Overview
1. Use a text or web resource to find an expression for the moment of inertia for a disk; determine the values of I for your aluminum disks. With its large central hole, the steel disk
Figure 3
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should be treated as a cylindrical tube. Using the appropriate expression, determine the value of I for the steel disk.
Examine the ?-t graph for your runs with the single aluminum disk (Run1) and the steel disk (Run 4). Determine the rate of change of the angular velocity, ?, for each disk as it slowed. Account for this change in terms of any unbalanced forces that may be acting on the system. Explain the difference in the rates of change of ? (aluminum vs. steel) in terms of the values you calculated in Step 1.
Examine the ?-t graph for Run 2. Determine the angular acceleration before you dropped the second disk onto the first.
Record the angular velocity just before and just after you increased the mass of the system. Determine the time interval (?t) between these two velocity readings.
In Logger Pro, drag-select the interval between these two readings. The ?x in the lowerleft corner gives the value of ?t.
5. The angular momentum, L, of a system undergoing rotation is the product of its moment of inertia, I, and the angular velocity, ?.
L = I? Determine the angular momentum of the system before and after you dropped the second
aluminum disk onto the first. Calculate the percent difference between these values.
- ?
6. Use the rate of change in ?, as determined in step 3, and the time interval between your two
readings to determine ?? due to friction alone. What portion of the difference in the angular speed before and after you increased the mass can be accounted for by frictional losses?
7. Repeat the calculations in Steps 3–6 for your third and fifth runs.
PART 2. ROTATIONAL DYNAMICS
Newton's 2nd law in its rotational form is expressed by ? = I? torque ? will be exerted on a thin rotating rod with attached point-like masses. Refer to Figure 4 for the set-up.
- PROCEDURE
Measure the mass of the mass hanger and the average mass of the two short cylindrical masses. Measure the radius of the largest of the 3-step pulley. Record the values in the Data Table.
Remove the disk from the Rotary Motion Sensor, securely fasten the thin rod to it, then replace it on the Rotary Motion Sensor. Attach a piece of string to the center of the rotating shaft. Wind the string around the largest pulley 3 to 4 times and attach the other end of the string to the mass hanger.
- . In the following experiment, the
Figure 4
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- Attach the Smart Pulley to the Rotary Motion Sensor and adjust its position so that the string passes over the Smart Pulley as shown in Figure 4.
Attach the two cylindrical masses to the rod symmetrically at distances r of about 7 cm from the center. You may need to adjust the position of the Smart Pulley so that the rod with masses can rotate freely.
Practice a few test runs by allowing the hanging mass to fall and collecting angular velocity data in order to determine the angular acceleration.
Choose 5 to 7 values for the distance r between 7 and 17 cm, and measure the angular acceleration in each case using the linear curve fit for the angular velocity curve. Record the angular accelerations in the Data Table.
ANALYSIS
1. Examine your data. Does the angular acceleration, ?, increase or decrease with r? 2. Derive the equation for the angular acceleration, ?, by applying Newton's 2nd Law both to
the falling mass and, in its rotational form, to the rotating system: MgR = (I + MR2 + 2mr2 )? . 0
M is the hanging mass, m is the mass of each cylindrical mass, R is the radius of the pulley that the rod is attached to, and I0 is the moment of inertia of all of the rotating system except the two cylindrical masses. Ask your TA for assistance if needed.
3. Complete the last two columns of the Data Table. Disconnect the Rotary Sensor and open a fresh LoggerPro file. Plot the magnitude of MgR (y axis) vs. r2 (x axis). Perform a linear fit.
4. What is the slope of the linear fit? How is the slope related to the mass, m, of each cylindrical mass? Determine m from the slope and compare it with the value obtained by weighing. Calculate the percentage difference.