A rocket is fired from the earth towards the sun. At what distance from the earth's centre is the gravitational force on the rocket zero? Mass of the sun`=2xx10^(30)kg`, mass of the earth`=6xx10^(24)kg`. Neglect the effect of other planets etc. (orbital radius=`1.5xx10^(11)m`).

Explanation

Let x = distance between the earth and sun = Radius of orbit.
Let P be a point at a distance from the earth where gravitational force due to sun and earth is equal and opposite and give rise to zero gravitational force.
`(GM_(S)m)/((x-r)^(2)) = (GM_(E)m)/(r^(2))`
where `M_(S)` is the mass of the Sun and `M_(E)` is the mass of the earth.
or `(M_(S))/((x-r)^(2)) = (M_(E))/(r^(2))`
`(2 xx 10^(30))/((x-r)^(2)) = (6 xx 10^(24))/(r^(2))`
`(x-r)/(r ) = sqrt((2 xx 10^(30))/(6 xx 10^(24))) = 577.35`
`x - r = 577.35 r`
`578.35 r = x`
`r = (x)/(578.35) = (1.5 xx 10^(11))/(578.35)`
`= 2.593 xx 10^(8)m`