# How does time dilation affect measurements of wheel rotation on a moving bus?

• B
• etotheipi
In summary, the conversation discusses a scenario involving a bus with a wheel painted with a yellow dot and two observers, Kip and Charles, measuring the time taken for each revolution of the wheel. They come to the conclusion that due to time dilation, the time interval between events calculated by Charles will be larger than that calculated by Kip. This is because the wheel appears as an ellipse in the ground frame and is time dilated. The conversation also touches upon the effects of length contraction and includes links to illustrations and articles for further understanding.
etotheipi
Here’s a simple scenario I came up with earlier, because I couldn’t make sense of a few things and so started to feel a bit sick.

There’s a bus driving along a road at say, ##v \, \text{ms}^{-1}##, and on one of the tyres someone has painted a bright yellow dot. The tyres have radius of ##1/(2\pi) \, \text{m}##, or in other words, they complete ##1## revolution in ##(1/v) \, \text{s}##. Kip, who’s attached to the bus (don’t ask me how…), uses a stopwatch to measure the time taken for each revolution of the wheel. That is, he’s just pressing the lap button every time the yellow dot reaches the top of its cycle. Charles is instead standing still on the pavement, but is also measuring the time taken for each revolution in the same way.

Because the speed of the bus relative to the pavement (and similarly the speed of the pavement relative to the bus) is just ##v \, \text{ms}^{-1}##, Kip and Charles should both calculate that the wheel - which is rolling - has the same angular speed ##\omega = v/r =2 \pi v \, \text{s}^{-1}##.

But the thing is, the events “yellow dot is at the top of it’s ##\text{i}^{\text{th}}## and ##\text{j}^{\text{th}}## cycle” respectively are of course at the same position coordinates as measured by Kip, so it’s easiest to just apply the so-called ‘time dilation formula’ which will tell you that the time interval between ##E_i## and ##E_j## calculated by Charles should be larger than that calculated by Kip.

That seems weird to me, because the effect of time dilation appears to be “invisible” here. What’s the missing piece of the puzzle?

Abhishek11235 and JD_PM
The wheel is round in the bus frame and length-contracted into an ellipse in the ground frame (relativistic wheels are weird - have a google, there are some nice videos of relativistic spoked wheels). That means that the wheel is a ##1\mathrm{m}/v## period clock in the bus frame but not in the ground frame. The ground frame would measure it to be time dilated.

You can also think about what if the paint were wet and left dots on the road as it rolls.

Last edited:
Safe, yeah I guess I just casually forgot that length contraction is a thing. Okay, makes sense now. Thanks!

vanhees71
etotheipi said:
Safe, yeah I guess I just casually forgot that length contraction is a thing.
The relativistic wheel is in fact the first graphic in the wiki article on length contraction:
https://en.wikipedia.org/wiki/Length_contraction

Demystifier and vanhees71
Looks trippy af haha

vanhees71, Motore, Ibix and 1 other person
A.T. said:
Looks even weirder if you also include the light signal delay, to show what you would actually see:
Looks even more weird, if an observer with 2 eyes sees the wheels and not one with only one eye. Unfortunately, I can't find the link to this again.

Edit: I found something on the subject:
paper said:
Gamow’s Cyclist: A New Look at Relativistic Measurements for a Binocular Observer
...
3. Measurements from Binocular Distortion

In this section, we introduce formalism with respect to binocular observations; herein, we define two types of observers:
• Class 1 An observer with a single aperture such as a camera
• Class 2 An observer with two apertures, capable of depth perception generated by visual parallax such as a human
Source:
https://arxiv.org/pdf/1906.11642.pdf

Last edited:
A.T.
The problem gets weird and requires assumptions about how the wheel physically distorts as we spin it up.

Scenario 1 is where the proper circumference of the bus wheel remains the same, which means it will leave paint marks on the road at equal intervals despite the speed of the bus and the reduced circumference of the ellipse as viewed by the road observer. In such a case, the radius of the wheel will contract, so the bus wheel gets smaller, compressed in both directions (vertical as well as horizontal). Relative to the bus frame, the wheel will need to spin more times for a given amount of road to go by, and the paint marks on the road get closer as the road contracts.
From the road frame, RPM is linear with speed. RPM seems to increase without limit from the bus frame, being linear with proper speed, not with road speed.

Scenario 2 is where the radius of the wheel remains fixed (a thin rubber tire stretching as it slides along a non-rotating rim of fixed radius). In this scenario, the proper circumference of the tire increases with the physical stretching and the paint splotches get correspondingly further apart (road frame) as speed increases, but remain a speed independent constant separation in the bus frame. Both observers see the same original height of wheel.
In the bus frame, RPM is linear with road speed, and so is limited.

In case anyone cares, I found a nice AJP article on the subject.

vanhees71

## 1. What is the meaning behind the lyrics of "The wheels on the bus go..."?

The lyrics of "The wheels on the bus go..." are a children's song that describes the sounds and actions of a bus ride. It is meant to be a fun and interactive way for children to learn about different parts of a bus and the noises they make.

## 2. Why is "The wheels on the bus go..." such a popular song for children?

"The wheels on the bus go..." is a popular song for children because it is catchy, easy to remember, and incorporates fun actions and sounds that children can participate in. It also teaches children about different parts of a bus and helps with their language and motor skills development.

## 3. Are there any variations of "The wheels on the bus go..."?

Yes, there are many variations of "The wheels on the bus go...". Some versions include additional verses about other parts of the bus, such as the wipers, horn, and doors. There are also versions that incorporate different modes of transportation, such as trains or airplanes.

## 4. Who wrote "The wheels on the bus go..."?

The origins of "The wheels on the bus go..." are unknown, as it is a traditional children's song that has been passed down through generations. However, it is believed to have originated in the United States in the early 20th century.

## 5. Is there any scientific or educational value to "The wheels on the bus go..."?

Yes, "The wheels on the bus go..." has educational value as it teaches children about different parts of a bus and the noises they make. It also helps with their language and motor skills development. Additionally, it can be used as a tool for teaching rhythm and music to children.

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