JEE MAIN - Physics (2021 - 24th February Morning Shift - No. 16)
Two stars of masses m and 2m at a distance d rotate about their common centre of mass in free space. The period of revolution is :
$${1 \over {2\pi }}\sqrt {{{{d^3}} \over {3Gm}}} $$
$$2\pi \sqrt {{{3Gm} \over {{d^3}}}} $$
$${1 \over {2\pi }}\sqrt {{{3Gm} \over {{d^3}}}} $$
$$2\pi \sqrt {{{{d^3}} \over {3Gm}}} $$
Explanation
The given situation is shown below
_24th_February_Morning_Shift_en_16_1.png)
The gravitational force between these two stars provide the required centripetal force for rotation in a circle about their common centre.
Assuming 2 m at origin, the centre of mass of the system lies at
$$x = {{2m \times 0 + m \times d} \over {2m + m}} = {d \over 3}$$
Hence, $${F_G} = {F_C}$$
where, FG is gravitational force between them and FC is centripetal force.
$$ \Rightarrow {{G{m_1}{m_2}} \over {{r^2}}} = 2m{\omega ^2}x$$
$$ \Rightarrow {{G(2m)(m)} \over {{d^2}}} = 2m{\omega ^2} \times {d \over 3} \Rightarrow {\omega ^2} = {{3Gm} \over {{d^3}}}$$
$$ \Rightarrow \omega = \sqrt {{{3Gm} \over {{d^3}}}} $$
We know that,
$$\omega = {{2\pi } \over T}$$
$$\therefore$$ $$T = {{2\pi } \over \omega }$$
$$\sqrt {{{3Gm} \over {{d^3}}}} $$
$$ = 2\pi \sqrt {{{{d^3}} \over {3Gm}}} $$ [using Eq. (i)]
The gravitational force between these two stars provide the required centripetal force for rotation in a circle about their common centre.
Assuming 2 m at origin, the centre of mass of the system lies at
$$x = {{2m \times 0 + m \times d} \over {2m + m}} = {d \over 3}$$
Hence, $${F_G} = {F_C}$$
where, FG is gravitational force between them and FC is centripetal force.
$$ \Rightarrow {{G{m_1}{m_2}} \over {{r^2}}} = 2m{\omega ^2}x$$
$$ \Rightarrow {{G(2m)(m)} \over {{d^2}}} = 2m{\omega ^2} \times {d \over 3} \Rightarrow {\omega ^2} = {{3Gm} \over {{d^3}}}$$
$$ \Rightarrow \omega = \sqrt {{{3Gm} \over {{d^3}}}} $$
We know that,
$$\omega = {{2\pi } \over T}$$
$$\therefore$$ $$T = {{2\pi } \over \omega }$$
$$\sqrt {{{3Gm} \over {{d^3}}}} $$
$$ = 2\pi \sqrt {{{{d^3}} \over {3Gm}}} $$ [using Eq. (i)]
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