How did they get this precession frequency of this gyroscope?

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Discussion Overview

The discussion revolves around the derivation of the precession frequency of a gyroscope influenced by an oscillating magnetic field produced by a solenoid. Participants explore the relationship between angular momentum, torque, and oscillation in the context of a fluid-filled torus and an Nb-plate. The focus is on understanding the mathematical expressions involved in this scenario.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant summarizes the setup involving an AC voltage solenoid, an AC magnetic field, and the resulting torque on angular momentum.
  • Another participant questions the presence of an extra θ0 term in the torque expression and suggests a potential missing π term in the derived rate of change of angular momentum.
  • A third participant reiterates the concern about the extra θ0 term and the missing π term in the calculations, indicating that their derived expressions do not match the original expression provided.
  • A fourth participant proposes a line of reasoning regarding the relationship between the change in angular momentum and the angle through which it moves, suggesting a differential approach to the problem.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the presence of the extra θ0 term and the completeness of the derived expressions. There is no consensus on the correct formulation or the missing components in the calculations.

Contextual Notes

Participants are working with assumptions about small oscillations and the relationships between angular momentum, torque, and angular displacement. The discussion highlights potential gaps in the mathematical derivation without resolving them.

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To summarize:

1. The solenoid is supplied with an AC voltage.

2. Produces an AC magnetic field out of the page

3. Exerts an AC magnetic force on the Nb-Plate that's placed just outside of one end of the solenoid

4. There is a fluid circulating in the torus, with angular momentum L in the z-direction (out of the page)

5. What is the torque exerted on L due to the oscillating Nb-plate which forces the torus to oscillate with it?

Directions: ∅ (Left-right), θ (In-Out of page)

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I'm not sure how they got an extra θ0 term in their torque about ∅. I know in small oscillations, the magnitude of change in L is

ΔL ≈ θ0L

Alternatively, considering change within half a period = T/2:

ΔL/Δt = 2θ0L/T = 2fθ0L

which also doesn't match their expression..
 
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No response yet!

Which extra θ0 term are you referring to because I do not see an extra.

Also, are you not missing a pi term in
ΔL/Δt = 2θ0L/T = 2fθ0L
 
256bits said:
No response yet!

Which extra θ0 term are you referring to because I do not see an extra.

Also, are you not missing a pi term in
ΔL/Δt = 2θ0L/T = 2fθ0L

Yeah, I did it two ways. Both don't match the expression they have derived..
 
I think I know their line of thought, correct me if I'm wrong:

In time Δt, the vector (θ0L) moves through an angle Δ∅. Hence

ΔL = (θ0L)Δ∅

Taking limits of Δt → 0,

dL/dt = (θ0)L(d∅/dt) = θ0Lω∅
 

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