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  • Two massive particles of masses M & m (M > m) are separated by a distance  They rotate with equal an

Two massive particles of masses M & m (M > m) are separated by a distance \ell . They rotate with equal angular velocity under their gravitational attraction. The linear speed of the particle of mass m is

Option: 1

\mathrm{\sqrt{\frac{G M m}{(M+m) \ell}}}


Option: 2

\mathrm{\sqrt{\frac{G M^2}{(M+m) \ell}}}


Option: 3

\mathrm{\sqrt{\frac{G m}{\ell}}}


Option: 4

\mathrm{\sqrt{\frac{G m^2}{(M+m) \ell}}}


Answers (1)

best_answer

The system rotates about the centre of mass. The gravitational force acting on the particle m accelerates it towards the centre of the circular path, which has the radius

\mathrm{\begin{aligned} & \mathrm{R}=\frac{\mathrm{M} \ell}{\mathrm{M}+\mathrm{m}} \\ & \Rightarrow \mathrm{F}=\frac{\mathrm{mv}^2}{\mathrm{r}} \Rightarrow \frac{\mathrm{GMm}}{\ell^2}=\frac{\mathrm{mv^{2 }}}{\frac{\mathrm{M} \ell}{\mathrm{M}+\mathrm{m}}} \\ & \Rightarrow \mathrm{v}=\sqrt{\frac{\mathrm{GM}{ }^2}{(\mathrm{M}+\mathrm{m}) \ell}} \end{aligned}}

Hence (B) is correct

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Gunjita

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