for PHYSICS GENIUSES!?

Question

Plaskett's binary system consists of two stars that revolve in a circular orbit about a cnenter of mass midway between them. This means that the masses of the two stars are equal. If the orbital velocity is 220000 m/s and the orbital period of each is 1546560 sec, find teh mass M of each star. please include details!!!

Answer 1

The mass of the stars is 1.5712 E32 kg. Their common center-of-mass is located halfway between their centers. This distance is also their orbital radius, and is 5.4151 E10 m. Plainly, then, distance between centers of these stars is 1.083 E10 m. Gravitational force between them is 1.4043 E32 N. Formula derivation follows:

Five relationships, all well known, are required for derivation:

1. Newton's Gravitation Law: F = G M m / d²
2. Newton's Second Law of Motion: F = ma
3. Acceleration of a body in circular motion: a = v² / r
4. Angular velocity definition: ω = v / r
5. Period definition, circular motion: ω T = 2 π

Since both stars have the same mass, F = G M² / d² ; also, F = Ma. Equating these expressions,

a = G M / d²

where d = 2r ; as, according to 3., a = v² / r,

G M / 4 r² = v² / r.

Solving for M,

M = 4 v² r / G.

But, from 5., T = 2 π / ω, and, since ω = v / r, T = 2 π r / v, whence r = v T / 2 π. Substituting this in the above expression,

M = 4 v³ T / 2 π G = 2 v³ T / π G.

This formula is in terms of known data, with no exception: G = 6.6726 E-11 m³/s² kg, v = 2.2 E5 m, and T = 1.5456 E6 s. Evaluating the formula with these values yields the result already given.

Additional remarks:
mac was close, indeed, but wrong because he uses rboatright's formula, which is incorrect; rboatright uses the same symbol, r, to denote distance between the stars' centers, and uses r also for orbital radius. Plainly, they aren't the same. Using the same symbol for different distances caused his error.

Answer 2

Classic problem in astronomy... how to determine the mass of a binary star system. This is the simplest version (equal masses and velocities...)

step by step...

the attraction of the two stars to each other (The gravitational force) is

Fg = GMm/r^2

the centripital force from the circular motion is

Fc = M V^2 / r

those forces must balance for there to be an ORBIT.... so...

Since we know the masses are the same, we can solve for M....

m = r * V^2 ./ G

now, we ALSO know the period t....

and since V = 2 * pi * r / t

we can solve for r as r = v * t / 2pi

So

m = (v * t / 2pi ) * V^2 / G

Substitute v = 220000 m/s and t = 1546560 sec

m = (220000 * 1546560 / 2 / 3.14) * 220000^2 / 6.7 * 10^(-11)

m = 3.91 × 10^9

Answer 3

d_a_r_a_n_c_e: Even with the junior high math I never use, I know how to get the radius from the information given.
distance = velocity x time
2.2 x 10^5 m/s = velocity
1.546560 x 10^5 s = time
3.402432 x 10^11 = distance = circumference of orbit
radius = circumference / 2 pi = 5.4151387133 x 10^10
I can't answer the actual question.

Answer 4

i agree with rboatrigh's final fomula M= (rv^2) / G but i disagree with his calculation.

The radius is 5.41x10^10m

M= 5.41x10^10*220000^2 / 6.67x10^-11

The answer should be 3.92x10^31 kg