Calculating Orbital Period of 3 Planets with Similar Mass

[woaini]

Homework Statement

The three planets (v1, v2 and v3) in the diagram all have similar mass and are in a line equally spaced so that v1 and v3 are orbiting around v2 synchronously. If the mass of each of the planets are M and the radius of the orbit is R, what is the orbital period?

Homework Equations

T=$\frac{2*pi*r}{v}$

v=$\sqrt{GM/r}$

Fg=GMm/r^2

The Attempt at a Solution

I am pretty sure you just need to set two equations equal to each other and solve for the variable T, but I am unsure which two equations this is. It would be appreciated if somebody could explain all this to me. Thank you.

 

[rude man]

One equation.
Equate the centripetal force of v1 to the gravitational pull exerted on v1 by v2 and v3.

 

[woaini]

One equation.
Equate the centripetal force of v1 to the gravitational pull exerted on v1 by v2 and v3.

$\frac{Mv^2}{R}$=$\frac{2*G*M*M}{R^2}$

v=$\sqrt{2GM}$

T=$\frac{2*pi*R}{v}$=$\frac{2*pi*R}{sqrt(2GM)}$

 

[rude man]

$\frac{Mv^2}{R}$=$\frac{2*G*M*M}{R^2}$

v=$\sqrt{2GM}$

T=$\frac{2*pi*R}{v}$=$\frac{2*pi*R}{sqrt(2GM)}$

You don't have the correct expression for the total gravitationl attraction of v2 and v3 on v1.

I also suggest changing mv^2/R to m(w^2)R.

 

[woaini]

You don't have the correct expression for the total gravitationl attraction of v2 and v3 on v1.

I also suggest changing mv^2/R to m(w^2)R.

How would I find the total gravitational attraction? Since they both have the same radius and masses, I assume that the gravitational forces would be the same and therefore have two times that.

 

[rude man]

How would I find the total gravitational attraction? Since they both have the same radius and masses, I assume that the gravitational forces would be the same and therefore have two times that.

Look at the picture. Is the distance from v1 to v2 the same as the distance of v1 to v3?

 

[woaini]

Look at the picture. Is the distance from v1 to v2 the same as the distance of v1 to v3?

$\frac{Mv^2}{R}$=$\frac{G*M*M}{R^2}$+$\frac{G*M*M}{2R^2}$

v=$\sqrt{3GM}$

T=$\frac{2*pi*r}{\sqrt{3GM}}$

 

[rude man]

Still not right. re-examine the second term on the right.

 

[woaini]

Still not right. re-examine the second term on the right.

$\frac{Mv^2}{R}$=$\frac{G*M*M}{R^2}$+$\frac{G*M*M}{2R^2}$

v=$\sqrt{5/4*GM}$

T=$\frac{2*pi*r}{\sqrt{5/4*GM}}$

 

[rude man]

$\frac{Mv^2}{R}$=$\frac{G*M*M}{R^2}$+$\frac{G*M*M}{2R^2}$

v=$\sqrt{5/4*GM}$

T=$\frac{2*pi*r}{\sqrt{5/4*GM}}$

What is the distance between v1 and v3? Ergo, what is the grav. attraction between them?

 

[woaini]

What is the distance between v1 and v3? Ergo, what is the grav. attraction between them?

Isn't the distance 2R?

Alright, I figured out I simplified to v wrong resulting in a wrong answer.

 

[tms]

Isn't the distance 2R?

Yes, but when you write it, you should write $(2R)^2$, not $2R^2$.

 

[woaini]

Yes, but when you write it, you should write $(2R)^2$, not $2R^2$.

Yes, so it's 4R^2

 

[rude man]

Yes, so it's 4R^2

So it is.