Theory of Machine
The velocity of a particle moving with simple harmonic motion, at any instant is given by (where ω = Displacement of the particle from mean position)

ω² √(x² - r²)
ω² √(r² - x²)
ω √(x² - r²)
ω √(r² - x²)

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Theory of Machine
For an isochronous Hartnell governor (where r₁ and r₂ = Maximum and minimum radius of rotation of balls respectively, S₁ and S₂ = Maximum and minimum force exerted on the sleeve respectively, and M = Mass on the sleeve)

(m.g + S₁)/(m.g + S₂) = r₁/r₂
S₂/S₁ = r₁/r₂
S₁/S₂ = r₁/r₂
(m.g - S₁)/(m.g - S₂) = r₂/r₁

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