how is precession of a heavy top possible?

How is Precession of a Heavy Top Possible?

Svilen Kostov and Daniel HammerDepartment of Geology and Physics

Georgia Southwestern State University

How is Precession of a Heavy Top Possible?

Svilen Kostov and Daniel HammerDepartment of Geology and Physics

Georgia Southwestern State University

Introductory physics is too simple; our task is to make it more complicated (a paraphrase of Søren Kierkegaard)

1. The heavy top-yesterday and today

‘Theorie des Kreisels’

Euler anglesand Euler-Lagrangeequations

the ‘tippy top’

a future physicist

2. The heavy top/gyroscope in torque induced precession

θ

φx

y

z

3. Precession?Q. What will the spinning gyroscope do after it is released from rest?

A. Many students predict this.

B. Most introductory texts prove this.

3. Precession?Q. What will the spinning gyroscope do after it is released from rest?

A. Many students predict this.

B. Most introductory texts prove this. (where did the vertical angular momentum come from?)

4. Precession and nutationA. Both!

“It has to go down a bit in order to go around” R. Feynman in ‘The Feynman Lectures on Physics, vol. 1’

5. A PASCO based lab

1.

2. 3.

4.

5.

1. Demonstration gyroscope (PASCO) with two digital rotational sensors

2.Mini drill (DREMEL) with rubber head to accelerate disk to target rpm values

3.Digital stroboscope (EXTEC Instruments) to measure initial spin rate by observing aliasing pattern of spot on disk

4.Supporting stands

5.Data acquisition device (‘Explorer’, PASCO)

6. The data

φ (rad)

θ (r

ad)

7. Theory/model

• Conservation of momentum

0L L 3

mgr

1 3

21

mgr

Lo

LΔL

Δθ

(“Steady precession”, from your favorite text)

(From diagram; for small angles)

(Keeping track of directions through moments of inertia)

8. Fit to the model

9. Summary• The standard derivation of precession velocity of a heavy

spinning top released from rest is incomplete (the external torque cannot start the precession)

• The above top necessarily “falls”; i.e. dips from the horizontal, since angular momentum in the y-z plane is conserved (the external torque at t= 0 is perpendicular to that plane)

• The dip angle is simply related to the precession velocity; both can be measured accurately using teaching quality equipment

• The experiment and analysis described in this work can easily be incorporated into a junior type lab for the physics and engineering major

10. References• R. A. Serway, Physics for Scientists and Engineers, 6th. ed.

(Thomson/Brooks/Cole 2009), p. 350• R. Feynman, R. Leighton and M. Sands, The Feynman Lectures in Physics

vol.1,(CIT 1963), p. 20-7• M. S. Tiersten et al. “Propagation of a Feynman error on real and inertial

forces in rotating systems”, Am. J. Phys. 66 (1998)• Tuleja et al. “Feynman’s Wobbling Plate,” Am. J. Phys. 75, 240 (2007)• Butikov E., “Forced Precession of a Gyroscope “ 2006

http://www.ifmo.ru/butikov/Applets/Gyroscope.html • S. Kostov and D. Hammer, “A Lab for Exploring the Precession and

Nutation of a Gyroscope”, AAPT Advanced Labs, www.compadre.org/advlabs