We did this as a final problem in our study of rotational energy. After working through a series of long equations from energy and kinematics, we discovered the final answer is surprisingly elegant:

UPDATE 12/27/2010: Thanks to Dan Fullerton’s class for catching our error. They analyzed the problem using a torque approach and came up with:

I double-checked our energy approach and now get the same answer as Dan. We must have made an algebra mistake somewhere. This is why I love physics — there is more than one way to solve a problem!

H is the drop height of the free-falling roll, h is the drop height of the unrolling roll, r and R are the inner and outer radii of the unrolling roll.

Using large rolls of paper towels, we tested our prediction. Here’s the result, captured in slow motion:

(Despite our mistake, the demo still works. From the video, it seems the students did not release the rolls simultaneously. Perhaps this compensated for our algebra error.)

They were just a tad excited when it worked. And yes, that class is all boys, much to my dismay.

What’s your favorite activity or demo for rotational motion?

We use the Casio EX-FH20. The video above was shot at 210 fps, which has worked fine for most cases. It takes great photos, too — has 2 settings for taking pictures in fluorescent lighting. There are cheaper models, too.

My favorite is the pulling the classic spool demo. Here’s a link to the description from UMD’s awesome physics question of the week website. You pull on the spool horizontally in such a way that it looks like the string will cause the spool to unwind and move away from you, but of course, N2 dictates that if there is a net force toward you, the spool moves toward you. And then there’s a an angle at which you can pull so that the force you exert creates no torque, and you can pull the spool without having it roll at all.

UMD also has a nice rolling vs sliding demo that gets at a similar idea you are getting at with the rolled paper demo.

Spent some time with this on our last day before the holiday break with my class, and we came up with a different solution. Sooo, picked up a copy of Ehrlich’s text (a great book at first glance!) and we still seem to be differing on our solution… not off by a whole lot, but I wonder where our derivations went off track?

Our solution shows H/h=3/2 + r^2 / (2R^2).

Basic derivation — set the times for the two drops to be equal, the real key is finding the linear acceleration of the unrolling TP roll. To do that, we find the moment of inertia of the unrolling TP roll as I=(M/2)(r^2+3R^2) by looking up the moment of inertia of a cylinder, and applying the parallel axis theorem. From here, we find the net torque as MgR, then solve for angular acceleration using Newton’s 2nd Law (Angular Version): Angular Acceleration = Net Torque / Moment of Inertia. Finally, we convert our angular acceleration to linear acceleration by multiplying the angular acceleration by the outer radius R, for an angular acceleration of a=(2gR^2)/(r^2+3R^2).

Have a great holiday!

(Don’t you love these problems that really make us delve into more than just the math, but really trying to understand what it means?)

Would love to get your thoughts on our derivation — might be a great opportunity to talk to each of our classes about the assumptions we made and what effects they may have on our final results! I tried to highlight our class’s derivations here: http://aplusphysics.com/flux/aplusphysics/unrolling-toilet-paper/

Looks like we made an algebra error. I ran through our derivation again (which uses energy instead of torque) and now I get the same solution as you, which is also the same solution in Ehrlich’s book. Thanks for trying it out with your students and catching our mistake! (In my video, it looks like our students do not release the rolls at exactly the same time, which may have compensated for our incorrect solution.)

Looking forward to trying out an energy approach –> didn’t even occur to me to use conservation of energy, but that’ll start our class off when the kids return after 1st of the year!

Still a terrific activity… my kids went through and verified their solutions, and had one group solve by conservation of energy to show a different route. Then, following your lead, we verified with the high speed camera. Slick!!! Thanks for a great activity that had my students excited and engaged.

I’ve also had great success with this demo with my students. I like to start the second semester of this course with Rotation, and agree that it is always my favorite to teach students. For those who use Logger Pro and Vernier equipment I strongly recommend getting the rotary motion sensor and accessory kit. I use this to build a strong conceptual understanding of angular acceleration, torque, and rotational inertia. I also enjoy using the CPO straight track with marbles (some hollow plastic, some solid metal) and having students predict the distance from the end of the table where the marble will land. The track is very well made with a rubber strip for reliable friction that the experimental results are very close to the theoretical predictions. Finally, I enjoy combining our work with SHM and rotation in a physical pendulum project.

Frank,

Which digital camera are you using for high speed?

Paul

Paul,

We use the Casio EX-FH20. The video above was shot at 210 fps, which has worked fine for most cases. It takes great photos, too — has 2 settings for taking pictures in fluorescent lighting. There are cheaper models, too.

My favorite is the pulling the classic spool demo. Here’s a link to the description from UMD’s awesome physics question of the week website. You pull on the spool horizontally in such a way that it looks like the string will cause the spool to unwind and move away from you, but of course, N2 dictates that if there is a net force toward you, the spool moves toward you. And then there’s a an angle at which you can pull so that the force you exert creates no torque, and you can pull the spool without having it roll at all.

UMD also has a nice rolling vs sliding demo that gets at a similar idea you are getting at with the rolled paper demo.

Love these, too! We’re starting torque this week. These would make a nice hook!

Great demonstration — I can’t wait to give this to my AP-C class as a challenge problem!

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Spent some time with this on our last day before the holiday break with my class, and we came up with a different solution. Sooo, picked up a copy of Ehrlich’s text (a great book at first glance!) and we still seem to be differing on our solution… not off by a whole lot, but I wonder where our derivations went off track?

Our solution shows H/h=3/2 + r^2 / (2R^2).

Basic derivation — set the times for the two drops to be equal, the real key is finding the linear acceleration of the unrolling TP roll. To do that, we find the moment of inertia of the unrolling TP roll as I=(M/2)(r^2+3R^2) by looking up the moment of inertia of a cylinder, and applying the parallel axis theorem. From here, we find the net torque as MgR, then solve for angular acceleration using Newton’s 2nd Law (Angular Version): Angular Acceleration = Net Torque / Moment of Inertia. Finally, we convert our angular acceleration to linear acceleration by multiplying the angular acceleration by the outer radius R, for an angular acceleration of a=(2gR^2)/(r^2+3R^2).

Have a great holiday!

(Don’t you love these problems that really make us delve into more than just the math, but really trying to understand what it means?)

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Would love to get your thoughts on our derivation — might be a great opportunity to talk to each of our classes about the assumptions we made and what effects they may have on our final results! I tried to highlight our class’s derivations here: http://aplusphysics.com/flux/aplusphysics/unrolling-toilet-paper/

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Dan,

Looks like we made an algebra error. I ran through our derivation again (which uses energy instead of torque) and now I get the same solution as you, which is also the same solution in Ehrlich’s book. Thanks for trying it out with your students and catching our mistake! (In my video, it looks like our students do not release the rolls at exactly the same time, which may have compensated for our incorrect solution.)

Looking forward to trying out an energy approach –> didn’t even occur to me to use conservation of energy, but that’ll start our class off when the kids return after 1st of the year!

Still a terrific activity… my kids went through and verified their solutions, and had one group solve by conservation of energy to show a different route. Then, following your lead, we verified with the high speed camera. Slick!!! Thanks for a great activity that had my students excited and engaged.

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I’ve also had great success with this demo with my students. I like to start the second semester of this course with Rotation, and agree that it is always my favorite to teach students. For those who use Logger Pro and Vernier equipment I strongly recommend getting the rotary motion sensor and accessory kit. I use this to build a strong conceptual understanding of angular acceleration, torque, and rotational inertia. I also enjoy using the CPO straight track with marbles (some hollow plastic, some solid metal) and having students predict the distance from the end of the table where the marble will land. The track is very well made with a rubber strip for reliable friction that the experimental results are very close to the theoretical predictions. Finally, I enjoy combining our work with SHM and rotation in a physical pendulum project.

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