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Consider a spherical gaseous cloud of ma...

Consider a spherical gaseous cloud of mass density `rho(r)` in a free space where r is the radial distance from its centre. The gaseous cloud is made of particle of equal mass m 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. If `rho(r)` is constant with time. the particle number density n(r)=`rho(r)` /m is : (g =universal gravitational constant)

A

`(k)/(2pi r^2 m^2 G)`

B

`(k )/(pi r^2m^2 G)`

C

`(3k)/(pi r^2 m^2 G)`

D

`(k )/(6pi r^2 m^2 G)`

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Knowledge Check

  • The kinetic energy K of a particle moving along a circle of radius R depends upon the distance s as K=as^2 . The force acting on the particle is

    A
    (a) `2a(s^2)/(R)`
    B
    (b) `2as[1+(s^2)/(R)]^(1//2)`
    C
    (c) `2as`
    D
    (d) `2a`

  • The kinetic energy K of a particle moving along a circle of radius R depends on the distance covered a as K=as^(2) . The force acting on the particle is

    A
    `2as^(2)//R`
    B
    `2as[1+(s^(2)//R^(2))]^(1//2)`
    C
    `2as`
    D
    `2aR^(2)//s`

  • Three particles of equal mass M each are moving on a circular path with radius r under their mutual gravitational attraction. The speed of each particle is

    A
    `sqrt((GM)/(sqrt(2)R ))`
    B
    `sqrt((GM)/(sqrt(3)R ))`
    C
    `sqrt((GM)/(2R ))`
    D
    `sqrt((GM)/(3R ))`

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