Gravitation problem -- Binary star system

[PhysicStud01]

Homework Statement

Homework Equations

The Attempt at a Solution

the solution says to equate the moments about C or equate the centripetal forces
but the moments, they used M1R1 = M2R2
how does the above represent the moment, and why is the moment even equal?similarly, why is the centripetal force equal?

thanks in advance

 

[SteamKing]

Homework Statement

View attachment 86837

Homework Equations

The Attempt at a Solution

the solution says to equate the moments about C or equate the centripetal forces
but the moments, they used M1R1 = M2R2
how does the above represent the moment, and why is the moment even equal?

The language here is a tad imprecise. Instead of "moment" in the sense of force times distance, the text is using "moment" to represent mass times distance.

In other words, point C represents the center of gravity of the binary system, if that means anything to you.

similarly, why is the centripetal force equal?

thanks in advance

What would happen to the stars if the net force acting on them was not zero?

 

[PhysicStud01]

The language here is a tad imprecise. Instead of "moment" in the sense of force times distance, the text is using "moment" to represent mass times distance.

In other words, point C represents the center of gravity of the binary system, if that means anything to you.What would happen to the stars if the net force acting on them was not zero?

there orbit would no longer be circular, i think

but then again, they are in different orbits (of different radius) - so, how is it equal.

also, why is the centripetal force equal?

 

[haruspex]

In other words, point C represents the center of gravity of the binary system, if that means anything to

Well, centre of mass.

there orbit would no longer be circular, i think

No, it's not related to that. There are no external forces on the system. What does that tell you about how the centre of mass of the system moves?

 

[PhysicStud01]

Well, centre of mass.

No, it's not related to that. There are no external forces on the system. What does that tell you about how the centre of mass of the system moves?

it also moves circular orbit? but how is this related?

 

[haruspex]

it also moves circular orbit? but how is this related?

If there are no forces acting on a mass, how does it move?

 

[PhysicStud01]

If there are no forces acting on a mass, how does it move?

if it's moving at constnat speed, it continues.
if it's at rest, it remains at rest.

but I'm lost here. how is this related to moment. why is the centripetal force equal in both cases?

 

[haruspex]

if it's moving at constnat speed, it continues.
if it's at rest, it remains at rest.

but I'm lost here. how is this related to moment. why is the centripetal force equal in both cases?

A centripetal force is that resultant force required to produce the centripetal acceleration. It is not an applied force, but is the resultant of applied forces. What is the applied force that produces it?

 

[PhysicStud01]

A centripetal force is that resultant force required to produce the centripetal acceleration. It is not an applied force, but is the resultant of applied forces. What is the applied force that produces it?

it's the gravitational force between the 2 stars, right.

 

[Bandersnatch]

it's the gravitational force between the 2 stars, right.

Alright. Write it down for each of the stars. Is it the same both? Is it different for each?

For the stars to be moving in circles, this force, acting on each of the stars separately, has to be equal to the centripetal force. Write down the centripetal force equation for each star. Are they the same? Are they different?

Next, for each star, equate the actual (applied) force that's acting on it (gravity), with the force (resultant) required to be moving in circles. You'll end up with two sets of equations. See if you can do any algebraic magic on them.

Show us your work.

 

[PhysicStud01]

Alright. Write it down for each of the stars. Is it the same both? Is it different for each?

For the stars to be moving in circles, this force, acting on each of the stars separately, has to be equal to the centripetal force. Write down the centripetal force equation for each star. Are they the same? Are they different?

Next, for each star, equate the actual (applied) force that's acting on it (gravity), with the force (resultant) required to be moving in circles. You'll end up with two sets of equations. See if you can do any algebraic magic on them.

Show us your work.

the gravitational force is the same. i understood the thing about centriptal force. thanks.
The solution is already available, i just wanted to understand it.but one thing i have not yet understood - why are the moment equal? why did they use to equation i stated at first? is this a simplification of something else, or is it a formula itself?

 

[haruspex]

the gravitational force is the same. i understood the thing about centriptal force. thanks.
The solution is already available, i just wanted to understand it.but one thing i have not yet understood - why are the moment equal? why did they use to equation i stated at first? is this a simplification of something else, or is it a formula itself?

Where is the mass centre of the system in relation to C?

 

[tms]

but one thing i have not yet understood - why are the moment equal? why did they use to equation i stated at first? is this a simplification of something else, or is it a formula itself?

Do what was suggested above: write down $F = ma$ for each star, and compare the equations. You will then see the answer to your question. Be careful with the radii.

 

[haruspex]

Do what was suggested above: write down $F = ma$ for each star, and compare the equations. You will then see the answer to your question. Be careful with the radii.

I believe PS01 understands why the forces are the same. The question is why the "mass moments" (mass times distance from C) can be immediately written down as being the same for each without having to think about forces.

 

[PhysicStud01]

Where is the mass centre of the system in relation to C?

sorry, i don't know about this one. i assume it is at a radius r from C, but i think that this centre also moves.

Do what was suggested above: write down $F = ma$ for each star, and compare the equations. You will then see the answer to your question. Be careful with the radii.

but what is F and a here? m is the mass of the planet into consideration, right.

I believe PS01 understands why the forces are the same. The question is why the "mass moments" (mass times distance from C) can be immediately written down as being the same for each without having to think about forces.

thanks. yeah, that's why i really want to know. but understanding what the others are saying will be a plus, i believe.

 

[haruspex]

i think that this centre also moves.

If there are no external forces acting on a system then the mass centre of that system does not accelerate.

 

[PhysicStud01]

If there are no external forces acting on a system then the mass centre of that system does not accelerate.

but the stars are moving, right. this is a change direction of velocity and thus there is an acceleration. anyway, when the positions of the stars are changing, shouldn't C be changing too?

 

[tms]

but what is F and a here? m is the mass of the planet into consideration, right.

$F$ is the only force the stars exert on each other. You get $a$ from the kinematics of uniform circular motion.

 

[PhysicStud01]

$F$ is the only force the stars exert on each other. You get $a$ from the kinematics of uniform circular motion.

so F is the gravitational force between them. but could you give me the formula for a

 

[haruspex]

but the stars are moving, right. this is a change direction of velocity and thus there is an acceleration. anyway, when the positions of the stars are changing, shouldn't C be changing too?

"The system" is both stars taken together. There is no external force on that system, so the mass centre of that system cannot accelerate. $\Sigma F_{ext} = ma$.
C in the diagram is defined as a fixed point. I'm endeavouring to prove to you that it is the mass centre of the system. M1R1 = M2R2 would follow immediately from that by definition of mass centre.

 

[PhysicStud01]

"The system" is both stars taken together. There is no external force on that system, so the mass centre of that system cannot accelerate. $\Sigma F_{ext} = ma$.
C in the diagram is defined as a fixed point. I'm endeavouring to prove to you that it is the mass centre of the system. M1R1 = M2R2 would follow immediately from that by definition of mass centre.

so, when the centre is fixed at C, we can write M1R1 = M2R2. how does this formula come? is it an actual formula, for example momentum = mv?

 

[Bandersnatch]

If you follow the steps I outlined earlier, you should arrive at the equation provided.

 

[tms]

so F is the gravitational force between them. but could you give me the formula for a

Yes, $F$ is gravity. For $a$, go back and look at uniform circular motion; there is an equation that relates the angular speed to the radius and mass.

 

[tms]

so, when the centre is fixed at C, we can write M1R1 = M2R2. how does this formula come? is it an actual formula, for example momentum = mv?

It comes from the definition of the center of mass.

 

[PhysicStud01]

It comes from the definition of the center of mass.

so, this is called moments?

 

[haruspex]

so, this is called moments?

Google moment of mass.

 

[PhysicStud01]

Google moment of mass.

OK. i tried and obtained moment of inetia. is this the same as moment of mass. but the formulae there seems complicated for this level of question though

 

[haruspex]

OK. i tried and obtained moment of inetia. is this the same as moment of mass. but the formulae there seems complicated for this level of question though

Moment of inertia is also known as the second moment of mass. That's because it involves an integral of the form $\int r^2.dm$. For finding mass centre use the first moment of mass, which has the form $\int r.dm$.

 

[Angie Tom]

I don't still quite understand how the centripetal forces are equal? Aren't the radii and the masses different?

 

[Angie Tom]

Anyone?

 

[gneill]

I don't still quite understand how the centripetal forces are equal? Aren't the radii and the masses different?

In this system the centripetal forces are provided by gravity. No other forces are acting. Can the force due to gravity acting on the two bodies be different?

 

[Angie Tom]

Yes, if the masses and radii are different

 

[Angie Tom]

In this system the centripetal forces are provided by gravity. No other forces are acting. Can the force due to gravity acting on the two bodies be different?

yes, if the masses and radii are different

 

[gneill]

yes, if the masses and radii are different

No. Review Newtonian gravity. There is only one distance that matters, and that is the separation between the centers of the two spherical bodies.

 

[Angie Tom]

No. Review Newtonian gravity. There is only one distance that matters, and that is the separation between the centers of the two spherical bodies.

I know the law! attractive force between any 2 point masses is directly proportional to the product of the masses and inversely proportional to the distance squared between them right?
Can u please clarify what exactly is point c

 

[gneill]

Point C is the center of mass of the system. It's the fixed point around which both bodies orbit.

 

[Angie Tom]

I meant Is there a body at point C? star...planet?

 

[Angie Tom]

Point C is the center of mass of the system. It's the fixed point around which both bodies orbit.

View attachment 92177

I meant is there a body at point c?

 

[gneill]

I meant is there a body at point c?

No, it's empty space.

The center of mass of two spherical masses lies along the line joining them, but there's no object associated with that center of mass: it's a mathematical point.

 

[Angie Tom]

Okay

No, it's empty space.

The center of mass of two spherical masses lies along the line joining them, but there's no object associated with that center of mass: it's a mathematical point.

Okay, so this means that the gravitational force is between the 2 masses and the centripetal force is directed from each mass to c right?

 

[gneill]

Okay

Okay, so this means that the gravitational force is between the 2 masses and the centripetal force is directed from each mass to c right?

Correct.

 

[Angie Tom]

Correct.

THANKS A MILLON

 

[Angie Tom]

Why does the normal on a mass have different values at the pole and at the equator?

 

[gneill]

Why does the normal on a mass have different values at the pole and at the equator?

This looks like a new question. Start a new thread for it, and use the template that will be provided in the edit window after hitting the "Post New Thread" icon.

 

[haruspex]

Okay

Okay, so this means that the gravitational force is between the 2 masses and the centripetal force is directed from each mass to c right?

I wouldn't word it that way. The mass centres and c are in a straight line, so a direction 'towards c' is the same as 'towards the other star's centre'. Besides, it is important to remember that centripetal force is a resultant force. Your wording might give the false impression that each star is somehow attracted to the common mass centre.

If two stars are on parallel and opposite courses, at right angles to the line joining them at some instant, each experiences attraction towards the other star so accelerates in that direction. Since that is at right angles to their directions of motion, that results in a change of direction, not a change in speed. That is, it will lead them to start to revolve around some points (not necessarily the same point) on the line joining them.
This situation may be transient; a moment later their arrangement no longer matches those conditions.

If the distance between them, their speeds and their masses are in the right relationship to each other, that point will be, for both, the common mass centre. If so, the arrangement is dynamically stable, i.e. they will continue to satisfy all thes conditions and continue to orbit around that common mass centre. But the attaction is always to the mass centre of the other star.

One more point, just to be clear. That each star is attracted to the mass centre of the other is a special feature of spherically symmetric bodies. In reality, each atom of each star is attracted to each atom of the other. But if each star is made of concentric spherical shells, each of uniform mass density, it turns out that the net attraction of each atom of one star is towards the mass centre of the other. The situation would be far more complex with two amorphous lumps of rock orbiting each other.

 

[Angie Tom]

I wouldn't word it that way. The mass centres and c are in a straight line, so a direction 'towards c' is the same as 'towards the other star's centre'. Besides, it is important to remember that centripetal force is a resultant force. Your wording might give the false impression that each star is somehow attracted to the common mass centre.

If two stars are on parallel and opposite courses, at right angles to the line joining them at some instant, each experiences attraction towards the other star so accelerates in that direction. Since that is at right angles to their directions of motion, that results in a change of direction, not a change in speed. That is, it will lead them to start to revolve around some points (not necessarily the same point) on the line joining them.
This situation may be transient; a moment later their arrangement no longer matches those conditions.

If the distance between them, their speeds and their masses are in the right relationship to each other, that point will be, for both, the common mass centre. If so, the arrangement is dynamically stable, i.e. they will continue to satisfy all thes conditions and continue to orbit around that common mass centre. But the attaction is always to the mass centre of the other star.

One more point, just to be clear. That each star is attracted to the mass centre of the other is a special feature of spherically symmetric bodies. In reality, each atom of each star is attracted to each atom of the other. But if each star is made of concentric spherical shells, each of uniform mass density, it turns out that the net attraction of each atom of one star is towards the mass centre of the other. The situation would be far more complex with two amorphous lumps of rock orbiting each other.

This is really helpful! Thanks a lot :D