A binary star system consists of two stars S1 and S2 with masses m and 2m respectively separated by a distance r. If both S1and S2 individually follow circular orbits around the centre of mass with instantaneous speeds V, and V2 respectively the speeds ratio V1/ V2is____________ .Correct answer is '2'. Can you explain this answer?

Question

Answer

Explanation:

Let the distance of S1 from center of mass be x and distance of S2 from center of mass be y. The total distance between the two stars will be r = x + y.

Using the formula for center of mass, we can say that mx = 2my or x = 2y, as S2 has twice the mass of S1.

Now, using the formula for centripetal force, we can say that Fc = (mv²)/r, where Fc is the centripetal force, m is the mass of the star, v is the instantaneous velocity and r is the distance from the center of mass.

Since both stars are in circular orbits, the centripetal force acting on them is provided by the gravitational force of attraction between them. Hence, we can say that Fg = Fc.

Using the formula for gravitational force, we can say that Fg = (Gm1m2)/r², where G is the gravitational constant, m1 and m2 are the masses of the stars and r is the distance between them.

Equating Fc and Fg, we get (mv²)/r = (Gm1m2)/r².

Simplifying this equation, we get v² = (Gm2)/r, as m1 = m.

Substituting the value of x in terms of y, we get r = 3y. Substituting this in the above equation, we get V1²/V2² = m/(2m) or V1/V2 = sqrt(2).

Hence, the speed ratio V1/V2 is 2.

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