Table of Contents
Rotational Dynamics (PHYS 141)
If you are in PHYS 121 or 131 you should go here.
In lecture, you learned how to calculate moments of inertia for certain geometries of solids. While these formulas are useful, it is common to encounter objects that have more complex geometries or even objects whose moments of inertia change with time. In these cases, it is difficult to find a single equation that can predict the object's behavior. However, we can still make empirical tests in the laboratory that can help us deduce what their behavior is, which can help to later inform us in constructing models.
In this lab, you will try to determine the moment of inertia of one aytpical object: the water inside an aluminum can, rotating about the axis passing through its center (from base to lid). In order to do this, you'll have to make very precise measurements, so the apparatus you'll use is fixed. As such, the focus of this lab will be primarily focused on data interpretation rather than experiment design.
In the first part of this project, you will get used to using the equipment by studying the motion of simple objects like disks, hoops, and cylinders. In the second part, you will extend these techniques to the water-filled can problem.
Part 1: Understanding the apparatus
Lab notebook template
This project's lab notebook is wide open – there are very few prompts, and instead your group will need to decide for itself what is important to record. Remember to record your techniques, your data, your observations and your thoughts as you go, not at the end of the period.
Designate a record-keeper (someone different from last week) and open the blank template below.
Brainstorming: Hypothesis generation
You have a can filled with water that can move around inside as the system spins. Can you come up with a potential model (or models) for what the moment of inertia should be for this system? Will the moment of inertia be simple – like a solid cylinder or a hoop – or will it be complicated – a time-dependent moment of inertia, or one that depends on rotation speed? Your group should record your initial thoughts in your lab notebook.
PHYS 131 and 141: Think about limiting cases. If the water is stationary what will the moment of inertia look like? If the water moves with the can as it spins, what will the moment of inertia look like? Try to be as specific as you can be (even including mathematical forms if you can think of them).
Apparatus
One way to measure the moment of inertia of an object is to apply a known torque to it, and to measure the resulting angular acceleration.
To do this, we will use an apparatus (shown in Fig. 1) which consists of disks which can freely rotate (by floating on a cushion of air) onto which we can place our different objects to study. The apparatus consists of an air table, steel and aluminum disks, pulleys, thread and small masses. A torque may be applied to the system by winding a thread around pulleys screwed onto the top of the upper disk and hanging a small mass over the low-friction air bearing.
In order to measure the angular displacement of a disk (and therefore any objects placed on it), the edge of each disk is marked with a series of 200 black lines which can be counted as they pass an optical sensor. This optical sensor is connected to a software program which can convert these detections into displacement, velocity, and acceleration as a function of time.
Figure 1: The rotational dynamics apparatus.
What does the lower disk do?
The lower disk doesn't do anything relevant right now. The apparatus we use can be used for many experiments, such as collisions between rotating objects, but nothing we do today needs the lower disk.
Getting used to the apparatus
To begin, you should practice with the software and setup to get an idea of how it works, how to interpret the plots provided, and what parameters change as the disk spins.
Step 1: Test for excessive friction
Before proceeding, let us test that the disk rotates freely on its air bearing. To do so, turn on the air and spin the upper disk. The optical bar reader should show no more than about 1% slowing between successive readings. If the disk slows more quickly, lift the top disk and clean the surfaces with the cleaning pad provided.
 Optical bar reader. If the disk is spinning and this doesn't show a reading, make sure the selector isn't set to “LOWER”. |
|
 Dirty Disk. Note the black smudges, particularly on the bottom. |  Clean disk! |
You should record your observations in your notebook.
Step 2: Using the Software
Make sure that the ¼-inch phone plug with the yellow band (from the optical sensor) is plugged into the DIG 1 input of the LQM. Open the Logger Pro configuration file titled rotdyn.cmbl which you should find on the computer desktop. (Or, you can download it from the link below).
To take data, click Collect 
. You should see data begin to appear in the data table and on the graphs. Once data collection has begun, rotate the upper disk through one full revolution. Check that the plot of the displacement shows the disk moving $2\pi$ (or $\approx 6.28$) radians as a way to verify that the equipment is working properly.
Why is displacement always increasing?
Note that the optical sensor only counts lines and cannot tell direction. As a consequence, we can only measure displacement, not distance. If you move the disk one revolution clockwise, then one revolution counter-clockwise back to its starting position, the reader will register $4\pi$ radians displacement.
To perform a measurement for this part, wind the thread around the spool, attach a mass to the end of the string, start the data collection, and then let go of the mass so that the disk rotates. Plots of angular displacement (in radians), angular velocity and angular acceleration should be displayed. If you started collection before releasing the disk, you should see that for some initial range of time the angular velocity should be zero. If not, the earlier part of the motion was lost and new data should be taken.
Step 3: Interpreting your data
Now that you've got some data, you should take a moment and interpret the different plots made by the software.
Ask yourself the following:
- Why does the platform (and upper disk and spool) start to rotate as the mass falls?
- What is the force on the mass?
- What is the torque acting on the platform?
- Is the acceleration constant? Why or why not?
How do I use calipers to measure things?
How do I use these weird scales?
If you can apply a constant torque $\tau$ to the object, the resulting angular acceleration $α$ is related to the moment of inertia $I$ as \begin{equation*} \tau = I\alpha. \end{equation*}
We'll be applying a torque by means of the gravitational force of our mass clip $mg$ acting on the spool at a radius $r$: \begin{equation*} \left\lvert \vec{F}\times\vec{r}\right\rvert = mgr = \tau. \end{equation*} From a measurement of the acceleration (either from the acceleration plot or from the slope from a fit of the velocity plot), determine the moment of inertia for the upper disc, spool, and platform together.
What equations will I need for rotational motion?
For our purposes, we'll define the following kinematic terms:
| Translational Motion | Rotational Motion | |||
|---|---|---|---|---|
| Position | $x$ | Angular Position | $\theta = \dfrac{s}{r}$ | $(1)$ |
| Velocity | $ v = \dfrac{dx}{dt} $ | Angular Velocity | $\omega = \dfrac{d\theta}{dt}$ | $(2)$ |
| Acceleration | $ a = \dfrac{dv}{dt}$ | Angular Acceleration | $ \alpha = \dfrac{d\omega}{dt}$ | $(3)$ |
The associated kinematic equations relating them are summarized as:
| Pure Translation | Pure Rotation | |
|---|---|---|
| $x = x_0 + v_0t + \frac{1}{2}at^2$ | $\theta = \theta_0 + \omega_0t + \frac{1}{2}\alpha t^2$ | $(4)$ |
| $v = v_0 + at$ | $\omega = \omega_0 + \alpha t$ | $(5)$ |
| $v^2 = v_0^2 + 2a(x-x_0)$ | $\omega^2 = \omega_0^2 + 2 \alpha (\theta - \theta_0)$ | $(6a)$ |
| $x = x_0 + \frac{1}{2}(v + v_0)t$ | $\theta = \theta_0 + \frac{1}{2}(\omega + \omega_0)t$ | $(6b)$ |
Note that the moment of inertia $\alpha$ plays the role of mass in our rotational equations, but there's a bit more to it. It also depends on how the mass of an object is distributed around the rotating axis: masses further away are harder to accelerate than ones close to the axis.
How can I get acceleration from Logger Pro?
You can get the angular acceleration from either the acceleration plot or the velocity plot. Use the instructions below to do so (click for larger screenshots)
| Acceleration Plot: | Velocity Plot: |
|---|---|
 |  |
| Select a range | Select a range |
 |  |
Click “Statistics”  | Click “Linear Fit”  |
 Read out the info on the plot |  Double-click the info box |
 Check “Show Uncertainty” |
|
 Read out the info from the fit |
While you could also determine the moments of inertia for each shape individually and add the results, there is no need to do so here. We can treat the upper disk, spool and platform as a single object.
While this setup is designed to minimize friction, it may still be present.
- How will you test for and/or measure frictional forces?
A hoop and a cylinder
Practice using the apparatus by measuring the moment of inertia of the two simple objects provided, rotated about the axis passing through the center:
 A hoop: $I = MR^2$ |  A disk or cylinder: $I = \frac{1}{2}MR^2$ |
where $M$ is the mass of the object and $R$ is its radius.
Note that when you rotate multiple objects together, their moments of inertia add: $I_{total} = I_1 + I_2$.
If your object is not centered, then your moment of inertia will differ from these values. The parallel axis theorem describes what you'd expect to observe instead.
The (relative) uncertainty in the measured moment of inertia is given by \begin{equation*} \frac{\delta I}{I} = \sqrt{\left(\frac{\delta \tau}{\tau}\right)^2 + \left(\frac{\delta \alpha}{\alpha}\right)^2}. \end{equation*} After expanding out the uncertainty in the torque $\tau$, this reduces to \begin{equation*} \frac{\delta I}{I} = \sqrt{\left(\frac{\delta m}{m}\right)^2 + \left(\frac{\delta r}{r}\right)^2+\left(\frac{\delta \alpha}{\alpha}\right)^2}, \end{equation*} where $m$ is the mass hung from the string, $r$ is the radius of the spool, and $\alpha$ is the angular acceleration.
Consider your measurement technique in order to account for frictional forces. If you cannot find good agreement with the predicted value, talk with your TA about possible systematic effects before you proceed. If you can't match the results for a known object, then there's no reason to expect that your measurements for an unknown object will be accurate.
Part 2: Measuring the moment of inertia of the water-filled can
Surveys
At the start of the quarter, you completed a pre-survey for the E-CLASS assessment. Since this is the final day of lab, it is time to complete the post-survey.
You may access the survey from your personal computer or phone. The survey is anonymous and your answers will not affect your grade in the course. Your name and student ID will be collected only in order to assign credit for completion. Please complete the survey before continuing on with the rest of the page!
Please complete the correct survey for your course!
| E-CLASS survey |
|---|
| E-CLASS for PHYS 121 |
| E-CLASS for PHYS 131 |
| E-CLASS for PHYS 141 |
The survey will close on Saturday, November 20 and cannot be reopened! Completion of the survey counts towards your lab notebook completion grade.
Measuring the moment of inertia
Now that you know how to use the apparatus, your task is to measure the moment of inertia of the water-filled can. To this end, you have access to additional cans that contain either air or sand. Using the materials at hand, characterize the moment of inertia for the water-filled can.
Look back on the models for the water behavior you proposed at the start of the lab. Your goal is to determine, as best as you are able, which model best represents the actual behavior you see in the experiment. Note that it is okay if you have inconclusive results or find that none of your models match the can's behavior, but be sure to justify your conclusions using the data you gather.
The rotational equations you may find useful are divided into three expandable sections below.
Kinematic Equations
For our purposes, we'll define the following kinematic terms:
| Translational Motion | Rotational Motion | |||
|---|---|---|---|---|
| Position | $x$ | Angular Position | $ \theta = \dfrac{s}{r}$ | $(1)$ |
| Velocity | $ v = \dfrac{dx}{dt} $ | Angular Velocity | $ \omega = \dfrac{d\theta}{dt}$ | $(2)$ |
| Acceleration | $ a = \dfrac{dv}{dt} $ | Angular Acceleration | $ \alpha = \dfrac{d\omega}{dt}$ | $(3)$ |
The associated kinematic equations relating them are summarized as:
| Pure Translation | Pure Rotation | |
|---|---|---|
| $x = x_0 + v_0t + \frac{1}{2}at^2$ | $\theta = \theta_0 + \omega_0t + \frac{1}{2}\alpha t^2$ | $(4)$ |
| $v = v_0 + at$ | $\omega = \omega_0 + \alpha t$ | $(5)$ |
| $v^2 = v_0^2 + 2a(x-x_0)$ | $\omega^2 = \omega_0^2 + 2 \alpha (\theta - \theta_0)$ | $(6a)$ |
| $x = x_0 + \frac{1}{2}(v + v_0)t$ | $\theta = \theta_0 + \frac{1}{2}(\omega + \omega_0)t$ | $(6b)$ |
Relating translation to rotation
Consider the case where an object is rotating about a point with a constant radius $r$. The derivative of $(1)$ is
| $\displaystyle \frac{d\theta}{dt} = \frac{1}{r}\frac{ds}{dt}$ | $(7)$ |
The term $ds/dt$ is equal to the tangential velocity ($v_T$) of the object, and we already know that $d\theta/dt$ is the rotational velocity $\omega$. Substituting these quantities into $(7)$ and rearranging, we find
| $v_T = r\omega$ | $(8)$ |
If we take the derivative of each side of $(8)$, we find that
| $a_T = r\alpha$ | $(9)$ |
where $a_T = d v_T / dt$ is the tangential acceleration and $\alpha = d\omega/dt$ is the angular acceleration.
Finally, if the motion of an object is purely rotational, we can relate the radial acceleration $a_R$ to the angular velocity:
| $a_R = r \omega^2$ | $(10)$ |
Torque, moment of inertia and angular momentum
We may define rotational analogs of force, mass, and linear momentum which will behave similarly to their translational counterparts.
| Pure Translation | Pure Rotation | |||
|---|---|---|---|---|
| Force | $\vec{F}$ | Torque ($\vec{\tau}$) | $\vec{\tau}= \vec{r}\times\vec{F}$ | $(11)$ |
| $\lvert \vec{\tau}\rvert = r F \sin(\theta)$ | $(12)$ | |||
| Mass | $m$ | Moment of Inertia ($I$) | $\displaystyle I = \sum_{i} m_i r_i^2 = \int_V r^2 dm$ | $(13)$ |
| Momentum | $\vec{p} = m\vec{v}$ | Angular Momentum ($\vec{L}$) | $\vec{L} = I \vec{\omega}$ | $(14)$ |
The relations for Newton's Law, Kinetic Energy, Work, and Power are tabulated below:
| Pure Translation | Pure Rotation | ||
|---|---|---|---|
| Newton's Law | $\displaystyle \vec{F} = \frac{d \vec{p}}{d t} = m \vec{a}$ | $\vec{\tau} = \dfrac{d\vec{L}}{dt} = I \vec{\alpha}$ | $(15)$ |
| Kinetic Energy | $K = \frac{1}{2} m v^2$ | $K = \frac{1}{2} I \omega^2$ | $(16)$ |
| Work | $W = \displaystyle \int \vec{F}\cdot d \vec{s}$ | $W = \displaystyle \int \vec{\tau} \cdot d\vec{\theta}$ | $(17)$ |
| Power | $P = \vec{F}\cdot \vec{v}$ | $P = \vec{\tau}\cdot\vec{\omega}$ | $(18)$ |
Report: Summary and conclusions
After your second lab period, you will need to write up your summary and your conclusions. This should be a separate document, and it should be done individually (though you may talk your group members or ask questions). Include any data tables, plots, etc. from the your lab notebook as necessary in order to show how your data support your conclusions.
The summary is just a retelling of the facts. What were the important things you did? How did you make measurements? What changed as you worked through the project? What are the take-away results?
The conclusion is your interpretation and discussion of your data. What do your data tell you? How do you data match the model (or models) you were comparing against, or to your expectations in general? (Sometimes this means using the $t^{\prime}$ test, but other times it means making qualitative comparisons.) Were you able to estimate uncertainties well, or do you see room to make changes or improvements in the technique? Do your results lead to new questions? Can you think of other ways to extend or improve the experiment?
Each of these above sections does not need to be long; one or two paragraphs for the summary and another one or two paragraphs of conclusion should be sufficient. What is important, however, is that your writing should be complete and meaningful. Address both the qualitative and quantitative aspects of the experiment, and make sure you cover all the “take-away” topics in enough depth. Don't include throw-away statements like “Looks good” or “Agrees pretty well.” Instead, try to be precise.
Remember… your goal is not to discover some “correct” answer. In fact, approaching any experiment with that mind set is the exact wrong thing to do. You must always strive to reach conclusions which are supported by your data, regardless of what you think the “right” answer should be. Never, under any circumstances should you state a conclusion which is contradicted by the data. Stating that the results of your experiment are inconclusive, or do not agree with theoretical predictions is completely acceptable if that is what your data indicate. Trying to shoehorn your data into agree with some preconceived expectation when you cannot support that claim is fraudulent.
REMINDER: Because of the Thanksgiving holiday, the final lab report submission will be pushed back until Week 9. All final lab reports (regardless of the day of the week you are normally in lab) will be due Tuesday, November 30 by 5:00 pm. Submit a single PDF on Canvas.