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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?
Solution
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 Ms
m2 = 4 × 109 Ms
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.
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