How far from Earth must a body be along a line joining the sun to the earth so that resultage gravitational pull on the body due to Earth and sun is zero ? Distance between sun and the Earth is `1.5 xx 10^(8) km`. Mass of sun `= 3.25 xx 10^(5)` times mass of Earth.

Explanation

Here, `M_(S) = 2 xx 10^(30) kg , M_(e) = 6 xx 10^(24) kg , r = 1.5 xx 10^(11) m`.
Let `x` be the distance of a point from the earth where gravitational forces on the rocket due to sun and earth become equal and opposite. Then, distance of rocket from the sun `= (r - x)`. If `m` is the mass of rocket then
`(GM_(s)m)/((r - x)^(2))` or `((r - x)^(2))/(x^(2)) = (M_(s))/(M_(e))` or `(r - x)/(x) = sqrt((M_(s))/(M_(e))) = sqrt((2 xx 10^(30))/(6 xx 10^(24))) = (10^(3))/(sqrt(3))`
or `(r )/(x) - 1 = (10^(3))/(sqrt(3))` or `(r)/(x) = 1 + (10^(3))/(sqrt(3)) = (sqrt(3) + 10^(3))/(sqrt(3))`
or `x = (rsqrt(3))/(sqrt(3) + 10^(3)) = (1.5 xx 10^(11) xx sqrt(3))/(sqrt(3) + 10^(3)) = (1.5 xx 1.732 xx 10^(11))/(1.732 + 1000) = 2.59 xx 10^(8) m`