If a graph is plotted between `T^(2) and r^(3)` for a planet, then its slope will be be (where `M_(S)` is the mass of the sun)

Options

A. `(4pi^(2))/(GM_(S))`
B. `(GM_(S))/(4pi)`
C. `4piGM_(S)`
D. `GM_(S)`

Answer

A

Explanation

(a) Time period of a revolution of a planet,
`T=( 2pi r)/(v)=(2pi r)/(sqrt((GM_(S))/(r )))=(2 pi r^(3//2))/(sqrt(GM_(S))`
where `M_(S)` is the mass of the sun
Squaring both sides, we get
`T^(2)=(4 pi^(2) r^(3))/(GM_(S))`
The graph between `T^(2)` and `r^(3)` is a straight line whose slope is is `(4 pi^(2))/(GM_(S))`.