SATHEE: Gravitation 2 Question 4

Gravitation 2 Question 4

7. A spherically symmetric gravitational system of particles has a mass density $ρ = \begin{matrix} ρ_{0} for r \leq R 0 for r > R \end{matrix}$ , where $ρ_{0}$ 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 distance $r$ from the centre of the system is represented by

(a)

(b)

(c)

(d)

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Answer:

Correct Answer: 7. (c)

Solution:

For $r \leq R, \frac{m v^{2}}{r} = \frac{G m M}{r^{2}}$

Here,

$M = \frac{4}{3} π r^{3} ρ_{0}$

Substituting in Eq. (i), we get $v \propto r$

i.e. $v - r$ graph is a straight line passing through origin.

For $r > R$ ,

$\frac{m v^{2}}{r} = \frac{G m \frac{4}{3} π R^{3} ρ_{0}}{r^{2}} or v \propto \frac{1}{\sqrt{r}}$

The corresponding $v - r$ graph will be as shown in option (c).

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Gravitation 2 Question 4

Answer:

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