Physics & Geometry: Solving Quaternary Star System Homework

[Bashyboy]

Homework Statement

A certain quaternary star system consists of three stars, each of mass m, moving in the same circular orbit of radius r about a central star of mass M. The stars orbit in the same sense and are positioned one-third of a revolution apart from one another. Show that the period of each of the three stars is given by

Homework Equations

The Attempt at a Solution

What I am having difficulty is with the geometry of this problem. I attached a diagram that the answer key provides. How am I to know that the three planets form a equilateral triangle, what betokens this. Likewise, why is the angle between two sides of the triangle 60 degrees?

 

[phinds]

positioned one-third of a revolution apart from one another

is that not clear?

 

[462chevelle]

think about it.
360/3=120
so if everything is equal. wherever each point is has to equal said number
and a triangle =180 and divided by 3 equals...

 

[tiny-tim]

Hi Bashyboy!

How am I to know that the three planets form a equilateral triangle, what betokens this. Likewise, why is the angle between two sides of the triangle 60 degrees?

If they move uniformly in a circle, the total force on one of of the outer planets from the other two must point towards the centre …

isn't it obvious then that the position must be symmetic?

(and the angles of an equilateral triangle must be 60° because the angles of any triangle must add to 180°)

 

[Bashyboy]

Well, I had already determined that a 120 degree angle was was maintained between adjacent planets. However, I was not sure if that related to anything.

Why must the angles of any triangle sum to 180 degrees, does this follow from some definition?

 

[tiny-tim]

Why must the angles of any triangle sum to 180 degrees, does this follow from some definition?

erm … you should be able to prove this in about 17 different ways!

(eg divide the triangle into two right-angled triangles)

you need to study an elementary geometry book!​

 

[Bashyboy]

I have another question, would the force of one planet on another produce some tangential acceleration?

 

[haruspex]

I have another question, would the force of one planet on another produce some tangential acceleration?

Yes, but by symmetry the tangential affects of each pair on the third cancel.

 

[Bashyboy]

Oh, I see. There are two tangential forces acting on each planet, each of which is equal and opposite to each other, is this correct?

 

[tiny-tim]

Oh, I see. There are two tangential forces acting on each planet, each of which is equal and opposite to each other, is this correct?

yes, but you're analysing this too much …

isn't it obvious that, if you have two planets of the same mass at the same distance, then the total force will be toward their midpoint?

 

[Bashyboy]

No, it is not immediately evident; however, after having analyzed the problem, I can see that. I don't think I am analyzing the problem too much, I want to understand every detail of every problem I solve.

 

[Bashyboy]

Tiny Tim, is what you say always true?

 

[tiny-tim]

isn't it obvious that, if you have two planets of the same mass at the same distance, then the total force will be toward their midpoint?

Tiny Tim, is what you say always true?

yes, because of symmetry …

if you reflect it in a mirror (through the midpoint), you'll have exactly the same …

so the force in the reflected situation must be same as the original force, in other words it must be its own reflection, in other words it must be in the mirror itself, ie towards (or away from) the midpoint

it's this concept of symmetry that you're missing …

many physics exam problems are deliberately constructed with a symmetry in, to help you and to save you time

it is perfectly acceptable in an exam to say "from symmetry, it is obvious that …"​

you need to think about symmetry a lot (sorry, but it isn't really a subject you can look up in books), until you're used to spotting it, and using it!