Jump to content
Sign in to follow this  
  • entries
  • comments
  • views

Unrolling Toilet Paper

Sign in to follow this  


In his Dec. 17 Action-Reaction blog post titled "Falling Rolls," one of my heroes of physics instruction, Frank Noschese, details an exercise from Robert Ehrlich's book Why Toast Lands Jelly-Side Down.

picture2.png?w=500&h=196The exercise, a rotational motion problem that challenges students to find the ratio of heights at which you can drop two identical toilet paper rolls, one dropped regularly, the other dropped by holding onto the end of the paper and letting it unroll, such that the two rolls hit the ground at the same time. It's a terrific, easy-to-replicate and demonstrate problem that pulls together a great number of rotational motion skills --> finding the moment of inertia, applying the parallel-axis theorem, identifying forces and torques from free body diagrams, and converting angular acceleration to linear acceleration. My students dove into the challenge with zest!

To begin the exercise, we set our variables (H=height for dropped roll, h=height for unrolled roll, r = inner diameter, R = outer diameter), then identified the time it takes for the dropped roll to hit the ground using standard kinematics:

png.latex?t_{drop} = \sqrt {{{2H} \over

Next, we did the same thing for the unrolling toilet paper roll:

png.latex?t_{unroll} = \sqrt {{{2h} \ove

Of course, if we want them to hit at the same time, the times must be equal, therefore we can show:

png.latex?{H \over h} = {g \over a}

Obviously, what we really need to focus our efforts on is finding the linear acceleration of the unrolling roll. To save ourselves some time, we started by looking up the moment of inertia for a cylinder:

png.latex?I = {\textstyle{1 \over 2}}M({

Using the parallel-axis theorem to account for the unrolled roll rotating about its outer radius we find:

png.latex?I = {\textstyle{1 \over 2}}M({

Next, we can use a free body diagram to identify the net torque on the roll as MgR, and use Newton's 2nd Law for Rotational Motion to find the angular acceleration:

png.latex? {{\tau }_{net}}=I\alpha \Righ

Since linear acceleration can be found from angular acceleration multiplied by the radius of rotation ®:

png.latex? a = \alpha R = {{2g{R^2}} \ov

Finally, since we're looking for the ratio of the dropped height to the unrolled height:

png.latex?\ {H \over h} = {g \over a} =

This conflicts with the results from Noschese's class, where they derived

png.latex? \frac{H}{h}=2+\frac{{{r}^{2}}

However, their demonstration based on their results is very convincing. Let's take a look at the difference in ratios using the two derivations:

For a toilet paper roll of inner diameter .0095m and outer diameter R=.035m (our school rolls from the janitor supply closet):

png.latex? \frac{H}{h}=2+\frac{{{r}^{2}}

png.latex? \frac{H}{h}=\frac{3}{2}+\frac

It appears that our derivation is correct, per our visual confirmation with a high speed video camera:

You can follow the original blog response at Physics In Flux.

Sign in to follow this  


Recommended Comments

There are no comments to display.

Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.

Add a comment...

×   Pasted as rich text.   Paste as plain text instead

  Only 75 emoji are allowed.

×   Your link has been automatically embedded.   Display as a link instead

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.

Terms of Use

The pages of APlusPhysics.com, Physics in Action podcasts, and other online media at this site are made available as a service to physics students, instructors, and others. Their use is encouraged and is free of charge. Teachers who wish to use materials either in a classroom demonstration format or as part of an interactive activity/lesson are granted permission (and encouraged) to do so. Linking to information on this site is allowed and encouraged, but content from APlusPhysics may not be made available elsewhere on the Internet without the author's written permission.

Copyright Notice

APlusPhysics.com, Silly Beagle Productions and Physics In Action materials are copyright protected and the author restricts their use to online usage through a live internet connection. Any downloading of files to other storage devices (hard drives, web servers, school servers, CDs, etc.) with the exception of Physics In Action podcast episodes is prohibited. The use of images, text and animations in other projects (including non-profit endeavors) is also prohibited. Requests for permission to use such material on other projects may be submitted in writing to info@aplusphysics.com. Licensing of the content of APlusPhysics.com for other uses may be considered in the future.

  • Create New...