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A spherical gas cloud in free space has mass density rho(r), r being the distance from the centre. It consists of identical particles of mass m moving in circular orbits about the centre, each with the same kinetic energy K, bound only by their mutual gravitation. If rho(r) is steady in time, the particle number density n(r) = rho(r)/m equals: (G = universal gravitational constant)
- K/(pi*r²*m²*G)
- K/(6*pi*r²*m²*G)
- 3K/(pi*r²*m²*G)
- K/(2*pi*r²*m²*G)
Correct answer: K/(2*pi*r²*m²*G)
Solution
From K = (1/2)m v² the speed is fixed. The circular-orbit balance gives the enclosed mass linear in r; differentiating yields the density and hence the number density.
Related JEE Advanced Physics questions
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