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Four particles each of mass 'm' are loca...

Four particles each of mass 'm' are located at the four corners of a sqaure of side a. If the system is revolving in a circle due to thei mutual force of attraction then the angular velocity and time period of each particle is

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The gravitational force between the masses provides the necessary centripetal force
`(Gm_1m_2)/(d^2)=m_(1)r_(1)omega^2 rarr(1)`
The distance of centre of mass from `m_1` is
`r_(1)=(m_2d)/(m_1+m_2)rarr(2)`
From (1) and (2) `(Gm_1m_2)/d^2 =(m_1m_2d)/(m_1+m_2).omega^2`
(or) `omega^(2)=(G(m_1+m_2))/d^3 ` (or) `omega=sqrt((G(m_1+m_2))/d^3)`
The resultant gravitational force on one of the particles
This gravitational force provides the necessary centripetal force
`mromega^2=(Gm)/(a^2)(sqrt2+1/2)implies`
`a/sqrt2omega^2=(Gm)/a^2((2sqrt2+1)/2)impliesomega=sqrt((Gm)/a^3(2+1/sqrt2)) and T =(2pi)/omega=2pisqrt((a^3)/(Gm)((sqrt2)/(2sqrt2+1))`
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