M-13  Gyroscope

OBJECTIVE:

To study gyroscopes and to check the relationship between angular momentum, applied torque and rate of precession.

REFERENCES:

Don't do this experiment before reading about angular momentum and gyroscopes; see also the Mitac Instruction Manual, especially pages 1-7, experiments 1-4, pages 11-14, and Sec. 6 on ``Operation''.

APPARATUS: Mitac Gyroscope, 150 g and 300 g snap on slotted masses, and stroboscope.

SUGGESTIONS:

1. Read the sections of the Mitac Manual referred to above, start the gyroscope, and experiment with it. For example, predict the direction of precession when a certain torque is applied; then check it experimentally.
2. When you have become familiar with the operation of the gyroscope perform the following quantitative experiment:

   i.

   Calculate the rotor's angular momentum from the following data:
   

   Outer diameter of rotor   20.0 cm

   Inner diameter of rotor   15.1 cm

   Thickness of rotor     2.54 cm

   Rotor webb thickness     0.435 cm

   Hub diameter     3.01 cm

   Mass of rotor     2.73 kg

   Rotor rpm 150 (Check rotor rpm with the stroboscope)

   ii.

   Place one of the supplied weights into the groove on the rotor axis so as to create a torque on the axis and cause the gyroscope to precess. From this weight and its position, calculate the torque. Predict the rate of precession by using the calculated torque and angular momentum. Check by experiment.

   iii.

   Ask your instructor what other quantitative measurement you should make. The experiments in the Mitac Manual are good possibilities.

Alternative M-13 Gyroscope:

OBJECTIVE: To study gyroscopic effects especially precession and nutation.

APPARATUS: Air bearing gyroscope, torque weights, timer, stroboscope.

INTRODUCTION: Review gyroscope treatment in text (or in handout).

SUGGESTED STUDIES:

1. Qualitative explorations: With air flow on to minimize friction, spin ball by twisting the torque shaft with thumb and forefinger. Observe precession about the vertical $Z$ axis, and observe nutation (torque shaft bobbing up and down) for different torque weights on the shaft and for different spin angular velocities, $\omega_s$. Upon what factors does the precession depend?

   Observe the three basic nutation patterns when the total torque weight, $m$, has:
   
   

2. Measure the angular acceleration, $\alpha$, from the air friction: For low $\omega_s$ use a stopwatch and time the revolutions of a mark on the rotating shaft. Plot spin angular velocity, $\omega_s$, versus time (remember to plot average $\omega_s$ at the middle of the time interval). Calculate $\alpha$ from the slope. For higher $\omega_s$ use the stroboscope. To avoid errors from subharmonics synchronizing with the mark on the shaft, start with the strobe at too high a frequency. If the synchronization is at the n$^{th}$ higher harmonic, you will see the shaft marks appear at $n$ different places.

3. Calculate mass of the unweighted torque shaft from observed precessional angular velocity, $\omega_p$. Spin the shaft at about $\omega_s \sim 2\pi$ rad/s and minimize nutation by starting the shaft with a small horizontal (or slightly upward) velocity to approximate the $\omega_p$ expected. Since the torque $\tau = (mgh)\sin\theta$ and $\omega_p = \tau/(L_s sin\theta)$, we predict $\omega_p$ is independent of $\theta$. Check this experimentally. With spin angular momentum $L_s = I_s \omega_s$ the result is
   
   $$\omega_p = \frac{mgh}{I_s \omega_s} \mbox{      or       } m = \frac{\omega_p \omega_s I_s}{gh} .$$
   
   

   For the shaft mass $m \ll M$ of the ball, $I_s \sim (2/5)MR^2$ where $R$ is radius of the ball. Measure the appropriate quantities and calculate $m$. Compare with $m$ and $h$ measured directly for an unattached torque shaft.

4. Sleeping Top: If
   
   $$\omega_s^2 \;\geq \; 4mgh \left[I_s \; + \; mh^2 \left(\frac{M}{m+M}\right) \right] / I_s^2 ,$$
   
   
   then one can show (see handout) that the gyroscope will spin without nutation if released carefully at $\theta$ = 0. You can observe the qualitative effect by starting with a higher $\omega_s$ and then gradually slowing down $\omega_s$ by touching the rotating ball symmetrically in the horizontal plane until $\omega_s$ becomes critical when the shaft will suddenly fall out of the vertical line. Calculate and check this critical $\omega_s$.

OPTIONAL:

Nutation frequency, $\omega_N$: One can show (see handout) that:

$$\omega_N = \frac{\omega_s I_s}{ I_s + mh^2 M/(m+M) } .$$

Check this expression when $\omega_s \sim 2\pi$ rad/s. If one has time, change m and h by adding torque weights and check the new ratio of $\omega_N / \omega_s$.

Amplitude of nutation: The handout shows that the amplitude of nutation varies as 1/L$_s^2$. Since $L_s = I_s \omega_s$, spinning the gyroscope faster should reduce nutation amplitude at the same time the nutation frequency increases. Check this at least qualitatively. Use strobe to measure high $\omega_s$.