MC-9  Angular Acceleration and Rotational Energy

OBJECTIVE:

To study rotational motion resulting from constant torque. Since there are only six flywheels, groups of up to four may be necessary.

FUNDAMENTAL CONCEPTS:

1. The equation that describes the rotational motion of an object moving with constant angular velocity are completely analogous to those of linear motion with: $\theta(t) = \theta_0 + \omega \cdot t$.
   If you make a plot of $\theta$ versus $t$, you find that it describes a straight line. The greek letter $\theta_0$ indicates the angular position of the object at time $t=0$. The letter $\omega$ is the slope of the line, and is equal to the angular velocity of the object.

2. The equations that describe the rotational motion of an object that moves with constant angular acceleration are: $\theta(t) = \theta_0 + \omega_0 \cdot t + \frac{1}{2} \alpha t^2$ and $\omega(t)= \omega_0 + \alpha t$.
   The greek letter $\theta_0$ again indicates the angular position of the object at time $t=0$. The letter $\omega_0$ is the the angular velocity of the object at time $t=0$, and $\alpha$ (angular acceleration) is the slope of the graph of $\omega$ vs $t$.

3. It is important to stress that the natural units of angular displacement are in radians and NOT degrees. One full revolution of an object corresponds to $2\pi$ radians or, equivalently, 360$^o$. Typically units for angular velocity are in rad/sec and for angular acceleration are in rad/sec$^2$. Notice that the unit dimensions for equivalent dynamical variables in rotational and linear motion do not match. A linear velocity can never be compared with a rotational velocity. If you are unclear with this distinction ask your lab instructor for further clarification.

APPARATUS:

Basic equipment: Heavy cylindrical flywheel mounted on a low friction bearing and having a thin hub about which one winds string to hold a weight hanger; weight hanger and slotted masses.

Computer equipment: Personal computer set to the M9 lab manual web-page; PASCO interface module; photogate sensor and extension jack.
NOTE: If a flywheel needs more than six grams on the end of string to maintain constant rotational velocity, notify the instructor.

EXPERIMENT I (suggested procedures):

1. Make sure the PASCO interface has been turned on. If not you will have to reboot the computer. Next make sure that the phone jack connector from the photogate has been plugged into the extension wire and that the extension wire phone jack is plugged into the first position in the PASCO interface module. The computer itself is configured to measure the on/off timing between two adjacent holes on the wheel (i.e., this gives four timing measurements) and assumes that there are six holes per revolution. All other dynamical measurements are based on these input parameters.

2. Make sure the infrared photogate is properly aligned with the fly wheel so that the little red LED sensor turns off when each of the six holes drilled in the flywheel move past the photogate.

3. To initiate the PASCO interface software you will need to click on the telescope icon in the ``toolkit.'' The bitmap image below gives a good idea of how the display should appear. Note that, while you are able to reconfigure the display parameters, the default values that are specified on start-up should allow you to do this experiment without necessitating any changes. Three instantaneous measurements, angular position ($\theta$), velocity ($\omega$), and acceleration ($\alpha$), are displayed simultaneously in combination with plots of the angular velocity and acceleration. Since $\omega$ is determined from the position data and $\alpha$ from $\omega$ the ``scatter'' in the data will become progressively more pronounced.

4. With NO string or weights attached to the flywheel spin the flywheel by hand so that you obtain approximately one revolution per second. Start the PASCO data acquisition by CLICKing on the REC icon. To stop it CLICK on the STOP icon. (Each data run gets its own data set in the ``Data'' display window. If there are any preexisting data sets you cannot reconfigure either the interface parameters or sensor inputs.)

5. Record data for about 60 revolutions of the flywheel and answer the following questions by using the cross-hair, magnify and rescaling features of the PASCO plot display. For the statistical analysis you can specify a ``region of interest'' by clicking to points inside the plot window. Your lab instructor can provide assistance if necessary.

   (i)

   What was the total angular displacement ($\theta$) of the wheel?

   (ii)

   What were the initial and final angular velocities? Do you observe any systematic variations?

   (iii)

   From these two measurements what was the average angular acceleration?

   (iv)

   Does this agree with the average from acceleration plot? USE the interactive statistical analysis feature of the plot software to find the mean acceleration.

   (v)

   What was the initial kinetic energy of rotation? (K.E. $= \frac{1}{2} I \omega^2$, I= $\frac{1}{2} M R^2$)

   (vi)

   What was the final energy of rotation and energy loss per revolution?

6.

Repeat the above experiment for 60 revolutions and qualitatively check (have one lab member count the number of revolutions by brute force) the computer calibration. Does everything seem consistent?

7.

Predict how many revolutions and how long it will it take for the flywheel to slow to 75% of its initial angular velocity of 1 rev/sec. What is the change in kinetic energy?

8.

Now test your prediction by performing this experiment.

EXPERIMENT II Suggested Procedures:

1.

Measuring the angular acceleration $\alpha$

(i)

Close the PASCO window and launch the next PASCO application by CLICKing on the telescope icon.

(ii)

Take a 2 m length of string and tie a multiple knot in one end so that it slides into the slot of the flywheel inner ring and cannot be easily pulled through. If necessary use a piece of tape, from the inside, to hold it more securely. On the other string end tie a loop so that either the 50 g mass hanger or a small weight may be hung.

(iii)

Find the friction correction by hanging small masses directly on the string (no hanger) until the tension ``f'' in the string just maintains flywheel rotation at a constant $\omega$ with the string hanging about halfway to the floor. Obtain the torque applied to the flywheel and compare the work done in one revolution ($W=\tau \theta$) to the energy lost in the Exp. I through the same rotation.
Are these two values comparable?

(iv)

Replace the small masses with the 50 g hanger and, with the string fully extended and hanging vertically, roll the string up by exactly four full turns of the flywheel. The hanger should not hit the ground at any time.

(v)

Add masses to reach exactly 400 g and start the data acquisition while releasing the 400 gm mass. Wait until the mass has both fallen and risen one time before stopping the scan. Print out one copy of each curve (three all told) by clicking on the ``File'' menu option and then clicking on the ``Print All Displays'' button. Use scissors to cut out each plot and enter it into you lab book labeling all regions of the position, velocity and acceleration plots.

(vi)

Determine the angular accelerations, $\alpha$, of the wheel (falling and rising) and the energy lost (from $\Delta \theta$) in this one cycle.
Does this compare favorably with the work done by the frictional torque?

(vii)

Repeat the experiment a few more times to ascertain the reproducibility of $\alpha$.

(viii)

Repeat the last step using a net mass of 200 g.

2.

Predict $\alpha$ after correcting for friction. One way to make the calculation follows:

$$I   = \frac{1}{2} M R^2 m   = T   = f   = r, R = \tau = (T - f)r = I\alpha F=ma (mg - T) = ma = mr \alpha \mbox{Solving these for $\alpha$ gives:   } \alpha = \frac{(mg - f)r}{I + mr^2} .$$

3.

Compare the measured $\alpha$'s with the computed values. Are they consistent within your estimated uncertainties?

4.

By drastically increasing the mass added to the hanger deduce the maximum angular acceleration that the flywheel could receive.

EXPERIMENT III (optional):
Second Measurement of the Flywheel Rotational Inertia

$$m < 10^{o} T T = 2\pi \sqrt{I/ \kappa}$$

where $I = I_0 + I_m$ is the total rotational inertia of the system about the axis of motion, and $\kappa$, the torsion constant, is

$$\kappa = \frac{\mbox{restoring torque}}{\mbox{angular displacement}} = \frac{mgR'\sin\theta}{\theta} \simeq mgR' .$$

By the parallel axis theorem, $I_m = I_c + mR'^2$ where $I_c$ is the rotational inertia of the added mass $m$ about its own c. of mass. Hence the rotational inertia of the flywheel, $I_0$, is:

$$I_0 = \left(\frac{T}{2\pi}\right)^ 2mgR' - I_m .$$

2.

Close the PASCO window and launch the third PASCO application window by CLICKing on the telescope icon immediately below.

3.

Measure the period of the pendulum and deduce the rotational inertia of the flywheel, $I_0$. Compare with the earlier calculated value.