Question 40 of 223

The mass density of a spherical body is given by $\rho \left(r\right)=\dfrac{k}{r}$ for $r\leq R$ and $\rho \left(r\right)=0$ for $r\gt R,$ 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:

ABCD

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Answer: (A)

For Inside point $\left(r\leq R\right)$

$g=E=\dfrac{G}{{r}^{2}}\int _{0}^{r}\left(4\pi {r}^{2}dr\right)\left(\dfrac{K}{r}\right)=\dfrac{GK}{{r}^{2}}\times \dfrac{4\pi {r}^{2}}{2}=2\pi GK=constant$

For outside point $\left(r\gt R\right)$

$E=g E=\dfrac{GM}{{r}^{2}} E=\dfrac{G}{{r}^{2}}\int _{0}^{R}(4\pi {r}^{2}d{r}^{'})(\dfrac{K}{{r}^{'}}) E=\dfrac{4\pi KG}{{r}^{2}}{[\dfrac{{{r}^{'}}^{2}}{2}]}_{0}^{R}$

$E=\dfrac{2\pi KG}{{r}^{2}}{R}^{2}$

$E\propto \dfrac{1}{{r}^{2}}$.