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A spherically symmetric gravitational sy...

A spherically symmetric gravitational system of particles has a mass density` rho={(rho_0,for, r,lt,R),(0,for,r,gt,R):}` where`rho_0` 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 distahce `r(0ltrltOO)` form the centre of the system is represented by

Text Solution

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The correct Answer is:
`C`
Case I:`rgeR`
Force on the rest mass can be given by `F=mgr/R` where `mg` is the force on the mass at the surface of the sphere.
`mgr/R=(mv^(2))/r`
`v=sqrt(g/R)r`
`v propr` for `rgeR`
Case II: `rleR`
When the test mass is outside the sphere


`:. (GMm)/(r^(2))=(mv^(2))/r,v=sqrt((GM)/r)=sqrt(g/r)R`
`impliesv prop1/(sqrt(r))`
Therefore, choice `C` is correct the graph is asshiwn in alternative
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