Hmm…if the wheel’s rotation slows at a constant rate, I’d expect those plots of beeps vs time to be parabolic. Plotting the beep rate vs time would be linear with the slope related to the angular acceleration and the intercept the initial beep rate. If we knew how many spokes are on the wheel we could convert the beep rates to radian per sec (or to revolutions per second or to degrees per second).

The angular acceleration should be independent of the initial rotation rate. One could check that be analyzing the data from the two spins. You might be able to accurately predict the time to stop and the number of rotations of a new third spin if you measured the initial rotation rate of that spin.

The data looks similar to the speed of a falling object experiencing a drag force linearly dependent on speed, or the charge on a capacitor charging through a resistor. C(1 – e^-t/k) C would be the final number of beeps on a given spin, and maybe k would be constant for any spin because it’s based on the arrow-peg resistance and not on the initial push. k is called the time constant for RC circuits, and determines how long it takes to get up to a certain % of the number of final beeps (after 1k seconds, you are at (1 – e^-1k/k) or 63% of the final number of beeps, after 2k seconds, you are at (1-e^-2k/k) or 86% of the final number, etc). This doesn’t immediately give you a way of calculating the number of beeps as a function of spin speed, but it could be part of the process.

Thinking out loud… it’s a case of rotational friction. The time for one revolution would give you the initial angular velocity.

Doing some sort of fit to the first curve (not sure what functional form would be most physically meaningful and 3 minutes of googling failed to tell me the answer) should give you some way of empirically measuring the angular deceleration friction from the ticker and whatever friction is between the bearing and the wheel.

Then the frictional factor applied to a new initial angular velocity might let you predict the curve for the second spinner and predict where it would stop… Unless the frictional factor is itself dependent on the angular velocity (which it well might be) in which case you’d need to determine that before you could make the prediction you’re looking for.

Overall this seems like a great problem for a college mechanics class, but maybe not so great for any high school math classes I know…

Sidenote:
Awesome book that describes a group of people in the 80’s with custom toe-controlled computers that would predict where the ball will fall on a roulette simulation. Also chaos theory. Great book.
http://www.amazon.com/Eudaemonic-Pie-Thomas-Bass/dp/0595142362

Note, all of my math is basically ignoring edge effects due to the tickers and other parts of the system. Viewing it as simply a dynamic system with velocity and deceleration.

Using GeoGebra, each of the two graphs do seem to fit parabolas pretty well, and there does seem to be a constant deceleration of very close to the same magnitude for both spins (since each quadratic has approximately the same leading term).

As an aside, I often use this clip – also from The Price is Right (the worst contestant ever):

http://www.youtube.com/watch?v=2QZpnBKZHVo

Ask: What maths comes from this video?

Some side questions:

There are 3 contestants spinning the wheel.

a) Spinner#1-
Assuming that the wheel is random, what strategy should spinner #1 take? (In other words, at what number should spinner #1 spin again? Where do we set the line?)

b) Spinner #2-
If spinner #1 goes over (loses), where do we set the line for spinner #2?

I spent a few hours working out probabilities of different TPIR games back in my University days. I definitely had some fun with wheel spinning strategies. Also fun for strategy talk that leads to very common math themes are, The Dice Game, The Clock Game, and Plinko just to name 3.

TPIR is definitely full of mathematical treasures. Now if we could only find someone who can pull some video off the internet and create some graphics.

Someone should also call CBS…lots of fun to be had.

It’s a rotational kinematics problem in physics. It involves using something called the moment of inertia as the “mass” of the wheel and doing position, velocity and acceleration in a polar coordinate system. The angular acceleration is found using the rotational equivalent of Newton’s laws of motion; Torque = (moment of inertia) x (acceleration) which is just F=ma for rotation. The acceleration should be approximately constant until you get to the last little bit where the higher order drag terms take over.

Just off the top of my head, if I were going use this in my math class I would have the woodshop build a large-ish solid wooden wheel and hang weights off it. I would steer them towards a plot of weight vs time and curve fit to get the acceleration experimentally. Maybe throw in standard deviation or just range to get error and have them use it to predict a stopping point within a certain number of spaces as the error.

I’m going with Rhett: frictional torque is likely constant, thus it slows with constant angular deceleration.

The graph of the beeps have 2 distinct features: A “happy” parabola while the player is accelerating the wheel, and a “sad” parabola after the player lets go and the wheel is decelerating.

If you time-shifted the graph so t=0 is when the player lets go (thus just the “sad” parabola part) and did a curve fit, I bet the x^2 coeffient (which represents 1/2 the deceleration) would be nearly the same for both spins.

[goes to analyze graph]

Sure enough!
http://fnoschese.files.wordpress.com/2012/01/priceisrightanalysis.png

There’s percent difference of just 1.9% in the x^2 coeffiecients for the two spins!

So the wheel slows down a constant, consistent rate (like Roy said). Great. But to make the prediciton on where it stops, you would still need to know the angular velocity when the player lets go and the starting position of the wheel.

Better question though:
When you are the player, you really can’t control the starting position of the wheel. So, given the starting position, determine the angular velocity you must impart to the wheel so that it slows down to the position you want.

What a rich problem space!

[Note: I don’t know if the x^2 coeffiecient would be the same for spins on different days. I wonder if they lube the wheel from time to time, which would decrease the frictional torque and thus decrease the deceleration.]

Students could easily replicate this in the lab using a record player. Is the angular deceleration constant as the record player slows down? Is the angular deceleration the same when the record player slows down from 16 RPM, 33 RPM, 45 RPM, and 78 RPM?

Despite Frank’s analysis I’m skeptical that the acceleration will be constant as the wheel slows to a stop. I agree with Sam (comment #3) that a model resembling air resistance is probably more accurate. If the torque slowing down the wheel was solely due to friction in the axle, then the acceleration would be constant. However, I think a significant amount of the torque is caused by the pegs on the wheel hitting the arrow. When the wheel is spinning faster, these torques are applied with more frequency.

@DavidGarcia: Very good point. But now we can debate about modeling and which model is “best.” Is the best model the one which is the most accurate portrayal of reality? Is the best model the simplest one that still yields accurate predictions? (Simpler models are cheaper and faster to use.)

We don’t include relativistic effects in our physics courses, but they are still there none-the-less. Same for ignoring air resistance at low speeds.

I wonder if the constant acceleration is accurate only when the initial velocity of the wheel is smaller than a certain value. If the initial velocity of the wheel is very large, then the repeated hits from the peg will slow the wheel down more rapidly than friction alone. Similarly, air resistance and relativity must be taken into account for some scenarios, but can be neglected for others.

More data please! I wonder if a gaming wheel found at carnivals (the local fire department or police department might actually have one) could be borrowed for data collection and modeling.

I thought about this on the way home for work. I’m certain that for a high school class Frank has the most accessible way to approach this. The trouble with real honest-to-goodness physics is that any measurement you make, no matter how accurate, is an approximation. You can always increase your accuracy by an order of magnitude or two but at the end of the day you have to pick the decimal place you stop worrying about accuracy for and move on. For my high school math classes that’s the hundreths place more often than not.

That wheel is big enough compared to the pivot to basically ignore complicated drag leaving a nice parabolic graph to discuss. Each beep corresponds to 1/20th of a trip around the big wheel leaving behind a solid position vs. time graph. To me the classroom question “where does it stop?” is estimated by either nailing down the rotational acceleration from that graph or using the first few points in the video to fit the rest of the quadratic. I think I’ve seen Dan put together something similar for predicting free-throw shots from the first half of the shot. In this case they would just need to use the curve fit to find the maximum number of beeps to know where it lands. The fact that the wheel is broken into nice neat 5% increments makes me a little excited to talk about measurement uncertainty without having to get extremely accurate in order for the students to feel like theirs is “right.

You’re question of “is it predictable” is close but veered away from the question that was obvious in my mind: “Do they cheat?” As a kid, ever since I learned that the people they call down to play are chosen by their producers, I always wondered whether they cheated on the wheel too. If the class turned into more of a detective case, I think that’d be more engaging for the kids. Predicting where it should land doesn’t seem realworld-applicable. No one is ever going to be able to train to always get a dollar on the show. But…finding out whether or not the show is fair, that (to me at least) is really interesting.

Maybe you could have the students convert the graphs into speed vs. time for several different contestants. If they are all linear with the same slope, it’s fair. And even if it is fair, you could ask them what a graph might look like if someone backstage, or even a computer program, were rigging the process.

It would be interesting to ask in a statistics class, “How predictable of a model can you build?” So for instance, the suggestion that many people have had “fit a parabolic model.” turns into the questions: “What is the best possible parabolic model you can fit?” and “Is the uncertainty/variance in that model so high that it’s basically rubbish?” which could also be varried to “Suppose I let you guess which three consecutive numbers the wheel might stop on. Now how good of a prediction can you make?”

Hey Dan, quit stealing the material from my Mathematics of Game Shows talk! (Just kidding, I’ve got a bunch of other questions from TPIR to think about instead.)

I’ll agree with the majority opinion: you can ‘simplify’ the problem by calling it a constantly-decelerating wheel. In this case it works like something thrown in the air: constant negative acceleration. And since “beeps” is basically “vertical distance”, it should be like a parabola.

It’s not actually continuous: there is a small constant deceleration due to the wheel itself, but a larger discrete deceleration when the wheel hits one of the pegs. So it actually works more like the way some kids draw a quadratic: down 7, over 1, down 5, over 1, down 3, over 1. Either way a quadratic fit should be very accurate.

Frank and Aaron are pretty much right on the money here, and you’d need a more accurate reading to know more.

The show’s wheel has crazy variation in how fast or slow it is show-to-show. When it’s ‘greased up’ it spins like crazy, while at other times it can be difficult to get the thing all the way around.

Dumb but related: what’s the least likely thing that can happen in a 3-player round of spins, and how likely is it?

Chaos Mathematics doesn’t have anything to do with a single spin. Chaos would have to do with trying to predict the landing place on several consecutive spins with a specific initial condition (and the limit in that predictability does become uniform as the number of spins increases). If initial speed is known (the average of the first couple beeps isn’t a terrible approximation) as well as the starting location, then predicting the ending spot within a couple locations is completely doable.

“It probably goes without saying I’m wondering, “Is it predictable?” What model underlies the showcase spinner? If you knew the initial position of the spinner and, say, the amount of time it took the spinner to complete one revolution, how close could you get to predicting its final position?”

Given the large mass of the wheel, the system is almost entirely defined by the rotational inertia of the wheel and the opposing torque of the bearings (due to friction) which appears to be substantial and constant. Drag (air) is insignificant and the wheel is very predictable (as the smoothness in your data shows). This rotational problem (as some posters have already noted) is identical (mathematically) to a ball thrown up in the air (constant acceleration). I am sure that with practice a player could “land” the wheel on a given spot with fairly high probability in the same manner that a basketball player can sink a shot from different positions on a court.

It makes me think of the book, the Quants. When they tried to predict the outcome of roulette. Everything else I was thinking was already said.