Calculating Orbital Period of 3 Planets with Similar Mass

In summary, the three planets (v1, v2 and v3) with similar mass and in a line equally spaced have an orbital period given by T = 2*pi*R/sqrt(5/4*GM), where R is the radius of the orbit and GM is the total gravitational attraction between the planets. This can be found by equating the centripetal force of v1 to the gravitational pull exerted on v1 by v2 and v3. The distance between v1 and v3 is 2R, resulting in a total gravitational attraction of 4R^2.
  • #1
woaini
58
0

Homework Statement



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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=[itex]\frac{2*pi*r}{v}[/itex]

v=[itex]\sqrt{GM/r}[/itex]

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.
 
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  • #2
One equation.
Equate the centripetal force of v1 to the gravitational pull exerted on v1 by v2 and v3.
 
  • #3
rude man said:
One equation.
Equate the centripetal force of v1 to the gravitational pull exerted on v1 by v2 and v3.

[itex]\frac{Mv^2}{R}[/itex]=[itex]\frac{2*G*M*M}{R^2}[/itex]

v=[itex]\sqrt{2GM}[/itex]

T=[itex]\frac{2*pi*R}{v}[/itex]=[itex]\frac{2*pi*R}{sqrt(2GM)}[/itex]
 
  • #4
woaini said:
[itex]\frac{Mv^2}{R}[/itex]=[itex]\frac{2*G*M*M}{R^2}[/itex]

v=[itex]\sqrt{2GM}[/itex]

T=[itex]\frac{2*pi*R}{v}[/itex]=[itex]\frac{2*pi*R}{sqrt(2GM)}[/itex]

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.
 
  • #5
rude man said:
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.
 
  • #6
woaini said:
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?
 
  • #7
rude man said:
Look at the picture. Is the distance from v1 to v2 the same as the distance of v1 to v3?


[itex]\frac{Mv^2}{R}[/itex]=[itex]\frac{G*M*M}{R^2}[/itex]+[itex]\frac{G*M*M}{2R^2}[/itex]

v=[itex]\sqrt{3GM}[/itex]

T=[itex]\frac{2*pi*r}{\sqrt{3GM}}[/itex]
 
  • #8
Still not right. re-examine the second term on the right.
 
  • #9
rude man said:
Still not right. re-examine the second term on the right.

[itex]\frac{Mv^2}{R}[/itex]=[itex]\frac{G*M*M}{R^2}[/itex]+[itex]\frac{G*M*M}{2R^2}[/itex]

v=[itex]\sqrt{5/4*GM}[/itex]

T=[itex]\frac{2*pi*r}{\sqrt{5/4*GM}}[/itex]
 
  • #10
woaini said:
[itex]\frac{Mv^2}{R}[/itex]=[itex]\frac{G*M*M}{R^2}[/itex]+[itex]\frac{G*M*M}{2R^2}[/itex]

v=[itex]\sqrt{5/4*GM}[/itex]

T=[itex]\frac{2*pi*r}{\sqrt{5/4*GM}}[/itex]

What is the distance between v1 and v3? Ergo, what is the grav. attraction between them?
 
  • #11
rude man said:
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.
 
Last edited:
  • #12
woaini said:
Isn't the distance 2R?
Yes, but when you write it, you should write [itex](2R)^2[/itex], not [itex]2R^2[/itex].
 
  • #13
tms said:
Yes, but when you write it, you should write [itex](2R)^2[/itex], not [itex]2R^2[/itex].

Yes, so it's 4R^2
 
  • #14
woaini said:
Yes, so it's 4R^2

So it is.
 

What is the formula for calculating orbital period?

The formula for calculating orbital period is T = 2π√(a^3/GM), where T is the orbital period, a is the semi-major axis, G is the gravitational constant, and M is the combined mass of the two bodies in the system.

How do I find the semi-major axis of a planet?

The semi-major axis can be found by measuring the distance from the center of the planet to the center of its orbit on the longest axis of the elliptical orbit. This is also known as the average distance between the planet and its star.

What is the gravitational constant?

The gravitational constant, denoted as G, is a physical constant that represents the strength of the gravitational force between two objects with mass. Its value is approximately 6.67 x 10^-11 N*m^2/kg^2.

Can the formula for calculating orbital period be used for any two bodies?

Yes, the formula can be used for any two bodies with mass in orbit around each other, such as planets and their moons, or a star and a planet. It is based on Kepler's Third Law of Planetary Motion.

How are orbital periods affected by the mass of the bodies?

The mass of the bodies does affect the orbital period, as seen in the formula. The larger the mass, the longer the orbital period. This means that planets with similar masses will have similar orbital periods, while a planet with a much larger or smaller mass than its neighbor will have a longer or shorter orbital period, respectively.

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