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The mass density of a spherical body is ...

The mass density of a spherical body is given by `rho(r)=k/r` for `r le R` and `rho (r)=0`for r > R , where r is the distance from the centre. The correct graph that describes qualitatively the acceleration, a, of a test particle as a function of r is :

A

B

C

D

Text Solution

Verified by Experts

The correct Answer is:
B
` rho (r ) = {{:((K)/(r )", "r le R),(0" , "r gt R):}`
Let us consider a unit mass inside the spherical body at distance r

Mass of radius `r(r le R)`
`M = underset(x = 0)overset(x = r)int rho xx 4pi x^(2). dx = underset(x = 0)overset(x = r)int (k)/(x) 4pi x^(2) dx`
`M = 2 pi k r^(2)`
Force of attraction,
`F = (GMm)/(r^(2)) = (G(2pi k r^(2))M)/(r^(2))`
`a = G(2pi k) = "constant " [r lt R]`
For `(r gt R)`
`F = (GMm)/(r^(2))`
`a = (Gm)/(r^(2)) rArr a prop (1)/(r^(2))`
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