# AP1 Unit 8: Rotating Bodies Model

We begin this unit with a simple paradigm activity similar to Unit 4‘s bowling ball games. We begin with a dot on a pulley. I begin rotating the pulley and tell the students to imagine the disk is spinning at constant speed. I ask them to figure out a way to describe the motion of the dot as a function of time. I give them a few minutes to talk as a group. Since my students math experience ranges from Algebra II to Calculus, I will usually have several students who come up with:

$x = r cos \left( \theta\right)$

$y = r sin \left(\theta\right)$

To ensure all the students remember the basic equation, we then discuss the parent function for sine and cosine.

$y = Acos \left(Bx + C\right) + D$

To do this, I first pull up Desmos and show the effects of changing each of the four variables. Here’s what happens when you increase $A$:

Here’s what happens when you increase $B$:

Here’s an increase in $C$:

And an increase in $D$:

We discuss the meaning of $A$ as the amplitude or vertical stretch of the graph, $B$ as the “frequency” or horizontal stretch of the graph, $C$ as the horizontal shift of the graph, and $D$ as the vertical shift. Since we have discussed frequency last unit, I relate that $B$ is not exactly frequency, but that it is related to frequency, and tell them that we will go into more detail later this unit.

From there, I grab a string with a mass tied at one end, I grab the other end of that string and attach it to the edge of the pulley. I then wind the string around the pulley until the mass is touching the side of the pulley. I then challenge them to come up with a way to describe the dot as a function of time as I let go of the pulley and the mass falls to the table. I give them a few minutes to struggle with coming up with a function. After of few minutes of struggle, I suggest looking at the world from a different perspective. I then ask them what other ways could we describe this beyond using $x$ and $y$?

I socratically lead them to using polar coordinates. Since the distance of the dot to the center of the pulley doesn’t change, the motion can be described as a one dimensional motion using only $\theta$. I attach a photogate to the stand, rewind the string, and display the results of $\theta$ vs. time. We usually get a graph that looks like this for $\theta$ vs. time:

<I always have trouble getting the timing of dropping the mass and starting the data collection, so my graphs are usually shifted like this one>

As we discuss the graph, the students usually notice that the graph looks similar to what we saw in Unit 2 other than the time shifted (in this example by a little less than 0.4 s). So I ask them again to mathematically describe this graph. They will usually then say that it should look like:

$x = \frac{1}{2}a_o t^2 + v_o t+ x_o$

When asked what the slope and curvature represent, to which they responded it should represent the “velocity” and “acceleration.” At that point I tell them that we call it “angular velocity” and give it the symbol: $\omega$ and “angular acceleration” which gets: $\alpha$. I then show them the graph of $\omega$ vs. time:

With this, they can see that we get a fairly linear relationship where the slope is the angular acceleration:

$\omega = \alpha_o t + \omega_o$

We then proceed through the same derivations to get to find the area under the curve.

$\Delta \theta = \frac{1}{2} \left(\omega_o + \omega\right) t$

Then use that to find the equation for the $\theta$ vs. time graph.

$\theta = \frac{1}{2}\alpha_o t^2 + \omega_o t + \theta_o$

After this day of work, we progress to worksheet 1, which deals with constant angular velocity problems, then worksheet 2, which deals with constant acceleration problems.

From there, we move on to the experiment for this unit. Here’s a picture of the apparatus:

The quantities that the students manipulate are the hanging mass (external force), the radius of the mass on the “twister” and the mass on the “twister.” In this picture that mass is pretty small, later washers and an additional nut will be on each have of the “twister.” The lab groups have to determine how each of those quantities effects the angular acceleration of the system. Most groups measure the displacement of the falling mass and record that time to determine the linear acceleration, then use the radius of the “plastic wheel” (aka pulley) to determine the angular acceleration. What we eventually find at the end of the experiment is:

$\alpha \propto \dfrac{r_{pulley}\times F_{ext}}{m_{system}~ r_{masses}^2}$

After discussion, I reveal that the constant needed to make this an equation relates to the distribution of mass around the axis of rotation, thus we get:

$\alpha = \dfrac{r_{pulley} \times F_{ext}}{k~m_{system}~r_{masses}^2}$

To help relate this to Newton’s Second Law, we notice that in the numerator, is a quantity that increases rotation, and in the denominator, one that inhibits rotations. The quantity in the numerator we call torque, $\tau$; the quantity in the denominator we call the moment of inertia, $I$. Thus, we can rewrite the equation as:

$\alpha = \dfrac{\tau}{I}$

Before we move on, I make the point to explain that the multiplication to find torque is likely the first time my students have truly encountered multiplication with an $\times$. In Unit 6, we used the dot product to find work. That dot product of force and displacement meant that the two quantities had to be in the same direction, or parallel. Similarly, the cross product means that the quantities multiplied must be perpendicular.

From this point I we relate what we see here to what we saw at the beginning of Unit 3, namely to change the motion of an object we have to change the momentum of the object. Thus, we relate linear momentum, $p = mv$, to angular momentum, $L = I\omega$. We also relate translational kinetic energy, $K_T = \frac{1}{2}mv^2$ to rotational kinetic energy, $K_R = \frac{1}{2}I\omega^2$.

From there, we progress to worksheet 3, which, just like Unit 3, looks at momentum conservation of closed systems. In the worksheet, students are introduced to angular momentum diagrams. Since they have already seen the entire momentum diagram:

We jump to the complete angular momentum diagram. During this worksheet, we then have blank torque vs. time graphs.

The students use this diagram to help solve problems such as a what happens to the angular speed of a star when it collapses.

In worksheet 4, the students are introduced to Free Body Diagrams, in which they draw extended bodies instead of a small circle. I instruct the students that since we are no longer dealing with a particle, they need to draw the forces on the object where that force acts on that object. This worksheet then looks at systems in which the torques are balanced, which would constitute the traditional static torque problems.

In worksheet 5, the students now deal with conservation of momentum in open systems. Some of the problems also require the students to include kinematics relationships from worksheet 2 to solve problems.

In worksheet 6, the students are given first problem with a metal ball at the top of a ramp. The problem walks them through determining how long it would take the ball to go through a photogate at the bottom of the ramp without friction, and if friction causes the ball to roll without slipping. I then set up that ball on that ramp and we test their predictions. We usually get results within $\pm$0.0003 seconds of the time predicted for the ball rolling. We then do a similar problem for a solid disk and a hoop, with similar results.

From there, we review and take the unit test.

SWBAT:

1. determine and correctly use the graphical and mathematical relationships between angular position, angular velocity, and angular acceleration with time.
2. develop mathematical models from graphs of angular acceleration vs. torque and angular acceleration vs. mass, and angular acceleration vs. distribution of mass.
3. use rotational energy as an energy storage mode to solve problems using conservation of energy.
4. create Angular Momentum diagrams.
5. use conservation of angular momentum to solve problems.