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एक गोलाकार पिण्ड का द्रव्यमान घनत्व है
$\rho( r )=\frac{ k }{ r }$ जब $r \leq R$ तथा
$\rho( r )=0$ जब $r > R$, जहाँ $r$ केन्द्र से दूरी है।
निम्न में से कौन सा ग्राफ़ एक परीक्षण कण के त्वरण $a$ को $r$ के फलन में गुणात्मक रूप से दर्शाता है ?
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.
