Two stars in a binary system orbit around their center of mass. The centers of the two stars are 7.40 1011 m apart. The larger of the two stars has a mass of 3.40 1030 kg, and its center is 2.50 1011 m from the system's center of mass. What is the mass of the smaller star?

To find the mass of the smaller star in the binary system, we can use the concept of center of mass. The center of mass is the point at which the two stars balance each other out.

First, let's define some variables:
M1 = mass of the larger star
M2 = mass of the smaller star
r1 = distance of the larger star's center from the center of mass
r2 = distance of the smaller star's center from the center of mass

According to the concept of center of mass, the sum of the mass of the two stars multiplied by their respective distances from the center of mass should add up to zero:
M1 * r1 + M2 * r2 = 0

From the given information:
M1 = 3.40 * 10^30 kg
r1 = 2.50 * 10^11 m
r2 = 7.40 * 10^11 m

Substituting these values into the equation, we get:
(3.40 * 10^30 kg) * (2.50 * 10^11 m) + M2 * (7.40 * 10^11 m) = 0

To solve for M2, we rearrange the equation:
M2 = -(3.40 * 10^30 kg) * (2.50 * 10^11 m) / (7.40 * 10^11 m)

Now, we can calculate the mass of the smaller star:
M2 = -(3.40 * 2.50 * 10^30 kg * 10^11 m) / (7.40 * 10^11 m)
M2 = -(8.50 * 10^30 kg * m) / (7.40)
M2 ≈ - 11.49 * 10^30 kg

Since mass cannot be negative, we discard the negative sign and take the absolute value of M2:
M2 ≈ 11.49 * 10^30 kg

Therefore, the mass of the smaller star in the binary system is approximately 11.49 * 10^30 kg.