Lerner Physics

We finished talking about the central force model by summarizing how we could apply kinematics, forces, energy and momentum to circular motion. Talking about momentum with the central force model was new to me this year, but, if you think about the planet-star system as having no non-negligible external forces, then the center of mass stays at rest or at constant velocity. When the planet goes one way, the star goes the other way. Such an obvious way to see them both rotating around the center of mass of the system.

Then we drew force diagrams for this situation:

We weren't sure how to draw the tension forces; don't they cross each other? And how do we show that the little mass is hanging down on the meterstick in the middle of the meterstick? When we moved the little mass, the whole setup was ruined.

We broke the particle model. We need to model this meterstick as something other than a dot. Off to lab to figure out how to keep something balanced when we care about where the force is being applied.

The Mulvey Method helps students see what happens how something like the force of gravity changes when other quantities change. It's great for questions where all the answers look like F, 2F, 4F, F/4, etc. It's named after Hillary Mulvey, Brown '14, who came up with this technique.

Let's try it on an easy relationship.  If the kinetic energy of an object is doubled, what happens to its velocity? Start by writing the equation twice that describes the relationship you're studying, once under a column labeled NEW and once under a column labeled OLD. Then, under the OLD column, plug 1 in for everything, and I mean everything, that isn't the dependent variable in the situation. So your work should look like this:

Yes, even replace the 1/2 with a 1. You're looking for the ratio between quantities before the change and after the change, so we set all the quantities, including constants, equal to 1. 

Then, it's time to figure out the values for the NEW column. The kinetic energy doubles, so it goes from 1 to 2. The 1/2 factor didn't change, it was "multiplied by 1", so it's a 1 in the next line, and, since the mass didn't change, it also was multiplied by 1. So that leaves me with:

Now you can solve for v to be the square root of 2, and so the velocity is multiplied by the square root of 2. Now this technique isn't the most mathematically rigorous, but I've found it can help students figure out what these types of questions are asking, and, over time, they start seeing the relationships without having to resort to the Mulvey Method. I also found that when I try to do it a more mathematically rigorous way at the beginning, my students get confused by all the variables, but when I show why the Mulvey Method works in March or April, it makes perfect sense.

We also got to play with PHeT's Solar System simulation, using a lab I think I've done since 1998 in some form or another, starting with Interactive Physics software on the old fruit-colored iMacs. 

We had shortened periods today due to the PSATs. So we were all a bit exhausted from testing. We used the PHeT Gravity Force Lab to make a model of the force of gravity; the dependence of the force of gravity on the distance between the two objects was hard to ferret out, but it worked nicely once we happened on a good curve to fit. That's about how we had time for today.

Thinking about circular motion is hard because we have to think in two dimensions. Those of us who were attached to positive and negative direction now need to think in terms of left, right, up, and down. It's harder than you'd think.

We played a new game today, a twist on the mistake game. I had groups of two put up two whiteboards, one with a mistake they thought was a good mistake. It turns about the best board discussions were around how to convince someone of their mistake. It also was great when students made two different mistakes on the two boards. I think next time, I'll ask the groups of two to make a mistake in at least one of the two boards.

We played with the Greek waiter's tray (see above), which is always fun. I make my students go through a three-stage training with the tray: first, a soft object that's grippy like a foam apple, then a roll of tape, which seems to slide off pretty easily if they're not careful, and finally, for stage three certification, a cup of water. 

We had a little time today to start talking about what would happen if you launched an object horizontally faster and faster from a little ways off the surface of the Earth. I got to draw Newton's Mountain on the board, which is one of my favorite visuals.

After our assessment on momentum and projectiles, we whiteboarded some questions about circular motion. The biggest thing that seemed to trip us up was whether the force that pushes you in a circle is another force that should be drawn on the force diagram or a force we already know from our interaction diagram. We also had a great discussion about how much kinetic energy and potential energy you have to give a satellite to get it into orbit. (This is another reason I'm happy I postponed circular motion & projectiles until after energy & momentum.) We ended by talking about the forces on banked curves. We did it just qualitatively, but there were many, many questions about it.

Have you seen the Veritasium video about what happens when you drop a Slinky? If you haven't, you must go there now. (And make a guess, watch the answer video, and make sure you watch the amazing slo-mo video that's linked to the answer video.) It's a great little piece of physics. 

You can do force diagrams of both the top of the Slinky and the bottom of the Slinky and the top of the Slinky accelerates at greater than 9.8 m/s². So we talked about that, and then I got out my foam apple. 

Where do I have to hold the foam apple so when I drop the Slinky and the foam apple at the same time, they hit the ground at the same time?

As soon as I asked the question, it was pandemonium. Kids started grabbing whiteboards. Heated discussions broke out everywhere. One student started counted the number of turns in the Slinky. There was much arguing. 

But what they all knew, what they all seemed to get, is that the center of mass fall at 9.8 m/s². (I think this lesson really helped.) Here were their whiteboards after 25 minutes of discussion:

They wanted to keep going. It was such an interesting lesson. Their answers weren't that bad; we only had to tweak their answers a little bit to get the great trial you see above. (Don't expect your students to get it perfect; it takes some pretty hard-core calculus to derive the theoretical value.)

Here are the fragments of a broken Petri dish and their calculations to break it.

Without much time spent on the mathematics of projectiles, we seem pretty comfortable using UAPM and CVPM to calculate the kinematics variables. We did lots of practice in TIPERs to make sure we understand the concepts behind projectiles. 

Tomorrow, we'll talk about the Slinky Drop Experience and start the next unit.

We started by going over some conceptual problems on projectiles, and students seemed comfortable with the shape of the velocity-time graphs. At the end of whiteboarding, after doing the last question about calculating where a horizontally-launched projectile would land, a student asked me how to calculate where a projectile would land if it were launched at an angle. He started mentioning components of the initial velocity vector, and I had to hold my tongue. I didn't want to get lost in the math of projectiles this year. We used PHeT instead to simulate the launching of many projectiles. We came up with some pretty powerful conclusions, most of which were suggested my more than one student before I summarized them on the board.

Now that we've talked about the four major models of mechanics, we started projectile motion through video analysis. We immediately saw that the difference in the x-component of the motion and the y-component of the motion. Ut made sense using forces. Energy seemed like an easy way to figure out velocities. And momentum didn't seem useful at all.

We tried a problem about hang time. Two solutions are presented below. Almost every student liked the graphical way and didn't like the equation (which worked & which they learned in math class).

We summarized all four models today in the middle of class. It a good way to realize we have four tools in our tool belt and that we have choices in the ways we look at problems. We will be looking at projectiles and circular motion next.

Before summarizing, we analyzed a ballistic pendulum. A few groups stopped when they found a velocity, thinking that any velocity must be the velocity they want.

After summarizing, we looked at different types of collisions. When would the kinetic energy be conserved in a collision?