at elevation of $9000 \,m$. The ball moves in circular orbit around earth. Acceleration due to gravity near the earth's surface is $g$. The magnitude of the ball's acceleration while in orbit is

If the value of g at the surface of the earth is $9.8$ $m/{\sec ^2}$, then the value of g at a place $480\, km$ above the surface of the earth will be ………. $m/{\sec ^2}$. (Radius of the earth is $6400\, km$)   

A mass falls from a helght $h$ and its time of fall $t$ is recorded in terms of time period $T$ of a simple pendulum. On the surface of earth it is found that $t =2 T$. The entre setup is taken on the surface of another planet whose mass is half of that of earth and radius the same. Same experiment is repeated and corresponding times noted as $t'$ and $T'$.