# When Do Points on Two Different Ferris Wheels Align?

• LaplacianHarmonic
In summary, the two points of intersection for the Ferris wheels are at the same time when they intersect if the Ferris wheels have the same number of rotations.
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

LaplacianHarmonic said:
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.

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

LaplacianHarmonic said:
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

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.

I am stuck

Hello?

LaplacianHarmonic said:
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?

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?

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...?

LaplacianHarmonic said:
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?

No.

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

w1 does not equal w2

LaplacianHarmonic said:
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?

Theta + 2ㅠN = phi

That's interesting

LaplacianHarmonic said:
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.

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.

LaplacianHarmonic said:
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.

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

LaplacianHarmonic said:
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?

## 1. What is a Ferris wheel?

A Ferris wheel is a large amusement ride consisting of a rotating wheel with passenger cars attached to the rim. It is designed to give riders a view from high above the ground.

## 2. How does a Ferris wheel work?

A Ferris wheel is powered by an electric motor that turns the wheel. The cars are attached to the rim and as the wheel turns, the cars move along with it. The weight of the passengers in the cars helps to keep the wheel balanced.

## 3. How tall are Ferris wheels?

The height of Ferris wheels can vary, but they are typically between 50-100 meters tall. The tallest Ferris wheel in the world is the High Roller in Las Vegas, which stands at 167.6 meters.

## 4. When were Ferris wheels invented?

The first Ferris wheel was invented by George Washington Gale Ferris Jr. for the 1893 World's Columbian Exposition in Chicago. However, a similar ride called the "pleasure wheel" was built in 1891 in Atlantic City, New Jersey.

## 5. Are there any safety concerns with Ferris wheels?

Ferris wheels are generally considered to be safe amusement rides, but accidents can happen. It is important for operators to regularly inspect and maintain the ride, and for riders to follow all safety instructions. In some cases, accidents may be caused by rider behavior, such as standing up or rocking the car, rather than a mechanical failure of the ride itself.

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