## III. Measuring LinearAcceleration and Io

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 Io.  This same procedure will be repeated in the next section to find values of Iadded.  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.      3. In DataStudio, click: create experiment, click in the Sensor window, and type s. Then drag and drop the smart pulley icon to the first digital sensor channel. Double-click the sensor and under the Measurement tab select velocity and nothing else. Now drag the graph onto the sensor, and choose the data to be displayed as velocity vs. time.

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 Io, 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 Io we have removed the weights and the wing nuts and thus Iadded is equal to 0 for this calculation.

### IV. Measuring Other Moments of Inertia

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

8.     Measure the angular acceleration of the system with the two 100-g masses attached, with the wing nuts, near the ends of the rod.  Record the radial position of the center of each 100-g mass.  As in step 6, use equation (8) to find Itotal = Iadded + Io.

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 experimental moment of inertia Itotal.     Plot Itotal as calculated from Eq. (8) verses R2What are the slope and intercept of your graph to 90% confidence?  Do your values agree with what you expect them to be?  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.

You need only print out two graphs: your original acceleration (v vs. t) graph and the moment of inertia graph: Itotal  vs. R2.