Circular Motion - Orbital Mechanics

[Jimmy87]

Homework Statement

Question 1. If two objects were in the same orbit around a planet they would both have the same orbital velocity regardless of their mass. Using the equation given, explain why an object with twice the mass experiences the same orbital velocity.

Homework Equations

Centripetal force = mv^2/r

The Attempt at a Solution

I know it has something to do with gravitational forces being proportional to mass but I am unsure how to relate it to the equation. If the object was twice as massive then it would experience twice the centripetal force which I thought would tend to increase the velocity?

I have seen other sources say things about planets/objects in orbit 'fall' like objects on Earth but wouldn't the velocity related to falling objects on Earth be different to the velocity of objects in orbit as their velocity is tangential which isn't influenced by gravity?

Any help is much appreciated! :)

 

[FactChecker]

hint: If their orbits are identical, the acceleration due to the centripetal forces must be identical. Use that.

 

[Jimmy87]

hint: If their orbits are identical, the acceleration due to the centripetal forces must be identical. Use that.

Thanks for your reply. So if your saying the acceleration is constant then v^2/r must be constant. Since r is constant then v^2 is also constant and hence v is constant? Is that right? What equation shows that acceleration is constant if the orbits are the same? How does all this relate to the independence if mass though?

 

[Doc Al]

Thanks for your reply. So if your saying the acceleration is constant then v^2/r must be constant. Since r is constant then v^2 is also constant and hence v is constant? Is that right? What equation shows that acceleration is constant if the orbits are the same?

To be in the same orbit means that v and r are the same.

How does all this relate to the independence if mass though?

Newton's 2nd law. What force holds an object in orbit?

 

[FactChecker]

Sorry. I think I may have underestimated what is expected. If "identical orbits" means identical paths and velocities, then the conclusion is true by definition. If "identical orbits" means identical paths, even if velocities are different, then it is not so simple. So my initial statement, "If their orbits are identical, the acceleration due to the centripetal forces must be identical." is really assuming what you are asked to prove.

Here is what I had in mind. Acceleration due to centripetal force is not constant, just independent of mass. Two masses, m1 and m1 with identical accelerations gives f1/m1 = a1 = a2 = f2/m2. Substitute your equation for centripetal forces, f1 and f2, and the mass cancels out, giving v1 = v2.

 

[Doc Al]

Sorry. I think I may have underestimated what is expected. If "identical orbits" means identical paths and velocities, then the conclusion is true by definition. If "identical orbits" means identical paths, even if velocities are different, then it is not so simple.

Yeah, I misread the question. By identical orbits they just mean the same radius. It's up to you to prove that the velocities must be the same, regardless of mass.

My previous 'tip' still holds: Go back to Newton's 2nd law. Solve for the velocity and see if it depends on the mass of the object.

 

[Jimmy87]

Thanks for the replies guys but I'm a bit confused now. Does being in the same orbit mean v and r are the same as the last two posts seem to contradict each other? Can you be in the same orbit with a different velocity?

 

[Jimmy87]

Sorry Doc Al I didn't see your latest thread as my page id not refresh please ignore my last post

 

[Doc Al]

Can you be in the same orbit with a different velocity?

That's for you to figure out. For a given orbital radius, can you find the velocity? What does it depend on?

 

[Jimmy87]

That's for you to figure out. For a given orbital radius, can you find the velocity? What does it depend on?

The equation for orbital velocity is the square root of GM/R

Is it suffice just to say that since R (same orbit), M (both orbiting same mass) and G are all the same then v must be the same by definition? Is that correct?

I am still unsure how to explain the fact that doubling the mass keeps the velocity to the same other than to state the obvious and say that it is missing in the equation and therefore must be independent.

I understand that doubling the mass of an object near the Earth's surface doubles the gravitational force but also doubles the inertia which results in the same acceleration (g) and hence mass independence. Not sure how to connect this to an orbital problem though?

 

[Doc Al]

The equation for orbital velocity is the square root of GM/R

Good.

Is it suffice just to say that since R (same orbit), M (both orbiting same mass) and G are all the same then v must be the same by definition? Is that correct?

Yes, but I wouldn't say 'by definition'. Assuming you can prove the expression for orbital velocity from first principles. Do that!

I am still unsure how to explain the fact that doubling the mass keeps the velocity to the same other than to state the obvious and say that it is missing in the equation and therefore must be independent.

I understand that doubling the mass of an object near the Earth's surface doubles the gravitational force but also doubles the inertia which results in the same acceleration (g) and hence mass independence. Not sure how to connect this to an orbital problem though?

It's the same reason that a falling object has the same acceleration regardless of mass. If you double the mass you double the force (gravity), which just cancels out (Newton's 2nd law) to give you the same acceleration.

 

[Jimmy87]

Good.


Yes, but I wouldn't say 'by definition'. Assuming you can prove the expression for orbital velocity from first principles. Do that!


It's the same reason that a falling object has the same acceleration regardless of mass. If you double the mass you double the force (gravity), which just cancels out (Newton's 2nd law) to give you the same acceleration.

Thanks for your help Doc Al! So if we know that from sqrt of GM/R means that v is the same for the same orbit (i.e. same R). Is it ok for me to use this and go on to say that this must mean that v^2/r must also be the same which means the centripetal acceleration must be the same? Then to finish off with the whole mass independence thing can I say that doubling the mass would tend to double the centripetal acceleration (which would then increase v) but since the inertia doubles the centripetal acceleration and thus the velocity remains constant. Is that logical?

 

[Doc Al]

Thanks for your help Doc Al! So if we know that from sqrt of GM/R means that v is the same for the same orbit (i.e. same R). Is it ok for me to use this and go on to say that this must mean that v^2/r must also be the same which means the centripetal acceleration must be the same?

Sure.

Then to finish off with the whole mass independence thing can I say that doubling the mass would tend to double the centripetal acceleration (which would then increase v) but since the inertia doubles the centripetal acceleration and thus the velocity remains constant. Is that logical?

Sounds good to me.