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 exact value of v_1/v_2, we can use Kepler's Third Law of Planetary Motion. This law states that the square of the period of revolution of a planet (or in this case, a star) around its center of mass is directly proportional to the cube of the semi-major axis of its elliptical orbit.
Since the fixed distance between the stars is the same in both cases, we can assume that the semi-major axis of the elliptical orbit is the same as well. Therefore, we can set up the following equation based on Kepler's Third Law:
(T_1^2 / T_2^2) = (a_1^3 / a_2^3)
Now, let's calculate the values of T_1 and T_2:
T_1 = Period of revolution for the star with mass 20m
T_2 = Period of revolution for the star with mass 14m
The period of revolution for an object in a circular orbit can be calculated using the formula:
T = 2π * (r / v)
where T is the period, r is the radius of the orbit, and v is the velocity of the object.
In this case, the fixed distance between the stars can be considered as the radius of the orbit. So, r_1 and r_2 are the same. Therefore:
T_1 / T_2 = v_2 / v_1 (since r_1 / r_2 = 1)
To find the exact value of v_1/v_2, we need to determine the values of T_1 and T_2.
Using the formula for the period of revolution, we have:
T_1 = 2π * (r_1 / v_1) (1)
T_2 = 2π * (r_1 / v_2) (2)
Now, substitute these expressions for T_1 and T_2 in the equation T_1 / T_2 = v_2 / v_1:
(2π * (r_1 / v_1)) / (2π * (r_1 / v_2)) = v_2 / v_1
Simplifying:
v_2 / v_1 = v_2 / v_1
Therefore, the exact value of v_1 / v_2 is 1.