A spherically symmetric gravitational system of particles has a mass density ρ = ρ 0 for r ≤ R and ρ = 0 for r R , where ρ 0 is a constant. A test mass can undergo circular motion under the influence of the gravitational field of the particles. Which figure represents its speed v as a function of distance r 0 r ∞ from the center of the system?

Question

A spherically symmetric gravitational system of particles has a mass density ρ = ρ 0 for r ≤ R and ρ = 0 for r > R , where ρ 0 is a constant. A test mass can undergo circular motion under the influence of the gravitational field of the particles. Which figure represents its speed v as a function of distance r 0 r ∞ from the center of the system?, , , ,

MARKS App · Accepted Answer

If M is the total mass of the system of particles, then similar to a satellite revolving around the earth, the orbital velocity of the test mass is given by v = G M r For r ≤ R , v = G ⋅ 4 3 π r 3 ρ 0 r i.e. v ∝ r 2 or v ∝ r i.e. v increases linearly with r up to r = R Thus ( 1 ) and ( 4 ) are wrong but ( 2 ) and ( 3 ) satisfy this. ∴ For r > R , v = G M r or v ∝ 1 r Thus is correctly shown in figure ( 2 ) . Figure ( 3 ) is wrong because for r > R , v is shown as constant.

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