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Topic Question Set

Q 1 :

A spherically symmetric gravitational system of particles has a mass density ρ={ρ0forr≤R0forr>R
where ρ0 is a constant. A test mass can undergo circular motion under the influence of the gravitational field of particles. Its speed v as a function of distance r(0<r<∞) from the centre of the system is represented by –                           [2008]

  • [IMAGE 274]

     

  • [IMAGE 275]

     

  • [IMAGE 276]

     

  • [IMAGE 277]

     

(3)

For r≥R

mv2r=m×[GMr2]  where M is the total mass of the spherical system.

∴ v∝1r                                            (∵M=(43πR3)ρ0)

For r<R

∴  mv2r=m[GMrR3] ⇒ v∝r

i.e., v–r graph is a straight line passing through origin.