How did they get this precession frequency of this gyroscope?

[unscientific]

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)

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..

 

[256bits]

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

 

[unscientific]

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..

 

[unscientific]

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_∅