Summer of Math Exposition

First the positive. You are highlighting something very important and confusing and that’s great. Also the animations are fun and friendly, and I like the little song. The color choices are good too.

Now some points for improvement.

When you point out that there’s no force required to keep the object at a constant velocity, I think it’s extremely important to point out here that this ignores friction and that it’s only the net force, not the force the person pulling must exert (which must still be enough to counteract the gravitational force that would otherwise decelerate the weight). Without saying these things, you’re planting a counterintuitive idea in the students’ minds but which actually is not fully true.

When you first bring up the straw, you are asking about the work required to move liquid from the top layer of where it was prior to inserting the straw, to the bottom of the straw, to outside of the cup, but I don’t think this makes sense. When you insert the straw, assuming the top of the straw is uncovered, it simply cuts through the liquid and fills to approximately the same height as it was before, but now also inside the straw. It requires no work to move the liquid here, because it was already there. The work involved is that of moving the liquid that was previously in the same location as the straw, but given a thin straw this would be negligible.

Later when you explain how the forces work with the straw, I am not totally certain I am interpreting you correctly, but I think you’ve also got this wrong. You say there is a force pulling the water upward but this is not true. The force that causes the water to move is the air pressure outside of the straw, which pushes down on the water at the surface of the cup. Sucking on the straw reduces the air pressure inside the straw, so the force in the straw is actually downward, but it’s just smaller in magnitude than the force from the outside air pressure, resulting in a net upward force. At minimum, I would not recommend drawing upward force arrows inside the straw, because this makes it appear that there’s an upward force inside the straw, which could be very misleading to students.

Finally, it’s worth pointing out that the justification for the two different integrals you point to is actually very deep. One justification appeals to changing force across a path: you can integrate a changing force over a displacement path because the energy used is “local” and adds up across time. On the other hand for the water example, the integral isn’t about the force adding up as the object moves across the path, but is about figuring out the total energy by adding up the energy contributions from each piece of a compound object. The fact that energy adds up in these two different ways is not obvious from the formula (in fact, arguably it is not contained in the formula at all), and would be worth highlighting more explicitly, besides just considering the calculation issues.

The breakdown of the formulas was pretty clear, and I get the main point of the video (that different things might be changing, so the dx might be part of the force or the displacement). But this video is about work, so it would be helpful to start with an intuitive definition of what work is, beyond just the integral. What does it mean, and why is this something we should want to compute?

Overall a nice video and I really enjoyed the animation style.

It was cool to have extra exercises in the end and I like the two simple examples explaining how the meaning of dx changes depending on the problem considered.

However, in the beginning it is claimed that math is all about interpretations and the anatomy of the formula is important, but when it comes to interpreting the formula the video does not really go into the interpretation and simply flashes the anatomy before moving on to the next example. I understand that one could pause the video and then see if the interpretations one has made on their own are correct. If the interpretations are not correct the video does not walk one through the solution or gives pointers. So the motivation in the beginning of the video is not really fulfilled by the end.

I really like the visuals and drawings that you made. Overall pretty clear explanation of a very important topic.

“recall that W = F * delta X, so a really small displacement would make the force negligible” at 2:30 is not accurate in the case described previously. You claimed that bringning the mass to some speed v quickly takes no work. But actually it takes delta E = 1/2 mv^2 work regardless if you do it super fast or slow (that’s because in your negligible argument, you forget that F will be really big).

Incredible video! You have a talent for explaining things. And the animations - so unique and nice!

What I liked was that you did more than just show how to calculate the work done in two interesting scenarios (which was entertaining in itself). But by comparing the two scenarios, you also encouraged the viewer to think about similar problems from different angles.

It is this style of teaching which really inspires people to SOLVE problems rather than learn solutions!

Entertaining and nicely illustrated. Nicely different from most of the other presentations.

Simple and useful, not so different from a frontal lesson. Just a remark: usually students encounter “Work” before knowing integration.

Really dissecting an integral to find that the dx is not always referring to the same thing was very cool.

The narration was catchy and drew me in. However, the formulas and handwriting could have been written more clearly. I felt there were some other issues throughout:

1. The illustration doesn’t show liquid ascending the straw.
2. The pulley example did not seem especially well motivated.
3. “Sucking force” is fictional and promotes a bad habit—it’s in fact atmospheric pressure pushing into an area of reduced pressure.
4. The formulas mixed centimeters and meters.