- Home
- Standard 11
- Physics
The mass density of a spherical body is given by $\rho \left( r \right) = \frac{k}{r}$ for $r \leq R\,\,$ and $\rho \left( r \right) = 0\,$ 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
Solution
Give that, mass desity $\left( {\frac{{mass}}{{volume}}} \right)$ of a
spherical body $\rho \left( r \right) = \frac{k}{r}$
$\frac{M}{V} = \frac{k}{r}\,for\,inside\,r \le R$
$M = \frac{{kv}}{r}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,…\left( i \right)$
Inside the surface of sphere Intensity
$I = \frac{{GMr}}{{{R^3}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,I = \frac{F}{m}$
${g_{inside}} = \frac{{GMr}}{{{R^3}}}\,\,\,\,\,\,\,\,\,\,\,or\,\,\,\,\,\,\,\,\,\,I = \frac{{mg}}{m} = g$
$ = \frac{G}{{{R^3}}}\frac{{kv}}{r}.r = constant\,\,\,\,\,From\,eq.\,\left( i \right),$
$similarly,\,{g_{out}} = \frac{{GM}}{{{r^2}}}$
Hence, option $(b)$ is correct graph.
