Built on Facts

You Spin Me Right Round

The National Weather Service does a very useful thing for people who live in an area expexted to experience severe weather danger. I have a little Firefox app in my browser that links to the NWS and advises me of the current conditions and forecast for the next few days, and as part of its mechanical duties it advises me of the various severe weather alerts that happen. They’re popping up at the rate of several times per week now, when the sky is a beautiful crystal blue. Why? Severe heat. Welcome to the southern summer!

I regret to say that I have an advisory of my own: a travel alert. I’m going to be on the road until Friday(ish) and thus posting might be sporadic or nonexistent. The latter is unlikely, I should be able to get at least a couple days in this week. But if not don’t worry. I’m not joining the ranks of the celebrity death outbreak.* Well I’m not a celebrity either, but I’m going to try to avoid the death bit too.

And since I’m on the road, I want to kick off the trip with a driving-related physics fact that few in my classes believe when I tell them. On a smoothly rolling wheel, the point in contact with the ground is stationary. It ain’t moving, no matter how fast the wheel is turning.

Why? The easy answer is that if it were, it would be skidding. A more meaningful answer is that a rolling wheel is really the combination of two motions: circular spinning and linear translation. For a wheel whose center is moving forward at velocity v, the wheel clearly must be spinning such that the points on its rim are also moving at v. The rolling itself makes the points on the top of the wheel spin in the forward direction at speed v, and the points on the bottom are obviously going in the opposite direction, velocity -v.

Add the overall forward speed v, and -v + v = 0 for the bottom of the wheel. You might not believe me, so let’s do the experimentalist thing and actually watch one in action. This nice and much more mathematically complete web explanation has the video.

Not surprising when you think about it, but maybe a little surprising before you think about it.

*Honestly it’s Billy Mays whose death actually managed to make me legitimately a bit sad. Go figure.


  1. #1 counters
    June 29, 2009

    Matt, could you link to that NWS Firefox app? I’m re-coding the website for the Cornell Chapter of the AMS, and I’d love to build an app like that into the site. Thanks!

  2. #2 Uncle Al
    June 29, 2009

    Tank treads are also stationary versus the ground. The driver and idler wheels are also stationary versus the ground where they touch. Nothing is linearly propagating! We don’t need bazookas, we need combat physicists. (Don’t knock it – guaranteed employment plus a Federal pension.)

  3. #3 Jason A.
    June 29, 2009

    I can’t get my head around the way you explained it, this part: “For a wheel whose center is moving forward at velocity v, the wheel clearly must be spinning such that the points on its rim are also moving at v.”
    Why would the translational velocity of the center of the wheel and the tangential velocity of the rim have to match? If we have a wheel not in contact with the ground, I can spin it as fast or as slow as I want and that has nothing to do with how fast I move it left to right.

    The way I think about is if the center is translating at v and the bottom edge is translating at zero (not skidding), then the top edge must be translating at 2v. The front and rear edges must be translating at v to match the center. So the tangential velocity must be something that varies the translational velocity from v (on the front and rear when the tangential is orthogonal to translational) and 0 or 2v (on the top and bottom when the tangential is parallel to translational). Therefore, the tangential velocity must be v.

    Huh, I guess I should have just believed you 😛

  4. #4 David A
    June 29, 2009

    What about the concept that the top of the wheel is actually moving faster than the bottom, to account for friction at the point of contact between the wheel and the surface against which it is rolling? No friction, no translational motion.

  5. #5 Matt Springer
    June 30, 2009

    It’s Forecastfox, which pulls its forecasts from AccuWeather.com. I’m not sure if they do their own forecasts or pull from NWS, but in any case their alerts are all from the NWS.

  6. #6 Matt Springer
    June 30, 2009

    Actually Jason I think you found a significant flaw in my wording. It would have been a lot more clear to say “…the wheel clearly must be spinning such that the points on its rim are also moving at v with respect to the axle.”

    With respect to the ground the translational considerations come into play, and your reasoning and mine works just as well.

  7. #7 Carl Brannen
    July 1, 2009

    Matt, somewhat off topic but might have the germ of a post in it:

    If you know anyone who believes that vapor trails are caused by dirty aircraft smoke, or are “chemtrails”, have them take a look at this beautiful photo.

  8. #8 rob
    July 1, 2009

    Matt: i used to get the same reaction about the bottom of the wheel. it is really amazing how many incorrect preconceptions people have about physics. some things that seem so simple really aren’t and can trick you.

    Carl: i found that photo yesterday and made it my desktop bitmap. very cool. there are several things to mention about it. it is an F-22. there are contrails coming off each wingtip due to the vortices and pressure change causing the water vapor to condense into visible droplets. also, the shockwave off the body of the jet is causing a triangular shaped vapor cloud. i am pretty sure the plane is super sonic, and the triangular shape is due to the Mach Cone. you should be able to measure the angle of the cone and deduce the speed of the jet. also, if you look between the horizontal stabilizers, you will see a saw tooth pattern that is where the engine outlets are. i think that pattern keeps that portion of the airframe from having a straight profile which helps reduce the radar return.

  9. #9 tom
    July 1, 2009

    go for forecastfox enahanced….it’s ‘better’.

    i remember discussing this topic (wheels) at the age of 9 or 10 while riding around the neighbourhood with friends….

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