When Do Points on Two Different Ferris Wheels Align?

[LaplacianHarmonic]

1. The problem statement, all variables
Ferris wheel 1 has constant angular velocity A with radius M described by parametric equations
X=Mcos(At)
Y=Msin(At) + M

Ferris wheel2 has constant angular velocity of B with radius N described by parametric equations
X= Ncos(Bt) + H
Y=Nsin(Bt) + N

At what time are the two point on each Ferris wheel right on top of each other if these two points start at angle 0 for both Ferris wheels?

Homework Equations

Parametric equations for the two Ferris wheels

The Attempt at a Solution

When the two points of each Ferris wheel intersect, the x and y coordinates for that point of intersection are the same from each Ferris wheel.

I equated the times for x coordinates of Ferris wheel 1 and Ferris wheel 2. Help me

 

[phinds]

I equated the times for x coordinates of Ferris wheel 1 and Ferris wheel 2.
Help me

Yes. Then what? If you don't show any work, there's nothing for us to help you with.

 

[LaplacianHarmonic]

I'm trying to learn how to write in latex properly to show the work.

 

[LaplacianHarmonic]

Can I post a picture of the notebook page and/or dry erase board with the actual work?

 

[phinds]

Can I post a picture of the notebook page and/or dry erase board with the actual work?

It's actually against the rules but if it is TOTALLY clear and utterly legible it might pass muster.

 

[LaplacianHarmonic]

 

[LaplacianHarmonic]

A bit difficult to set up the functions to find the time at which both x and y coordinates for the intersection of the two Ferris wheel come out from the 2 sets of parametric equations.

 

[LaplacianHarmonic]

I am stuck

 

[LaplacianHarmonic]

Hello?

 

[haruspex]

these two points start at angle 0

How are you interpreting that? What angle? Is this any different from the obvious fact that when t=0 we have At=0 and Bt=0?

I cannot follow your working. Please take the trouble to type it in. That allows reviewers to target comments at individual lines.
Having equations left and right on a page makes it hard to follow the sequence.

I do not understand why you want to work in terms of X and Y, or introduce new labels for radii (they are given).
If the two points coincide at some time t, what two equations can you write involving t?

 

[LaplacianHarmonic]

We use x and y to find the 2 points of intersection.

X1 = R1 [cos (w1 t)] =
R2 [cos (w2 t)] + H

And then solve for t hoping t will match...?

 

[haruspex]

We use x and y to find the 2 points of intersection.

X1 = R1 [cos (w1 t)] =
R2 [cos (w2 t)] + H

And then solve for t hoping t will match...?

You also have an equation for matching y. Can you manipulate them to eliminate w1?

 

[LaplacianHarmonic]

No.

Y1 = R1 [sin (w1 t)] + R1
= R2 [sin (w2 t)] + R2

 

[LaplacianHarmonic]

w1 does not equal w2

 

[haruspex]

No.

Y1 = R1 [sin (w1 t)] + R1
= R2 [sin (w2 t)] + R2

Ok.
Write each equation so that on one side you only have the cos or sine of w1t, and no reference to w1 on the other side. No x or y either.
What algebraic relationship do you know between cosine and sine?

 

[LaplacianHarmonic]

Theta + 2ㅠN = phi

That's interesting

 

[haruspex]

Theta + 2ㅠN = phi

That's interesting

I don't understand. Is that a response to my question about the relationship between the sine of an angle and the cosine of the same angle?
What I'm looking for is equivalent to Pythagoras' Theorem.

 

[LaplacianHarmonic]

This point of intersection at same time is more rare than initially perceived.

It can never occur and almost always occurs once if the conditions aren't perfect.

 

[haruspex]

This point of intersection at same time is more rare than initially perceived.

It can never occur and almost always occurs once if the conditions aren't perfect.

Perhaps, but let's just concentrate on solving theequations to find out.

 

[LaplacianHarmonic]

The equations are different between intersecting at same number of rotations and intersecting at different number of rotations.

 

[haruspex]

The equations are different between intersecting at same number of rotations and intersecting at different number of rotations.

The equations you posted in posts #11 and #13 do not depend on the number of rotations. There may be multiple solutions, but for now please concentrate on how to solve that pair of equations. Can you make some progress with that? Can you do what I proposedin post #15?