Rolling without Slipping Demo

[erobz]

Homework Statement

Rolling without Slipping Demo - Walter Lewin

Relevant Equations

$\sum F = ma$
$\sum \tau = I \alpha$

So I'm watching this demo posted in another problem by @PeroK by Prof. Walter Lewin



and I can't help but see that in the part where he is demonstrating the independence of acceleration cylinder length that the (presumably) shorter aluminum cylinder is edging out the win...Not intended, but visible. So, think it must be drag? However, I come to the equation:

$$ \frac{3}{2} m \frac{dv}{dt} = mg \sin \theta - \frac{1}{2} C_d \rho_{air} A_{proj.} v^2 $$

Then use the mass of the cylinder to eliminate $A_{proj.}$:

$$ A_{proj.} = \frac{4}{\rho_{cyl} \pi D} m $$

$$ \frac{3}{2} \cancel{m} \frac{dv}{dt} = \cancel{m} g \sin \theta - \frac{1}{2} C_d \rho_{air} \frac{4}{\rho_{cyl} \pi D} \cancel{m} v^2 $$

$$ \frac{3}{2} \frac{dv}{dt} = g \sin \theta - \frac{1}{2} C_d \rho_{air} \frac{4}{\rho_{cyl} \pi D} v^2 $$

And voila...I'm not left with an EOM dependent on $L$ (the length of the cylinder). Hmmm, so what is the explanation for what I'm seeing...given just this one observation (was it a fluke)?

 

[TSny]

And voila...I'm not left with an EOM dependent on $L$ (the length of the cylinder). Hmmm, so what is the explanation for what I'm seeing...given just this one observation (was it a fluke)?

Without repeating the experiment several times, we don't know if the result is consistent. It appears to me that the longer cylinder did not roll straight down. It veered toward us a bit. But I don't know if this could account for the difference.

 

[erobz]

Without repeating the experiment several times, we don't know if the result is consistent. It appears to me that the longer cylinder did not roll straight down. It veers toward us a bit. But I don't know if this could account for the difference.

I see what you are saying there, the gap from the edge of the board does also appear to be closing. I guess we can't really tell what the other is doing. So we need to get ahold of Prof. Lewin, or his cylinders...

 

[erobz]

It's also clipped. Not that it matters per-se, but perhaps the attendee challenged the result too and there was another run that got tossed. At least that's what I think Columbo would say...

7:07 end of the run

7:15 apparent pick up with demonstration. Notice the sticker that appears.

 

[TSny]

Interesting. Good eye!

 

[kuruman]

So we need to get ahold of Prof. Lewin, or his cylinders...

And his incline. Are we sure that the cylinders roll without slipping? Are we sure that the coefficient of static friction is the same across the length of the cylinders?

The cylinder farther from the viewer is clearly winning despite what Lewin claims.

 

[erobz]

And his incline. Are we sure that the cylinders roll without slipping? Are we sure that the coefficient of static friction is the same across the length of the cylinders?

The cylinder farther from the viewer is clearly winning despite what Lewin claims.

View attachment 352276

I was hoping someone from here had the demo from their teaching career already. I think I have some material laying around, (smaller radius steel axle), maybe I'll cut it and make some of my own. However, I don't have a lathe so I would be concerned about the precision of the cut ends.

 

[kuruman]

I was hoping someone from here had the demo from there teaching career already.

I have such a thing but not quite what you are looking for. Start at about 13:40 in the video shown below. Note that the sides of the cylinders are wrapped with black tape to increase the coefficient of static friction. It is a 20-year old video, so the quality is not great. There is a sad and happy face drawn on the flat sides of the cylinders facing the audience. Guess which is which.



The falling plank, another nice demo that is not often done although it never fails, is at about 33:30.

 

[erobz]

So in the meantime, I tried to investigate the veering off course. I'm sure this is crude, but:

At best I think I can say that for the experiment $a \leq g \sin \theta$.

I estimate $\theta \approx 5^{\circ}$, lets say it's at most $10^{\circ}$. I just took a snip, pasted it into PowerPoint and drew some reasonable triangle and got the ratio (height to length of the ramp read out from the line properties - It seems reasonable)

Time down the ramp, approx. $1.5 ~\text{s} \pm 0.5 ~\text{s}$

From a top down view I would estimate the right most cylinder veers $1 \text{cm}$ per $1 \text{m}$. Meaning from the side I'd expect the leftmost cylinder to appear to be in the lead by (trying to make this error propagation as large as possible:

The length of the ramp I estimate as $1 \text{m} \pm 0.3 \text{m}$. Keeping consistent signs for that length in the numerator/denominator of the last fraction: Again, that the largest I can make it keeping consistent signs for error on the ramps length.

I it looks like we are seeing a lead of maybe $2 ~\text{cm}$ in the video. Is that worth a crap? I hope I'm not too annoying beating this horse, and rather I do the experiment, but I don't actually have the material after all. Furthermore, I'm also checking my limited understanding of error propagation. I think you (scientists) usually use statistical techniques, but I don't see any harm in just going for the throat of that error. The question I'm asking is have I got the throat, are there different numbers anyone would want to see computed, etc...