Gravitation Part 1

The given situation is shown below The gravitational force between these two stars provide the required centripetal force for rotation in a circle about their common centre. Assuming 2 mat origin, the centre of mass of the system lies at
x=‌

2m×0+m×d

2m+m

=‌

d

3

Hence,
FG=FC
where, FC is gravitational force between them and FC is centripetal force.
⇒‌‌‌

Gm1m2

r2

‌‌=2mω2x
⇒‌‌‌

G(2m)(m)

d2

‌‌=2mω2×‌

d

3

⇒ω2=‌

3Gm

d3

ω‌‌ω‌‌=√‌

3Gm

d3

We know that,
‌ω=‌

2π

T

‌T=‌

2π

ω

⇒ T‌‌=‌

2π

√‌ 3Gm d3

‌‌=2π√‌

d3

3Gm

[using Eq. (i)]