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Acceleration of a freely falling sphere with added mass

A sphere of density

ρ_{s}

falling in inviscid fluid of density

ρ

has effective inertia

(ρ_{s} + ρ /2) V

and net weight

(ρ_{s} - ρ) V g

. Hence

\dot{U} = \frac{ρ _{s} - ρ}{ρ _{s} + ρ /2} g .

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