Question 192: Gravitation

Question

Four identical particles of mass M are located
at the corners of a square of side 'a'. What
should be their speed if each of them revolves
under the influence of other's gravitational field
in a circular orbit circumscribing the square?
![](https://imagex.cdn.examgoal.net/mdsTt65wD7MG10kSf/qTMRFYU6mebUjS95b6xKpBlg9U8TA/twYhxAAuoZ473gkVXZSvST/image.png)

Accepted Answer

Net force on mass M at position B towards centre of circle is
${F_{BO\,net}} = {F_{BD}} + {F_{BA}}\sin 45^\circ  + {F_{BC}}\sin 45^\circ $
$ = {{G{M^2}} \over {{{(\sqrt 2 a)}^2}}} + {{G{M^2}} \over {{a^2}}}\left( {{1 \over {\sqrt 2 }}} \right) + {{G{M^2}} \over {{a^2}}}\left( {{1 \over {\sqrt 2 }}} \right)$
[where, diagonal length BD is $\sqrt2$a]
$ = {{G{M^2}} \over {2{a^2}}} + {{G{M^2}} \over {{a^2}}}\left( {{2 \over {\sqrt 2 }}} \right) = {{G{M^2}} \over {{a^2}}}\left( {{1 \over 2} + \sqrt 2 } \right)$
This force will act as centripetal force.
Distance of particle from centre of circle is ${a \over {\sqrt 2 }}$.
Here, ${F_{centripetal}} = {{M{v^2}} \over r} = {{M{v^2}} \over {{a \over {\sqrt 2 }}}} = {{\sqrt 2 M{v^2}} \over a}$ ($\because$ $r = {a \over {\sqrt 2 }}$)
So, for rotation about the centre,
${F_{centripetal}} = {F_{BO(net)}}$
$ \Rightarrow \sqrt 2 {{M{v^2}} \over a} = {{G{M^2}} \over {{a^2}}}\left( {{1 \over 2} + \sqrt 2 } \right)$
$ \Rightarrow {v^2} = {{GM} \over a}\left( {1 + {1 \over {2\sqrt 2 }}} \right) = {{GM} \over a}(1.35)$
$ \Rightarrow v = 1.16\sqrt {{{GM} \over a}} $
![](https://app-content.cdn.examgoal.net/fly/@width/image/1l383442o/6d39c8d1-54cc-4b00-8e6b-e0333f02708a/b70faa00-d4bc-11ec-ad28-abe411cf4979/file-1l383442p.png?format=png)

Answer

$1.21\sqrt {{{GM} \over a}} $

Answer

$1.16\sqrt {{{GM} \over a}} $

Answer

$1.41\sqrt {{{GM} \over a}} $

Answer

$1.35\sqrt {{{GM} \over a}} $

Gravitation

Four identical particles of mass M are located at the corners of a square of side 'a'. What should be their speed if each of them revolves under the influence of other's gravitational field in a circular orbit circumscribing the square?

Choose the correct answer: