What is the period of rotation for three stars forming an equilateral triangle?

[DRC12]

Homework Statement

Three stars, each with the mass of our sun (1.9891E31 kg), form an equilateral triangle with sides long. The triangle has to rotate, because otherwise the stars would crash together in the center. What is the period of rotation?

Homework Equations

F₁=GM²/r²
ƩF=0
F₂=ma
a=4π²r/T²

The Attempt at a Solution

F₁=2.639E26
Because the sum of the forces is equal to zero F₁=F₂
2.639E26=(1.9891E31)*a
a=.000133m/s²=4π²/T²
T=5.45E8 s

Do I have to do something else because there's three stars instead of two?

 

[gneill]

What is the length of the sides of the triangle?

It's difficult to judge what calculations you're performing when you only show results; show more of your work.

Yes, you need to take all the masses into account. Think summation of vectors.

 

[DRC12]

The length of the sides of the triangle is 1E12m.
F₁=(6.67E-11)(1.9891E30kg)/(1E12)²=2.639E26
Because it's a equilateral triangle all angles are 60 so I when summing the vectors I made one side at 0^o and the other 60^o above it so the coordinates of the first side is F_x=2.639E26 F_y=0 and the coordinates of the other side is F_x=2.639E26 cos(60)=1.32E26 F_y=2.639E26 sin(60)=2.29E26.

the total F_x=3.959E26 and F_y=2.29E26 F=(F_x²+F_y²)^.5=4.57E26

a=F/m so a=4.57E26/1.9891E30=2.3E-4
a=4π²r/T²
2.3E-4=4π²*1E12/T²
T=4.14E8

The correct answer: 3.15E8

 

[gneill]

The length of the sides of the triangle is 1E12m.
F₁=(6.67E-11)(1.9891E30kg)/(1E12)²=2.639E26
Because it's a equilateral triangle all angles are 60 so I when summing the vectors I made one side at 0^o and the other 60^o above it so the coordinates of the first side is F_x=2.639E26 F_y=0 and the coordinates of the other side is F_x=2.639E26 cos(60)=1.32E26 F_y=2.639E26 sin(60)=2.29E26.

the total F_x=3.959E26 and F_y=2.29E26 F=(F_x²+F_y²)^.5=4.57E26

a=F/m so a=4.57E26/1.9891E30=2.3E-4
a=4π²r/T²
2.3E-4=4π²*1E12/T²
T=4.14E8

The correct answer: 3.15E8

Ah. Well the radius of the circle around which the stars are rotating is not the same as the length of the triangle sides.

 

[DRC12]

Yes that was the problem thanks for the help