Two particles undergoing circular motion

[Vibhor]

Homework Statement

Homework Equations

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 ?

Please help me with this problem .

Thanks .

 

[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?

 

[Vibhor]

I think you should use the relative angular velocity between the two particles.

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

Presently I do not have the answer .

 

[cnh1995]

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

Presently I do not have the answer .

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.

 

[Vibhor]

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

 

[haruspex]

You can calculate the angular velocities of both the particles using the given information (in rad/s).

Revolutions per minute would be simpler.

 

[Vibhor]

Revolutions per minute would be simpler.

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 ?

 

[cnh1995]

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

Yes.

 

[PeroK]

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 ?

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.

 

[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 ?

 

[PeroK]

@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 ?

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.

 

[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 .

Thanks for your patience .

 

[haruspex]

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 ?

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?

 

[jbriggs444]

Homework Statement

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]

 

[Vibhor]

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.

Thanks for coming to my rescue

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?

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 ?

 

[haruspex]

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 ?

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

 

[Vibhor]

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

Wouldn't that be 10/3 min ?

 

[haruspex]

Wouldn't that be 10/3 min ?

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

 

[Vibhor]

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

In that case Q doesn't revolve around P .

 

[haruspex]

In that case Q doesn't revolve around P .

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.

 

[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 ?

 

[haruspex]

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 ?

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.

 

[Vibhor]

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.

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 ?

 

[haruspex]

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 ?

Right. Epicycloid?

 

[Vibhor]

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.

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

 

[haruspex]

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

Yes, but in P's frame that's just some arbitrary point.

 

[Vibhor]

OK...Thanks a lot . As usual you were fabulous with your replies .

 

[Vibhor]

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.

One last thing , refer to @jbriggs444 post #14 , if P moves in outer radius and if we connect P and Q by an imaginary line , don't you think , that line completes one full rotation as P and Q move. Doesn't that count as one revolution of Q around P ( despite Q moving in smaller radius ) ??

 

[haruspex]

don't you think , that line completes one full rotation as P and Q move. Doesn't that count as one revolution of Q around P ( despite Q moving in smaller radius ) ??

With P in the outer circle, in P's rotating reference frame, that line would wiggle about a bit, but return to its original position without completing a revolution.
In the lab frame, the line would complete a revolution, but you would describe it as P going around Q, I think.