# Modeling Rotational Collisions

This lab is designed to align with AAOT science outcome #1: Gather, comprehend, and communicate scientific and technical information in order to explore ideas, models, and solutions and generate further questions.

## Materials:

• writing utensil
• calculator
• digital device with spreadsheet program
• digital device with internet access

## ObjectiveS

1. Apply established formulas to calculate the moment of inertia of objects with a high degree of symmetry.
2. Apply the Law of Conservation of Angular Momentum to model a collision involving rotating objects and experimentally determine the moment of inertia of an object with complex shape.
3. Apply the Angular form of Newton’s 2nd Law to model the torque caused by friction and incorporate frictional effects into a more complete collision model.
4. Compare the results of the two models.
5. Use the output of the models to then analyze energy transfer during the collision.

## Experimental Methods

The following video shows the necessary experiments being performed. The data collected during the experiments seen in the video are contained in an online spreadsheet. Copy and paste the data into your own sheet to perform your own analysis.

## A simplified collision MOdel

### Moments of Inertia

If we assume the disk and ring have uniform density we can model their moments of inertia using  known formulas.

1) Write the formula for the moment of inertia of a ring.

2) Use the values provided in the video to calculate the moment of inertia of the ring.

3) Write the formula for the moment of inertia of a disk

4) Use the values provided in the video to calculate the moment of inertia of the disk.  Show your work.

### Collision Model

If we neglect air resistance, friction in the axle of the rotating base, friction in the sensor pulley, and the small moment of inertia of the sensor pulley, then we can model the base, ring, and disk as an isolated system and apply the Law of Conservation of Angular Momentum to the collision.

5) Symbolically write out the Law of Conservation of Angular Momentum applied to the collision. Include the rotating support, disk, and ring in your system. Use the symbols $I_r$, $I_d$, $I_b$ for the moment of inertia of the ring, disk, and base, respectively. Use $\omega_i$ and $\omega_f$ for the initial and final angular velocities.

6) Symbolically solve your conservation equation for the moment of inertia of the base. Show your work.

## DAta Analysis

We can see that our model equation for the moment of inertia of the base requires the angular velocity of the base immediately before and after the collision as inputs.

7) Plot the angular velocity vs. time data for the collision experiment. Be sure to name your graph and label the axes, including units.

8) Find the angular velocity values immediately before (initial) and immediately after the collision (final). Record here:______________________

9) Assuming the rubber belt does not slip, the tangential speeds of the sensor pulley and the apparatus axle must be the same. Symbolically write out the formula for the tangential speed of the sensor pulley in terms of angular speed and pulley radius. Do the same for the support base axle.

10)  Set the tangential speeds equal and symbolically solve for angular speed of the base in terms of the angular speed measured by the sensor. Don’t assume that the radius of the pulley and the radius of the base axle are equal.

11) If radius of the base axle and the radius of the pulley were not the same then we would need to measure them so we could use your previous equation to find the true angular velocity of the base from the measured pulley angular velocity. In this case the they are the same, so we can cancel them out of the previous equation. After making that cancellation we find that the angular velocity of the base and the the angular velocity measured by the sensor are _______________.

12) Enter the velocity values and the disk and ring moments of inertia into our model equation solved for the base moment of inertia to find that value. Show your work.

## A more Complete Collision Model

Our model of the collision assumed that friction in the axle of the support was negligible, but at this point we don’t have any evidence to support the validity of that assumption. We will use the rotational form of Newton’s 2nd Law and rotational kinematics to model the effects of friction and air resistance on the velocity and then incorporate that effect into our Angular Momentum model to estimate how the

\begin{equation}

\tau_{net} \Delta t = \Delta L

\end{equation}

If we expand the change in angular momentum we have

\begin{equation}

\tau_{net} \Delta t = L_f -L_i

\end{equation}

The final and initial angular momenta can be calculated from the moments of inertia and the final and initial velocities.

\begin{equation}

\tau_{net} \Delta t = (I_b + I_d + I_r)\omega_f – (I_b + I_d)\omega_i

\end{equation}

We can obtain the collision interval $\Delta t$ from the velocity data, but we don’t know the torque caused by friction and air resistance. However, we can use Newton’s Second Law to replace the frictional torque in the equation. For example, the frictional torque must be equal to the moment of inertia of the base + disk multiplied by the angular acceleration they experience when the system spins down due to friction. $\tau_f = (I_b + I_d)\alpha_f$.

\begin{equation}

(I_b + I_d)\alpha_f \Delta t = (I_b + I_d + I_r)\omega_f – (I_b + I_d)\omega_i

\end{equation}

We can analyze the angular velocity data prior to the collision to determine the angular acceleration caused by friction and air resistance. At that point the only unknown variable in the previous equation will be the moment of inertia of the base.

13) Symbolically solve the previous equation for the moment of inertia of the base. Show your work.

14) Do you expect that your model that ignores friction will produce a higher or lower value for the inertia of the base as compared to the model that does account for friction? Explain your reasoning. [Hint: Think about how friction changes the angular velocity and if that would make it appear as though the base inertia was larger or smaller. Also compare the equations produced by the two models, how would differences in the equations change the result for the base inertia calculation?]

## DAta Analysis

To use the model equation your derived above we need to determine the acceleration of the spinning base +disk under the influence of friction and air resistance, or $\alpha_f$.

15) Plot a the angular velocity vs. time data for the period prior to the collision when the base + disk are spinning down due to friction alone. Be sure to name your graph and label the axes, including units.

16) Fit a line to the data and record the fit equation  and R2 value here:

17) Examine the fit equation to determine the acceleration due to friction.[Hint: What part of the velocity vs. time graph represents the acceleration?]

18) Enter the acceleration value into your model equation for the base moment of inertia of the base. Also enter the known moments of inertia for the disk and ring, the measured initial and final collision velocities, and the  collision interval. Calculate the moment of inertia of the base.

## Conclusions

19) Compare the results of the simplified model and the advanced model. Did the model ignoring friction produce an base inertia value that was larger or smaller than the frictional model in the way that you predicted? Explain.

20) Calculate a percent difference between the output of the two models. Show your work.

21) If we want to maintain 5% accuracy in determining moments of inertia using our experimental setup can we use the simplified model or should we use the more complete model instead?

## Further Questions

22) In addition to accounting for friction, what next steps would you take to further improve the collision model (make the collision model more complete)? Explain why the changes would improve the model and how you would determine any new unknown parameters added to the model by your changes.

23) What type of collision did we analyze? [Hint: The objects started separated and then stuck together.]

24) Do you expect kinetic energy to be conserved during this collision? Explain.

25) Calculate the percent change in kinetic energy during the collision. Show your work and work symbolically until the end.

26) Where did the kinetic energy go? 