Two particles of equal mass $(m)$ each move in a circle of radius $(r)$ under the action of their mutual gravitational attraction. Find the speed of each particle.

If a tunnel is cut at any orientation through the Earth,a ball released from one end will reach the other end in time ........ $\text{min}$ (neglect Earth's rotation).

$A$ spiral galaxy can be approximated as an infinitesimally thin disc of uniform surface mass density located at $z=0$. Two stars $A$ and $B$ start from rest from heights $2z_{0}$ and $z_{0}$ (where $z_{0} \ll$ radial extent of the disc),respectively,and fall towards the disc,cross over to the other side,and execute periodic oscillations. The ratio of the time periods of $A$ and $B$ is

Imagine a narrow tunnel between two diametrically opposite points of the Earth. $A$ particle of mass $m$ is released in this tunnel. The time period of oscillation is ..........

$A$ geostationary satellite is at a height $h$ above the surface of the Earth. If the Earth's radius is $R$,which of the following statements is correct?

If the distance of Mercury from the Sun is $0.4$ times the distance of the Earth from the Sun,find the solar constant of Mercury in $cal/min \cdot cm^2$. The solar constant of the Earth is $2 \, cal/min \cdot cm^2$.