Question

If G is the universal gravitation constant and is the uniform density of a spherical planet, then,

• A. Time period of a planet will be independent of density of the planet
• B. The shortest period of rotation of the planet will have very high density
• C. The shortest period of rotation of the planet will have very low density
• D. The shortest period of rotation of the planet depends on the radius of the planet

Answer 1

The correct option is A Time period of a planet will be independent of density of the planet

From the Kepler's laws of periods:-

The square of the orbital period of a planet is proportional to the cube of the semi-major axis of the elliptical orbit of the planet .

The period $T$ and radius $R$ of the circular orbit of a planet about the Sun are related by $T^2=\frac{4{\pi }^2{R}^3}{G{M}_S}$ ,

where $M_S$ is the mass of the Sun. Most planets have nearly circular orbits about the Sun. For elliptical orbits, the above equation is valid if $R$ is replaced by the semi-major axis, $a$.

$\Rightarrow T^2\propto R^3$ (Since $\frac{4{\pi }^2}{G{M}_S}=Constant$ )

Time period of planet is only depends on radius of orbit.

$\therefore$ Time period of planet will be independent of density of the planet.