What Are Common Misconceptions in Understanding Circular Motion and Orbits?

[lazybum122]

Ok I have 2 related questions

Qns1

Homework Statement

Two satellites, A and B, orbiting around Earth have the same kinetic energy. Satellite A has a larger mass than satellite B. Which of the following statements is false?
A Satellite A has a larger period.

B Satellite A has a larger orbital radius.

C Satellite A has a smaller total energy.

D Satellite A has a smaller angular velocity.

Homework Equations

Ok if I'm not wrong we're supposed to use mv^2/r = GMm/r^2?
Then v=rw and w=2(pi)/T

The Attempt at a Solution

Ok so i was thinking since KE is constant, velocity of A should be smaller than B.
Then.. what? Ok v^2/r = GM/r^2, but I'm stuck here cause gravitational acceleration and r can vary.
Answer is C btw.


Then there is another similar question:
Qn2

Homework Statement

Two stars of masses M and 2M move in circular motion about their common centre of mass. Which of the following statements is true?
A. Both stars move with the same speed
B. Both stars move with the same angular velocity
C. Both stars move with the same radius
D. Such a motion is not possible.

Homework Equations

Gravitational field strength = GM/r^2 = v^2/r = rw^2

The Attempt at a Solution

No idea really. I thought all of them were untrue. The answer was B. But how can that be possible? Satellites orbit around Earth with different angular velocities no? Unless the question is saying that the stars exert some gravitational force on the centre of mass also? If that makes a difference.

Oh and i would like to clarify something. What is the condition for a uniform circular motion?
Is it that the resultant force must be just enough to provide for the centripetal force?

 

[Delphi51]

Welcome to PF, lazybum.
Kind of a silly name - you have done most of the work on this already, and that is unusual on this forum this fall.

since KE is constant, velocity of A should be smaller than B.
Then.. what? Ok v^2/r = GM/r^2, but I'm stuck here cause gravitational acceleration and r can vary.

Great start! To answer (b), I would solve that equation for r. Knowing that A's velocity is smaller, and that GM is the same for both, you can quickly tell if r is larger or smaller for A.
For (a), you must derive your equation again using the formula for Fc that has a T in it instead of a v.
For (d) you'll just need a formula relating angular velocity to regular velocity. And for (c) you need the total of kinetic energy and gravitational potential energy (I think the latter is something like GM/r - check it out).

For the second problem, you must take a look at
http://en.wikipedia.org/wiki/Multiple_star
Scroll half way down and look at the video of the binary system in the problem.

I would say the condition for uniform circular motion is that there be a centripetal force exactly equal to mv²/R and no other forces.

 

[lazybum122]

Haha it was some online nick that I stuck to since I was really young. Too lazy to think of a new name everytime i signed up at forums so yeah there you have it. So in a sense its somewhat true ^^

Anyway thanks for the help. I've got the first question figured out now.

For the second question, looks like i mistook the meaning of "centre of mass". I thought it literally meant a mass in the centre of their orbit =.=

But anyway i haven't learned much about binary star systems. The wiki page doesn't seem to show how to prove that angular velocity is the same for both stars. Any idea how to prove? I was thinking the gravitational force provides the centripetal force for both stars. Since the gravitational force on both stars are the same, GMm/r^2 = M(r1)w^2 = m(r2)w^2, where
M is the mass of the first star,
m is the mass of the second,
r is the distance between the 2 stars
(r1) and (r2) is the orbital radius of M and m respectively

so we have M(r1)w^2 = m(r2)w^2
still doesn't prove anything

 

[Delphi51]

Oops, I see a little mistake in that; GMm/r^2 = m(r2)w^2
should say Gm1*m2/(r1+r2)^2 = m(r)w^2 where (r1 + r2) is the constant distance between the stars and r is either r1 or r2 depending on which w you want to find. Same for m.

And that r² -> (r1+r2)² error applies to our earlier deductions!
Back to the drawing board. Say we call R = r1+r2, which is the same for both orbits. Then
Fc = Fg
mv²/r = Gm1*m2/R²
For star 1, m1*v²/r1 = Gm1*m2/R²
v² = G*m2*r1/(R²*m1)
For star 2, v² = G*m1*r2/(R²*m2)
Can't tell which is larger!

Getting too late at night to sort this out!

 

[lazybum122]

no worries I don't think its that important at the moment. I think we just haven't learned it yet and will learn it in the future.

Anyway as for the error you pointed out, isn't it just a matter of labeling?
In my working i used r = (r1) + (r2)

But that aside, GMm/(r1+r2)^2 = m(r)w^2 won't seem to prove anything cause we don't know how r varies M?. As in the equation can be simplified to GM/(r1+r2)^2 = (r)w^2. But from here we cannot conclude much about the relationship between r and w because GM/(r1+r2)^2 is not the same for both stars since M will be replaced by m if we consider the other star.

k not sure if I'm making sense. But its like if we assume that w is fixed (which it actually is), then it will mean that the star with the smaller mass will have the larger area, which is true. So the value of M affects the value of r. So we have 3 variables and cannot conclude how the other 2 relate to w.

And your second part about "A". I'm not sure what you mean by A. Are you confusing it with the first question?

 

[Delphi51]

Sorry about last night - I can't do physics when I'm tired.
I got the two questions merged together for one thing.

The first question seems easy this morning. Much can be done with the fact that they have the same KE, so clearly the one with the larger mass has to have the smaller velocity to achieve that.

The second question is tricky because we have to deal with the fact that the distance between them (in the gravity force formula) is the sum of the two radii. I can't get anywhere without assuming that the periods are the same. As you say, with that assumption, the 4 parts of the question are fairly easy. I think equal periods could be proven from the given that they are in circular orbits. If the periods were unequal, something would soon go very wrong with the circular motion.

I said question A or part A because it seems to me that each of the 4 parts must be answered as true or false.

Very interesting question and collaboration! Thank you.

 

[lazybum122]

No problem. Thanks for helping!