question_answer

Consider two solid spheres of radii x\[{{R}_{1}}=1\text{ }m,\]

\[{{R}_{2}}=2\text{ }m\] and masses \[{{M}_{1}}\]and \[{{M}_{2}},\]respectively.

The gravitational field due to sphere  and  are shown. The value of\[\frac{{{M}_{1}}}{{{M}_{2}}}\] is

[JEE MAIN Held On 08-01-2020 Morning]

A)  \[\frac{1}{6}\]             

B)       \[\frac{1}{2}\] C)  \[\frac{2}{3}\]              D)       \[\frac{1}{3}\] Correct Answer: A Solution : [a] From the diagram

Gravitation field at the surface

\[E=\frac{Gm}{{{r}^{2}}}\]

\[\therefore {{E}_{1}}=\frac{G{{m}_{1}}}{{{r}^{2}}_{1}}\]

and \[{{E}_{2}}=\frac{G{{m}_{2}}}{{{r}^{2}}_{2}}\]

\[\therefore \frac{{{E}_{1}}}{{{E}_{2}}}={{\left( \frac{{{r}_{2}}}{{{r}_{1}}} \right)}^{2}}\left( \frac{{{m}_{1}}}{{{m}_{2}}} \right)\]

\[\therefore \frac{2}{3}={{\left( \frac{2}{1} \right)}^{2}}\left( \frac{{{m}_{1}}}{{{m}_{2}}} \right)\]

\[\Rightarrow \left( \frac{{{m}_{1}}}{{{m}_{2}}} \right)=\frac{1}{6}\]