$A$ study of binary stars is most helpful in

Two spherical bodies of mass $M$ and $5M$ and radii $R$ and $2R$ are released in free space with initial separation between their centres equal to $12R.$ If they attract each other due to gravitational force only,then the distance covered by the smaller body before collision is (in $R$)

Mean solar day is the time interval between two successive noons when the sun passes through the zenith point (meridian). $A$ sidereal day is the time interval between two successive transits of a distant star through the zenith point (meridian). By drawing an appropriate diagram showing the earth's spin and orbital motion,show that the mean solar day is $4\,\text{min}$ longer than the sidereal day. In other words,distant stars would rise $4\,\text{min}$ early every successive day.

The mass density of a spherical body is given by $\rho(r) = \frac{k}{r}$ for $r \leq R$ and $\rho(r) = 0$ for $r > R$,where $r$ is the distance from the centre. The correct graph that describes qualitatively the acceleration,$a$,of a test particle as a function of $r$ is

Four particles,each of mass $M$,move along a circle of radius $R$ under the action of their mutual gravitational attraction as shown in the figure. The speed of each particle is:

State whether the following statements are true or false:
$(a)$ The intensity of the Earth's gravitational field increases as we move higher above the Earth's surface.
$(b)$ There is no atmosphere on the Moon because the escape velocity on the Moon is small.
$(c)$ $A$ satellite orbiting the Earth is in a state of weightlessness.