# Rotating Wheels and Point Line-up

• r731
In summary, the conversation discusses a problem involving three rotating wheels and how to determine when the three points on the wheels line up. It is determined that the first and second wheels will line up when the first wheel has performed an integral number of turns, and the third wheel will also line up if it has performed an integral number of turns based on the ratios of velocities of the first and third wheels. This can be proven using number theory and would also occur if pi were replaced by a rational number.
r731
Hello all,

I'm trying to obtain solution(s) for problems of the following nature:
Suppose there were three wheels of equal size, rotating on the same axis, one point marked on the circumference of each wheel, and these three points lined up in one straight line. If the second wheel rotated twice as fast as the first, and if the speed of the third wheel was 1/π of the speed of the first, the initial line-up would never recur.

To keep things simple, I begin with two wheels instead, A and B.
If A has the frequency f_A, and B f_B, then the location (in terms of degrees) of each marked point is computed as follows:
f_A * 360 * t % 360;
f_B * 360 * t % 360;

where t is the elapsed time.

And,
360*f_A - 360*floor(f_A*t) = 360*f_B - 360*floor(f_B*t),
which simplifies to,
f_A - floor(f_A*t) = f_B - floor(f_B*t)
can be solved for t; the solutions of which represent the time instants during which the two points (each of which is marked on one wheel) line up.

Is there any simpler way other than using %-operator and floor()?
I was especially thinking of graphs plotted on a coordinate system, where the intersection points would be solutions.

It's a number theory problem.

It is easy to derive that the first and second wheels line up when and only when the first has performed an integral number of turns - call that number n. For the third wheel to also line up it must have also performed an integral number of turns.

Using the ratios of velocities of the 1st and third wheels, work out how many turns the third wheel must have made when the 1st has made n.

Can that number be an integer, given what is known about pi in number theory?

Now if pi were replaced by a rational number, say 3.14159, would the initial line-up occur?

How to prove these kinds of stuff?

Yes it would occur. Say the angular velocities of the three wheels are ##1, \frac{a}{b}, \frac{c}{d}## where ##a,b,c,d## are integers and ##b,d>0##. Then the dots will line up after every ##kn## revolutions of the first wheel where ##k## is any positive integer and ##n## is the least common multiple of ##b## and ##d##.

Can you see why?

## 1. What is the purpose of rotating wheels in a vehicle?

Rotating wheels allow a vehicle to move forward by converting the energy from the engine into rotational motion. This motion is transmitted to the wheels, which then provide the necessary traction to move the car forward.

## 2. How do rotating wheels affect the handling of a vehicle?

The size and shape of rotating wheels can greatly impact the handling of a vehicle. Larger wheels provide better stability and traction, while smaller wheels are more responsive and agile. The alignment of the wheels also plays a role in the handling of a vehicle.

## 3. How do you know if your rotating wheels are properly aligned?

If your vehicle pulls to one side while driving, or if you notice uneven wear on your tires, it may be an indication that your wheels are not properly aligned. You may also feel vibrations or shaking in the steering wheel.

## 4. Can rotating wheels affect the fuel efficiency of a vehicle?

Yes, rotating wheels can have a significant impact on the fuel efficiency of a vehicle. Larger wheels require more energy to rotate, which can result in lower fuel efficiency. Properly inflated tires and wheel alignment can also improve fuel efficiency.

## 5. How often should rotating wheels be replaced?

The lifespan of rotating wheels can vary depending on the type of vehicle and driving habits. On average, tires should be replaced every 3-4 years or every 50,000-60,000 miles. However, it is important to regularly check the tread depth and condition of your tires to ensure safe and efficient driving.

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