Two stars in a binary system orbit around their center of mass. The centers of the two stars are 7.80 x 10^11 m apart. The larger of the two stars has a mass of 3.50 x 10^30 kg, and its center is 3.00 x 10^11 m from the system's center of mass. What is the mass of the smaller star?
m2= m1/d/x-cm *-1
m2= 3.50e30 kg/ 7.80e11/ 3e11 * -1
m2= 2.9e30
To solve this question, we'll use the concept of conservation of angular momentum in a binary system.
The formula for angular momentum is: L = Iω
Here, L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
In a binary system, the angular momentum of each star is equal and opposite, so we can write:
L1 = -L2
To calculate the moment of inertia, we need to know the mass and the distance of each star from the system's center of mass.
Given data:
Mass of larger star (m1) = 3.50 x 10^30 kg
Distance of larger star from the center of mass (r1) = 3.00 x 10^11 m
Distance between the centers of the two stars (d) = 7.80 x 10^11 m
To find the moment of inertia for each star, we use the formula:
I = mr^2
For the larger star:
I1 = m1 * r1^2
For the smaller star (m2) with distance r2, its moment of inertia can be written as:
I2 = -m2 * r2^2 (Since it's in the opposite direction)
Now, let's use the concept of conservation of angular momentum:
L1 = I1 * ω = -I2 * ω
Since L = Iω, we can write:
I1 * ω1 = -I2 * ω2
Dividing the above equation by ω2:
(I1 * ω1) / ω2 = -I2
Now, let's substitute the expressions for moments of inertia:
(m1 * r1^2 * ω1) / ω2 = -(-m2 * r2^2)
Simplifying the equation, we get:
(m1 * r1^2 * ω1) / ω2 = m2 * r2^2
Given that the distance between the stars' centers is d = r1 + r2, we can write:
r2 = d - r1
Let's substitute r2 in the equation:
(m1 * r1^2 * ω1) / ω2 = m2 * (d - r1)^2
Now, let's solve for m2:
m2 = ((m1 * r1^2 * ω1) / ω2) / (d - r1)^2
Here, we have the values for m1, r1, and d. We need to calculate ω1 and ω2 before substituting these values into the equation.
To find the value of ω1, we can use the relation between angular velocity ω and linear velocity v:
v1 = r1 * ω1
From this equation, we can isolate ω1:
ω1 = v1 / r1
Since we don't have the linear velocity v1, we can calculate it using the centripetal force equation:
F1 = (m1 * v1^2) / r1
The gravitational force between the two stars provides the centripetal force for both of them. So:
F1 = F2
Using the formula for gravitational force:
F1 = G * ((m1 * m2) / d^2)
F2 = G * ((m2 * m1) / d^2)
Equating F1 and F2:
G * ((m1 * m2) / d^2) = G * ((m2 * m1) / d^2)
Canceling out common terms:
m1 * m2 = m2 * m1
Since m1 and m2 are not equal to zero, we can conclude that the masses of the two stars are the same:
m1 = m2 = m
Now, let's substitute m in place of m1 and m2:
m^2 = m * m
Taking the square root of both sides:
m = √m^2
m = |m|
Hence, the mass of the smaller star is the same as the mass of the larger star.
Therefore, the mass of the smaller star is 3.50 x 10^30 kg.