Question

The mass density of a spherical galaxy varies as `"K"/"r"` over a large distance 'r' from its center. In that region, a small star is in a circular orbit of radius R. Then the period of revolution, T depends on R as :

विकल्प

• T² ∝ `1/"R"^3`

• T² ∝ R

• T∝ R

• T² ∝ R³

Answer 1

T² ∝ R

Explanation:

Given: Mass density of a spherical galaxy is ρ = `"K"/"r`, radius of the orbit of a small star inside the galaxy is R .

To find: The relation between R and the time period of revolution T for the small star.

Mass of the galaxy:

M = `int rho"dV" = int_0^"R"rho(4pi"r"^2) "dr"`

M = `4pi"K"int_0^"R""rdr" = 2pi"KR"^2`

Let m be mass of the star.

Force of gravitation on the star due to the galaxy of mass M:

F_G = `"GMm"/"R"^2`

The force of gravitation is balanced by the centripetal force:

F_c = mω²R

`"GMm"/"R"^2` = mω²R

`"G"/"R"^2xx2pi"KR"^2` = ω²R

2πKG = `(4pi^2)/"T"^2`R

T² = `(2pi"R")/"KG"`

T² ∝ R