# Explaining Rolling Motion

Although rolling wheels are everywhere, when most people are asked “what is the axis of rotation of a wheel that rolls without slipping?”, they will answer “the axle”. It is an intuitively obvious answer shared by 3/4 or more of the students in an introductory physics class. It is also the wrong answer. Here I describe an incremental teaching method that compels students to come face-to-face with their preconceptions concerning rolling motion. I then offer some speculation on how these preconceptions could have taken hold and suggest ways to assess how well this method works.

##### Setting the stage

I begin by defining the axis of rotation of a rigid body as the set of all points that are (instantaneously) at rest while all the other points rotate about it with angular speed ## \omega ##. I use a meter stick mounted vertically and free to rotate about a nail going through a hole drilled at some distance from the midpoint (see figure below).

By pulling with a string below and above the pivot (solid and dashed lines) I establish the rule that relates the sense of rotation to the point of application of a force directed to the left: **Clockwise ## \rightarrow ## below the pivot, counterclockwise ## \rightarrow ## above the pivot**. Although this seems to be a trivial observation that everyone can understand and accept intuitively, I write it on the board for the students to see and so that I can refer to it later.

##### Theory

First I establish the size of the majority by asking “how many of you believe that the axis of rotation of a rolling wheel is at its center?” Then I proceed to show that the axis of rotation axis is actually the instantaneous point of contact (IPOC) with the table. The argument is simple: if the wheel is to roll without slipping, the point on the wheel that is in contact with the table must be instantaneously *at rest* with respect to the corresponding point on the table. If not, the two points will rub past each other and the wheel will be slipping. Therefore, by the definition of axis of rotation, the IPOC is the axis of rotation while the other points of the wheel rotate about it. So much for the theory. At this point, I ask how many students (a) believe that the axis of rotation is the IPOC, (b) believe that the axis of rotation is the center or (c) don’t know what to believe. Few students vote (a) while a healthy percentage plays it safe with (c). Preconceptions are based on misunderstood or misinterpreted observation. To dislodge them, theoretical arguments need to be bolstered by correctly guided observation.

##### Verification

I bring out yo-yo 1. It is a discarded cable spool furnished with a side arm, a short piece of PVC tubing attached to the center of the spool. The other end of the short piece is attached, through an elbow joint, to a longer PVC piece hanging vertically. The contraption is placed near the edge of the table with the side arm hanging straight down (see figure below).

First I pull on the string at a level below the table top which results in clockwise rotation of yo-yo 1. According to the rule on the board, the force is exerted at a point below the pivot. No problem with anyone. Next I raise the string to a point below the center of the wheel, but above the table top (dashed line). Before I pull on the string, I remind the rule to the audience, **Clockwise ## \rightarrow ## below the pivot, counterclockwise ## \rightarrow ## above the pivot**. When I pull, the rotation is counterclockwise. I conclude that the pivot must be somewhere between the two positions. Since theory predicts that it is at the point of contact, I place the string at the same level as the table top. When I pull, there is no rotation. Note: If you try this, be careful not to pull too hard, else the yo-yo will rotate about the vertical axis that has a non-zero lever arm. A rubber mat makes the contact less slippery.

##### The reality check

After reinforcing the theory with experiment it is time to use the theory to predict an outcome and verify the prediction by experiment. If the IPOC is indeed the axis of rotation and if the entire wheel rotates about with angular speed ## \omega ##, then the center of mass will have linear speed ## V_{CM} = \omega R, ## where *R* is the radius of the wheel. I predict that the top point on the rim of the wheel, which is diametrically across the point of contact, must have speed ## V_{top} = \omega (2R)=2 V_{CM}##. To validate this prediction, I place a soda can on the table and a meter-stick on top of the can parallel to the table top. The 0 cm mark is lined up with the top of the can. The starting point of both the center of the can and the edge of the meter-stick are marked with a small object placed on the table. I then push the meter stick forward so that the can rolls without slipping on both the table and the meter-stick. I stop when the top of the can is at about the middle of the meter-stick and mark the new positions of the can’s center and the 0 cm mark. It is clear that, in the same amount of time, the 0 cm mark is twice as far from the center of the can. Therefore, the top point of the can moves at twice the average speed of the center in agreement with the theory.

##### Are we there yet?

A theoretical prediction has been made and an experiment verified it. I ask the students if they they are now convinced that the POC is the axis of rotation and most of them are. But wait, there is more! To see if they really, really got it, I bring out yo-yo 2. This is another spool with string wrapped around the *inner* cylinder. First I wrap the string so that it comes off the cylinder parallel to the table from the top (figure below left). When I ask, an overwhelming percentage of students say that the cylinder will roll to the left, in the same direction as the pull. Hurray, that is what happens when I pull.

I then flip yo-yo 2 over, so that now the string comes off the cylinder parallel to the table from the bottom (figure above, right). When I ask, an overwhelming percentage of students say that the cylinder will roll to the right, in the opposite direction as the pull. Alas, that is not what happens when I pull. Some students audibly gasp when they see yo-yo 2 unexpectedly move to the left. At this point, I feign exasperation, point to **Clockwise ## \rightarrow ## below the pivot, counterclockwise ## \rightarrow ## above the pivot** on the board, and ask, “Is the force not acting above the point of contact in both cases? What more can I do to get you to accept that the point of contact is the axis of rotation?”

##### Whence the preconception?

It is a good guess that the preconception at play here is the belief that the axis of rotation is always the center of the wheel. This is coupled to the inability to transform from a moving to a stationary frame consistently. If one rides a bicycle and looks straight down at the front wheel, one sees a stationary axle with the wheel turning around it. Specifically, the bottom of the tire touching the ground is seen moving backwards while the top of the tire moving forward. One also sees the ground rush past backwards, but one dismisses this observation as merely an appearance because one “knows” who is “really” moving and it is not the ground. Thus, one fails to realize that the point of contact of the tire and the ground are moving at the same speed and, therefore, that they are at rest with respect to each other. So when one sees someone else riding a bicycle, one erroneously thinks that the point of contact is moving, just like it was moving when riding the bicycle. The preconception arises as a result of incomplete and misinterpreted observation.

I also hypothesize that the preconception involved with yo-yo 2 arises from intuitive ideas adopted by the observation of unrolling toilet tissue or paper towels, items intimately familiar to everyone. Imagine yo-yo 2 (previous figure left) being a roll of toilet tissue in its dispenser. If pulled as shown in the figure left, it will spin counterclockwise *in place*. No problem here. Now, if the roll is placed on a table top and then pulled, the intuitive prediction is that the roll will spin counterclockwise and as a result roll to the left. That the axis of the roll is no longer constrained to be immobile, thus no longer serving as the axis of rotation, is not taken into account. Furthermore, when the experiment is performed, the result matches the prediction. This validates in one’s mind not only the prediction, but also the inappropriate analysis that led to it. One then carries this faulty reasoning to “flipped over” yo-yo 2 (previous figure right.) If pulled, “obviously” the string will unravel clockwise resulting in a translation of the roll to the right. The new scientifically established rule on the board has apparently been set aside in favor of intuitive beliefs compiled after years of observing unwinding rolls of paper.

##### Follow-up

The effectiveness of the method can be quantified by using clickers to monitor the ebb and flow of student understanding as it develops. I have not done that, but I test the persistence of the preconception by putting a multiple choice question on the next hourly test.

Example: A yo-yo has inner diameter D and outer diameter 2D. It is puled to the left by an inextensible string and rolls on a horizontal surface without slipping as shown in the figure below. At some point in time, the string is moving with velocity v. At that same time, what is the velocity of point P at the top of the outer wheel?

(A) | (B) | (C) | (D) |

4v to the left | 4v to the right | 3v to the left | 3v to the right |

The correct answer is (A). Students who answer (B) or (D) are likely to be confusing the rolling motion with the spin of a paper roll about a fixed axis. Students who answer (C) or (D) are likely to be clinging to the idea that the axis of rotation is at the center of the yo-yo. When I ask this question, 40%-50% of the students get it right. Answer (B) comes second at about 20%. I don’t have baseline data from a control group of students who have answered the question without the benefit of the demonstrations, but I believe the method works.

Explaining Rolling Motion

Continue reading the Original PF Insights Post.

Here are some puzzles similar to the spool for your students:

Great first Insight @kuruman!

Good job, thank you.

How would you respond to a student who raises the issue of "frame of reference" for which the definition of "axis of rotation" is made?

For example if a student said in the context of a coin rolling on a table at a constant velocity, "What about from the frame of reference of the center of gravity of the coin? The table is moving in a straight line relative to the COG, the edge is moving without slipping at the same speed as the table, but the coin is rotating about the COG which is stationary (translationally). Why in this frame of reference should the COG not be considered the axis of rotation?"

How would/should you respond?

Indeed, in the frame moving with the coin, the COG is the axis of rotation. The fact that in the moving frame of reference it is and in the stationary frame it is not gives rise to the preconception that I attempt to explain in the first paragraph of "Whence the preconception?" Let me rephrase my hypothesis. In the moving frame of a rolling wheel, the axis of rotation is the center of the wheel. An observer in this moving frame "knows" that (s)he and not the ground is moving. Put the same observer at rest on the ground. When (s)he sees a wheel roll by, (s)he carries over the mental image of the axis of rotation being at the COG and doesn't even consider that this axis of rotation in the stationary frame is never at rest, not even instantaneously. On one hand most people are unaware of the physics definition of the axis of rotation. On the other hand, they find it more convenient to think of the COG as always being the axis of rotation. They are intuitively familiar with axes of rotation that are stationary relative to them (spinning turntable, opening door, etc.), but have a difficult time visualizing an instantaneous axis of rotation. You and I and everybody else are perfectly content driving our cars and riding our bicycles without giving a hoot about where the axis of rotation is. Nevertheless, students in a physics class ought to know the correct way of looking at rolling motion.

As referred to in AT's post, in the insights "verification" section, one reason for which way the cable spool moves in response to where a force applied to the side arm has to do with effective gearing. The side arm could be replaced with an inner cylinder 1/2 the radius of the cable spool, and geared to rotate at 3 to 4 times the rate of the spool. In this case a horizontal tension applied to a string wrapped under and around the inner cylinder, would result in the spool moving in the opposite direction of the string, even though the force is being applied at 1/2 the radius above the ground that the spool rolls on. This is the same reason that the second walkway cart in AT's second video moves in the opposite direction. Not mentioned in that video is the reason the carts move at +/- 1.0 the speed of the walkway is that the gear ratios are 2:1 or 1:2. With gear ratios of 1.5:1 or 1:1.5, the carts would move faster, and with gear ratios of 3:1 or 1:3, the carts would move slower.

I agree but would add that different problems can have different correct ways of looking at them. Øyvind Grøn's paper 'Space geometry in rotating reference frames: A historical appraisal', ( http://areeweb.polito.it/ricerca/relgrav/solciclos/gron_d.pdf ) is a useful compendium of various different approaches.

Grøn also gives a calculated solution to one relativistically rolling wheel problem in the paper. The part C 'optical appearance' plot, at retarded points in time, provides

I have only seen one complete verification of Grøn's part C plot (on another forum) and the process is interesting considering the basic parameters; no z axis (no Born rigidity issues), x, y (all in t) only with wheel, axle (or carriage) and road frames at any specific time. The specific time gives us the location of the axle anywhere over one complete rotation and the velocity of the wheel gives us the length contracted location of the emission point wrt the axle and its location and also the fixed observer (camera) location on the road ahead. The photons just have to travel straight from their emission point to the camera at c.

One major elements missing from the part C figure are the straight line photon paths from each emission point to the camera. The length of these lines can represent the actual time that each photon travels for and, for each emission point, this time must also equal the time that the wheel will take to roll from its emission point axle location to the camera point along the road. As a final cross check you can compare the axle locations and times between consecutive emission points to see if they match the angular velocity of the emission point/spoke tip in part and in all.

Note that while the COG is used in the calculations its use here is as a hypothetical central axle for hypothetical relativistically rolling wheel spokes that had emission points at their tips. Grøn describes the emission points as being on a rolling ring, not a wheel with spokes, although the plot results appear the same.

This issue may be a little more subtle than your explanation so far. Since you don't explicitly say anything about the reference frame when you ask the original question, an answer framed from the point of view of someone on the bicycle is just as correct as an answer framed from the point of view of someone on the ground.

The preconception you mention, is not necessarily a misconception, it is simply a perception from a different point of view, a different reference frame.

> one “knows” who is “really” moving and it is not the ground

It almost sounds like you are thinking that the person on the surface of the earth is in a fixed, preferred, reference system. In reality, all we know about the reference frame of the bicycle and the reference frame on the surface of the earth is that they are in motion relative to one another. Neither is truly stationary, and neither is preferred.

Neither of these reference frames is preferred, and the laws of physics are the same as measured from either frame. These are incredibly important concepts in physics and this example could be a powerful and memorable example for students. So rather than talking about which axis of rotation is correct or incorrect, include how it varies with reference frame.

In the last diagram. Does pulling on the string make the length of the straight part shorter?

Last diagram being the one in the multiple choice question. Yes, the string gets shorter.

To follow up on Alex Kluge post about frame of reference, take the case where a coin is rolling on a treadmill such that the center of mass of the coin is not moving with respect to ground, but instead the treadmill is moving with respect to the ground.

> spool trick

If the spool was on rails and the hub had a larger radius than the sides of the spool, then pulling the string from under and back from the spool would result in the spool moving forwards. The hub could also be geared so that it rotated faster than the spool, so that string speed is greater than spool rolling speed, and again the spool would move forwards. This would be the same as the second walkway cart at 1:28 into the second video of AT's post.

> point of rotation from ground based frame of refernce

To follow up on my prior post, from a ground frame of reference, a point on a wheel follows the path of a cycloid, where the instantaneous acceleration at the point of contact would be zero, and It might be mathematically convenient to use the point of contact as the pivot point, but this conflicts with the tension within the wheel, which corresponds to the centripetal acceleration of all points about the center of mass, regardless of which inertial frame of reference is being used. In my opinion, both views are mathematically correct for a ground based frame of reference, rotation about the contact point, or a combination of linear motion at the center of mass and rotational motion about the center of mass.