Question

Solve the following problem.

Two galaxies of masses 9 billion solar mass and 4 billion solar mass are 5 million light-years apart. If, the Sun has to cross the line joining them, without being attracted by either of them, through what point it should pass?

Answer 1

The Sun can cross the line joining the two galaxies without being attracted by either of them if it passes from a neutral point. Neutral point is a point on the line joining two objects where effect of gravitational forces acting due to both the objects is nullified.

Given that;
m₁ = 9 × 10⁹ M_s

m₂ = 4 × 10⁹ M_s

r = 5 × 10⁶ light years

Let the neutral point be at distance x from m₁.

If sun is present at that point,

`("Gm"_1"M"_"s")/"x"^2 = ("Gm"_2"M"_"s")/("r - x")^2`

`therefore (9 xx 10^9)/"x"^2 = (4 xx 10^9)/("r - x")^2`

Taking square roots on both sides,

`3/"x" = 2/("r - x")`

∴ 3r - 3x = 2x

∴ 3r = 5x

∴ x = `"3r"/5 = (3 xx 5 xx 10^6)/5 = 3 xx 10^6` light years

The Sun has to cross the line from a point at a distance 3 million light years from the galaxy of mass 9 billion solar mass.