Homework Help: Two particles undergoing circular motion

1. Dec 9, 2016

Vibhor

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

I am actually stumped by this seemingly simple problem . Since both the particles are moving , in order to calculate angular velocity of Q with respect to P looks to be difficult as the relative velocity of Q w.r.t P is changing at all times .

Am I missing something simple ?

Thanks .

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2. Dec 9, 2016

cnh1995

I think you should use the relative angular velocity between the two particles. Also, particle Q should move along the outer circle. Is the answer provided?

3. Dec 9, 2016

Vibhor

Could you explain how to calculate without being given any other information ?

Presently I do not have the answer .

4. Dec 9, 2016

cnh1995

You can calculate the angular velocities of both the particles using given information (in rad/s). The difference between them will be the relative angular velocity. Using this velocity, you can compute the time required by Q for one revolution around P, provided that Q is in the outer circle.

5. Dec 9, 2016

Vibhor

Oh ! I was misreading the data . Strangely I was reading minutes as m/s .

Last edited: Dec 9, 2016
6. Dec 9, 2016

haruspex

Revolutions per minute would be simpler.

7. Dec 9, 2016

Vibhor

Is relative angular velocity simply the difference between the angular velocities of two particles ?

I thought it was tangential component ( perpendicular to the line joining the two ) of relative velocity divided by the distance between the two points .

Or are the two equivalent ?

8. Dec 9, 2016

Yes.

9. Dec 9, 2016

PeroK

You should draw a diagram of the motion of the two particles. Perhaps plot their approximate position every 30s or 1min and see how one moves with respect to the other.

PS I would change Q's rotation to 4mins for the diagram to make things simpler.

10. Dec 9, 2016

Vibhor

@PeroK , @Chestermiller

I understand how relative angular velocity is difference between the angular velocity of the two particles . The problem is solved .

Could you explain mathematically how the two definitions of angular velocity in post#7 are equivalent ?

11. Dec 9, 2016

PeroK

You have to imagine P "looking" in the same direction all the time. Alternatively, choose P's reference frame, in which Q rotates anti-clockwise (but not in a circle). Or, Q's reference frame in which P moves clockwise. This is better, actually, than drawing both points moving. And probably the simplest way to look at it.

I don't really understand your second definition of relative angular velocity.

12. Dec 9, 2016

Vibhor

Sorry . I was not able to convey the second definition properly .

I will take an example .Suppose there is a rigid rod moving in space such that at an instant one end A is moving with velocity v1 in the direction making an angle 30° anticlockwise with +x axis .The other end is moving with velocity v2 in the direction making an angle 45° anticlockwise with +x axis . How would you calculate angular velocity of point A w.r.t B ?

This is an example to demonstrate the second definition in post#7 .

By no means I intend to test your skills .You are far more knowledgeable than me

Please look at the second definition in light of the above setup and see if you could explain how the two definitions are equivalent .

13. Dec 9, 2016

haruspex

They are different.
The first definition supposes the two angular velocities are each defined in terms of some other point as a common centre of rotation. The second takes each as reference for the other.

I can see your difficulty. Let us suppose Q is outer, but instead of P moving in steady circles it just moves randomly about inside Q's circle. It takes Q 5 minutes to go around P. How is this changed by P moving in steady circles also? It isn't. The answer is still 5 minutes.
If P is outer, Q does not go around P at all, and that is nothing to do with Q's being slower.
In my view, the question setter has outsmarted himself.

To rescue the question, you have to think of P and Q as observers who always face ahead in their motion, and ask how many times Q appears to P to have gone around. I.e. use P as a rotating frame of reference. Does the answer to that depend on which is outer?

14. Dec 9, 2016

jbriggs444

To me, the problem as posed is not solvable. Let us define the rate of "revolves around" by imagining a telescoping, straight, massless rod connecting the two particles end to end and asking how many times this rod rotates 360 degrees end over end per unit time on average. The system will return to its starting state in ten minutes. So it suffices to find the number of rotations made by this rod over a ten minute span and divide by ten minutes.

Suppose that the radius of Q's circle is larger then P's. Then in 10 minutes, the rod will have made two complete rotations due to Q's motion while P's gyrations are irrelevant.

Suppose, contrariwise, that the radius of P's circle is larger. Then in 10 minutes, the rod will have made five complete rotations due to P's motion while Q's gyrations are irrelevant.

Since we are not told which radius is larger, the problem has no unambiguous answer.

[Missed seeing @haruspex say much the same thing]

15. Dec 9, 2016

Vibhor

Thanks for coming to my rescue

I think irrespective of which one of P and Q is in outer radii , from P's frame Q appears to be rotating with angular speed 0.3 rev/min anticlockwise . Is that right ?

16. Dec 9, 2016

haruspex

Yes, but that is not what the question asks. It asks for the time "for Q to make one revolution about P".

17. Dec 9, 2016

Vibhor

Wouldn't that be 10/3 min ?

18. Dec 9, 2016

haruspex

If Q is on the outer path, yes, but what if it is on the inner circle?

19. Dec 9, 2016

Vibhor

In that case Q doesn't revolve around P .

20. Dec 9, 2016

haruspex

Right. Using P's rotating frame of reference, Q will still appear to complete a circuit in 200 seconds, but it does not go around P.

21. Dec 9, 2016

Vibhor

With reference to post #15 and #13 , in P' s frame Q is rotating with angular speed 0.3 Rev/min , but as you mentioned in post #13 , the angular speed measured is w r.t the common center of rotation . Right ?

22. Dec 9, 2016

haruspex

Yes, the relative rate around the common centre is .3 rpm.
If Q is outer, in P's rotating reference frame, P sees Q as going around P, backwards, maintaining a constant rate in a circle about some point (but not a constant angular rate about P). Regardless, P sees Q as having completed one lap of P in 10/3 minutes.
If P is outer, in P's rotating reference frame, P sees Q as maintaining a constant rate in a circle about some point, but P is outside that circle, so Q never completes a lap of P.

23. Dec 9, 2016

Vibhor

Brilliant !

How would Q appear to move as seen from P i.e trajectory of P w.r.t Q ? I think it is not a circle . Right ?

24. Dec 9, 2016

haruspex

Right. Epicycloid?

25. Dec 9, 2016

Vibhor

"about some point" is the common center of rotation i.e center of the two circles ??