By Team Aakash Byju's | 17th November 2022

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How to Find the Particle Number Density n(r)?

Consider a spherical gaseous cloud made of particles of equal mass m, has a mass density ρ(r) in a free space where r is the radial distance from its centre.

(Question Continues)

r

The particles are moving in circular orbits about their common centre with the same kinetic energy K. The force acting on the particles is their mutual gravitational force.

(Question Continues)

If ρ (r) is constant in time. The particle number density n(r)  =  ρ(r)   is?  (G =  universal gravitational constant)

C

2

A

K

B

D

K

K

3K

m

2

πr

m

G

2

2

πr

m

G

2

2

πr

m

G

2

6

πr

m

2

G

2

Detailed Explanation

Arrow

K

2

2

πr

m

G

2

The correct answer is A.

The net gravitational force due to all particles should provide the necessary centripetal force for the circular motion in the spherical gaseous cloud.

Therefore, F   = F

C

G

=>

GMm

r

2

-

mv

2

r

or

GMm

r

2

-

2

1

2

1

mv

(           )

2

=>

GMm

r

2

=

2k

1

Where K is the kinetic energy

=>

From the previous equation, let us express mass in terms of the kinetic energy and radius of the spherical gaseous cloud.

=

M

Gm

2Kr

=>

dM

dr

=

2k

Gm

But for an elemental sphere in the gaseous cloud, the elemental mass will be

Since mass = density × velocity, we can write the previous expression as:

=

ρdV

Gm

2K

dr

=>

4πr  drρ

dr

=

2k

Gm

Since the elemental volume in the spherical cloud of thickness dr is the product of surface area 4πr and the thickness dr.

2

dr

Isolating the density, we get

=

ρ

Gm

K

2

2π

r

=

Now, divide both sides by m to find the particle number density.

=>

ρ

m

Gm

K

2

2π

r

2

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