Question

The gravitational force of a star on an orbiting planet 1 is f1. planet 2, which is three times as massive as planet 1 and orbits at twice the distance from the star, experiences gravitational force f2. part a what is the ratio f2f1? you can ignore the gravitational force between the two planets.

Answer 1

Let us consider two bodies having masses m and m' respectively.

Let they are separated by a distance of r from each other.

As per the Newtons law of gravitation ,the gravitational force between two bodies is given as -
`F = G(mm')/(r^(2) )` where G is the gravitational force constant.

From the above we see that F ∝ mm' and
`F\alpha (1)/(r^(2) )`

Let the orbital radius of planet A is
`r_(1)` = r and mass of planet is
`m_(1)`.

Let the mass of central star is m .

Hence the gravitational force for planet A is
`f_(1) =G (m_(1)*m )/(r^(2) )`

For planet B the orbital radius
`r_(2) =2r_(1)` and mass
`m_(2) = 3 m_(1)`

Hence the gravitational force
`f_(2) =G(m m_(2) )/(r^(2) )`

`f_(2) =G(m*3m_(1) )/([2r_(1)] ^(2) )`

`= (3)/(4) G(mm_(1) )/(r_(1) ^(2) )`

Hence the ratio is
`(f_(2) )/(f_(1) ) = \frac{(3)/(4)G mm_{1/r_(1) ^2}  }{Gmm_(1)/r_(1) ^2 }`

`=(3)/(4)` [ ans]