There is a planet which is $8$ times more massive and $27$ times denser than the Earth. If $g^{\prime}$ and $g$ are the accelerations due to gravity on the surfaces of the planet and the Earth respectively,then:

$A$ planet of radius $R = \frac{1}{10} \times$ (radius of Earth) has the same mass density as Earth. Scientists dig a well of depth $\frac{R}{5}$ on it and lower a wire of the same length and of linear mass density $\lambda = 10^{-3} \ kg \ m^{-1}$ into it. If the wire is not touching anywhere,the force applied at the top of the wire by a person holding it in place is (take the radius of Earth $= 6 \times 10^6 \ m$ and the acceleration due to gravity on Earth $g_e = 10 \ m \ s^{-2}$) (in $N$)

If the radius of the earth shrinks by $1.5\%$ (mass remaining same),then the value of acceleration due to gravity changes by ....... $\%$.

If the radius of the earth shrinks by $1 \%$,its mass remaining the same,then the acceleration due to gravity on the earth's surface would

$A$ simple pendulum performing small oscillations at a height $R$ above the Earth's surface has a time period of $T_1 = 4 \ s$. What would be its time period $T_2$ if it is brought to a point at a height $2R$ from the Earth's surface? Choose the correct relation ($R =$ radius of Earth).

The percentage decrease in the weight of a rocket,when taken to a height of $32 \ km$ above the surface of the Earth,will be $.....\%$
(Radius of Earth $= 6400 \ km$)