Q 11-07-198JEE MainJEE Main 2017 (9 Apr)Medium
The mass density of a spherical body is given by $\rho(r) = \dfrac kr$ for $r \le R$ and $\rho(r) = 0$ for $r > R$, where $r$ is the distance from the center. The correct graph that describes qualitatively the acceleration, $a$, of a test particle as a function of $r$ is:
Answer: (A) see figure
Mass inside radius $r \le R$:
$$M(r) = \int_0^r\frac kr'\,4\pi r'^2\,dr' = 2\pi kr^2$$
$$a = \frac{GM(r)}{r^2} = 2\pi Gk = \text{constant}\quad(r \le R)$$
Outside, $a = \dfrac{GM}{r^2}$ falls off as $1/r^2$. So $a$ is constant up to $R$ and then decreases: graph (1).
Solution by Sreeraj P, M.Sc Physics