Gravitational acceleration on the surface of a planet is $\frac{\sqrt 6}{11}g$ , where $g$ is the gravitational acceleration on the surface of the earth. The average mass  density of the planet is $\frac{2}{3}\, times$ that of the earth. If the escape speed on the surface of the earth is taken to be $11\, kms^{-1}$, the escape speed on the surface of  the  planet in $kms^{-1}$ will be

At a height of $10 \,km$ above the surface of earth, the value of acceleration due to gravity is the same as that of a particular depth below the surface of earth. Assuming uniform mass density for the earth, the depth is ............. $km$

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 $T'$.

Assuming earth to be a sphere of a uniform density, what is the value of gravitational acceleration in a mine $100\, km$ below the earth’s surface ........ $m/{s^2}$. (Given $R = 6400 \,km$)

Given below are two statements:

Statement $I:$ Acceleration due to earth's gravity decreases as you go 'up' or 'down' from earth's surface.

Statement $II:$ Acceleration due to earth's gravity is same at a height ' $h$ ' and depth ' $d$ ' from earth's surface, if $h = d$.

In the light of above statements, choose the most appropriate answer form the options given below

The acceleration due to gravity at a height $1\, km$ above the earth is the same as at a depth $d$ below the surface of earth. Then  $d\,=$ ......... $km$