In
this section, we will use the computer to measure the angular acceleration and
then use this value of **a** to calculate the moment of inertia I_{o}. This same procedure will be repeated in the next section to
find values of I_{added}. A
smart pulley connected to the computer is used to measure the linear velocity v.
As the vertical shaft rotates, the spokes of the pulley repeatedly break
the photogate beam. The time intervals during which the beam is broken are read
by the computer and are saved so that they can be imported into a spreadsheet. A
plot of linear velocity versus **total elapsed time** will yield the linear
acceleration. Conversely, a plot of
angular velocity versus total elapsed time will yield angular acceleration,
where we relate the two accelerations by: a=aR.

1.
Measure the **diameter** of the vertical shaft with the Vernier
calipers.

2.
Connect the smart pulley to the Pasco Interface and then open the
DataStudio software.

3.
In DataStudio, click *create experiment*,
then click on digital channel #1. Select the smart pulley icon. Double-click
the icon to attach it to the interface. Under the Measurements tab select
velocity and nothing else. Now drag the graph onto the sensor.

4.
In this initial procedure ** remove the 100-g masses and wing nuts**
from the rotation apparatus, but leave the threaded rod mounted on the rotating
shaft. (Subsequent measurements
will place two 100-g masses and wing nuts on the threaded rod at various
distances from the shaft.) Then
wrap the cord with the attached 50-g weight hanger around the shaft so that the
weight hanger hangs just below the pulley.
Add another 100 g to the 50-g hanger so that the total is 150 g.
The cord between the shaft and the pulley should be as nearly horizontal
as possible.

5.
Once you have completed the experiment look at the v vs. t graph. We want
the acceleration and SE of the acceleration. We could import all of this data
into Excel, but we are going to use DataStudio instead. There is a button on the
left that will allow you to view the data full screen
.
Click in the v vs. t graph and choose the Fit button
,
and a **linear fit.** You may need to delete some spurious points at the
beginning or at the end. With the graph highlighted, print out your graph (under
the File menu). Determine the acceleration to 90% confidence.

6.
In order to calculate I_{o}, or any subsequent value of I, use equation
(8) with your measured value of a and the values of m and R determined
previously. Note that to find I_{o} we have removed the weights and the
wing nuts and thus I_{added} is equal to 0 for this calculation.
Report your value of I_{o} to 90% confidence.

7.
Measure once the total mass of each 100-gram mass together with the pair of
wing nuts.

8. Measure
the acceleration of the system with the two 100-g masses attached, with
the wing nuts, near the ends of the rod. You should use the same method as you
did for I0. Record the acceleration to 90% confidence, but do not print out the
graph or data. Record the radial position of the center of each 100-g mass. Life is easier if the radii are the same. As in step 6, use
equation (8) and your value of the acceleration to find I_{total}.
Use equation (5) to calculate I_{added}.

9. Repeat step 8 for at least four more positions of the 100-gram
masses. You should now have 5 (or more) pairs of values for the experimentally
determined moment of
inertia I_{total. }Plot
I_{total} as calculated from Eq. (8) versus I_{added}. (Here r
is the radial position of the center of one of the 100-g masses.)** What are the slope and intercept of your graph
to 90% confidence? (Use the linest command.) Do your values agree with what you expect them to be
(do null tests)?
What
might be some reasons for the discrepancy?
**

**Note: make sure you understand how the various masses and distance
measurements go into your calculations for the moment of inertia. (There
are several masses and several radii discussed here; don’t get them mixed up!)**

**You need only print out two graphs: your original acceleration (v vs. t)
graph and the moment of inertia graph:
I _{total} vs. **
I