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:

$\parbox{5.3in}{ \item[i.] Calculate the rotor's angular momentum from the following data:}$

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) .

$\parbox{5.3in}{ \item[ii.] Place one of the supplied weights into the... ...you should make. The experiments in the Mitac Manual are good possibilities. }$

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).

CAUTION: The air bearing surfaces and torque shaft can easily suffer damage so handle carefully: e.g.  when the air pressure is on (and ball friction negligible) the glued-on torque shaft may break off if the shaft bangs against the stationary ball holder.

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

$$\omega_p = \frac{mgh}{I_s \omega_s} \hspace{.25in} \mbox{or} \hspace{.25in} m = \frac{\omega_p \omega_s I_s}{gh} .$$

  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_s². 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 $\omega_s$.