The Block and Cylinder Problem

The students had handed in their solutions to the tenth problem set. The instructor, David, took a quick clicker poll to find that most students had the wrong answer to problem #3 (from Close, Gomez & Heron, 2013). With a great deal of new material planned for the lecture, David told the class they would discuss this “very briefly.”

A block and a cylinder, each of mass M, are on a level, frictionless surface. There’s a string tied to the center of the front of the block, and another string wound many times around the cylinder. Each string exerts the same force. The cylinder has radius R and height 2R; the block has sides 2R.

a)  If they both start from rest, which one gets to the finish line first?
b)  Find the angular acceleration of the cylinder.

Textbox Section

A.J. and Maayan express a common idea: the cylinder's energy splits between linear and rotational motions. So the cylinder has less linear kinetic energy, and the block wins the race. Maayan compares this situation to a previous question: A ball rolling down a ramp was slower than a block sliding.

Michael's argument is two-fold: 1) in the prior question, friction was necessary for the ball to roll, and 2) Newton's second law says the block and cylinder must tie. Michael speaks frequently in the class, and other students have their hands raised.

Oona argues that it takes pulling more rope to move the cylinder than it takes to move the block, because rope unwinds, and more rope would mean more time.

Textbox Section

Josh works to respond to the arguments that the block wins, but he seems to have trouble articulating his reasoning. David tries to help.
 
Josh agrees with Oona that rope comes off the cylinder quickly and easily, but he thinks the block and cylinder can still tie. His argument responds to others’ reasoning about energy: The block and cylinder do not have the same energy, because you do more work pulling the cylinder.
 
Off camera, many students have their hands raised. David summarizes both arguments (not shown) and prepares to take another vote.

Textbox Section

Like Josh, Jacob thinks different amounts of work are done on the block and cylinder: It does not matter that some energy goes into the rotation of the cylinder; the cylinder simply ends up with more total energy than the block.

Textbox Section

Shivani and Stanley challenge Michael’s argument because they believe that something different happens to an object when a force is applied on an edge instead of along the center. While Shivani struggles to make her point clear to David, Stanley tries to help. He compares this situation to that of a rock hitting a stick off-center, from a problem in assignment #8.
 
Connor then reiterates Oona’s argument: Getting the cylinder to the finish line takes a “greater distance” of rope than the block, so if you are pulling the ropes at the same time, the block will get there first.

Textbox Section

When David suggests leaving this question for next time, Brian signals off camera that he would like to speak. Brian introduces a new type of argument into the discussion: the results of an at home experiment. He used two identical cans of a soft drink, tying string around one (like a block) and wrapping string around the other. In his experiment, the can with the tied string won.
 
Nitin responds to Brian by saying that the rope comes off the cylinder much faster, and so he does not understand how he can apply the same force to both objects.
 
After this clip, David provides several explanations of why the two objects tie (not shown). Six minutes later, Connor jumps back in.

Textbox Section

Often in introductory physics, the difference between a problem result and the “real world” is that the problem, like this one, excludes friction. Connor’s question fits both with his experiment and with this typical explanation of why “physics class” has a different result from “the real world.”
 
Connor’s previous contributions make more sense in light of his experimental design. By using headphone wire, he could indeed pull on both the “block” and “cylinder” at the same time using the jack end of the headphones. This set up actually guarantees that a smaller force is applied to the cylinder because the wire can unwind instead of being firmly attached. The problem with his set up was not friction, but the unequal forces on the two objects.
 
David continues pressing to move on, and eventually he does, picking up his notes from the podium.