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Consider a spherical gaseous cloud of mass density ρ(r) in free space where r is the radial distance from its center. The gaseous cloud is made of particles of equal mass m moving in circular orbits about the common center with the same kinetic energy K. The force acting on the particle is their mutual gravitational force. If ρ(r) is constant in time, the particle number density n(r) = ρ(r)/m is (G is universal gravitational constant).
- 3K / πr²m²G
- K / 2πr²m²G
- K / πr²m²G
- K / 6πr²m²G
Correct answer: K / 2πr²m²G
Solution
The gravitational force provides the centripetal force for circular motion. Using the density ρ(r) and equating forces, the particle number density n(r) is derived as K / 2πr²m²G.
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