In a binary star system in which 2 stars orbit each other about their centre of mass, the mass of one star is 20m while the other star is m. In another binary star system, the masses of the star are 14m and m. The fixed distance between the stars in the both cases are the same. Let v_1 be the velocity of 20m mass. Let v_2 be the velocity of the 14m mass. Find the exact value of v_1/v_2 ?

To find the ratio of velocities (v_1/v_2) in the two binary star systems, we need to use the principle of conservation of angular momentum.

In a binary star system, the total angular momentum is conserved, which means that the sum of the angular momenta of the two stars remains constant.

Angular momentum (L) is given by the product of the moment of inertia (I) and the angular velocity (ω): L = I * ω.

In this case, the moment of inertia is proportional to the mass of each star, and the angular velocity is inversely proportional to the distance between the stars. Since the distance between the stars is fixed in both cases, we can assume that the angular velocities are the same.

Now, let's calculate the angular momentum for each star system and equate them.

For the first binary star system:
Angular momentum of the 20m mass star (L_1) = I_1 * ω
Angular momentum of the m mass star (L_2) = I_2 * ω

Since the distance between the stars is the same in both systems and the angular velocities are the same, the ratio of the angular momenta is directly proportional to the ratio of the moment of inertia, which is directly proportional to the ratio of masses.

So, we have:
L_1 / L_2 = (mass of the 20m star) / (mass of the m star)
L_1 / L_2 = 20m / m
L_1 / L_2 = 20

Now, the angular momentum of each star is given by the product of its mass, velocity, and distance to the centre of mass. So, we have:
L_1 = (20m) * v_1 * d
L_2 = m * v_2 * d

Where d is the fixed distance between the stars.

Substituting these values into the previous equation:
(20m * v_1 * d) / (m * v_2 * d) = 20

The distance (d) and mass (m) cancel out, leaving us with:
v_1 / v_2 = 20

Therefore, the exact value of v_1/v_2 is 20.