### Block and Cylinder – Background

#### The Course

Physics 11 is a calculus-based introductory course. David Hammer teaches his version with an emphasis on students’ “learning how to learn,” as the syllabus explains. Lectures are responsive to student thinking, with frequent clicker polls and time for discussion. Students watch video “prelectures” and complete conceptual “checkpoint” questions beforehand, from FlipItPhysics.

There is a set of five problems each week. The following is a problem from set #8, which the students had completed three weeks earlier, which comes up in the Block and Cylinder discussion.

5) After break we’re going to be getting into rotational motion, but there’s no reason you can’t get started, based on the physics we’ve done and on what you’ve experienced in the world. So, here’s a start. (No need to read ahead!)

A stone slides across the ice collides with a stick. Compare two ways the stone could hit the stick. One way, the stone hits the end of the stick; the other way, the stone hits the center of the stick. Suppose the stone has the same velocity in each case. Just qualitatively — this is about getting a physical sense of what happens — compare what happens as a result of the two collisions with respect to

a) The angular velocity of the stick — that is, how quickly the stick spins around its center of mass.

b) The velocities (regular old dx/dt) of the stick and stone — that is, of their respective centers of mass.

c) The possibility of damage to the stick — does the stone dent it?

d) Now go get a penny and a pencil, or something similar, and try to hit the pencil with the penny first in the center and then on the end. See if what happens agrees with your answers to the parts above.

Teaching Assistants read and score problems mainly for sensible reasoning and exposition, rather than for correctness, as part of the course’s emphasis on learning how to learn. David posts his solutions as reading assignments, after students hand in theirs.

#### Leading up to this day

The episode we examine took place during the 11th week of the semester, with two more classes scheduled prior to the final exam. Several days had been canceled for snow during the semester, so there was some pressure for time.

The previous week introduced rotational dynamics, including the following slide from the *smartPhysics* prelecture.

The slide shows the result of a worked example: The speed of a ball *rolling* down a ramp is less than that of a block sliding down a similar ramp without friction. By reasoning students mention in this case study, for the block, all of the initial gravitational potential energy goes into the linear motion, but for the ball some of the energy goes into rotation.

#### That morning

Students handed in problem set #10 as they walked into class. For the first 13 minutes, they discussed the following question from that assignment:

4) A bar of length L and mass M is held up by two cords, tied at either end. But one cord slips, and the bar starts to fall as shown. For an instant, the bar starts to rotate around its left end, due to the torque by its weight.

a) Find the angular acceleration of the bar as it starts to fall.

b) And then consider this reasoning: The tip should have the acceleration g, because at that end of the bar it’s just like freefall. Does that reasoning agree with your answer in part a? If not, how do you account for the discrepancy?

The discussion was typical: Students presented arguments and the class came to a consensus. The instructor elaborated on the solution for a couple of minutes immediately before the first clip begins.