A closed cylinder of length ℓ containing a liquid of variable density ρ ( x ) = ρ 0 ( 1 + αx ) . Find the net force exerted by the liquid on the axis of rotation. (Take the cylinder to be massless a

Question

A closed cylinder of length ℓ containing a liquid of variable density ρ ( x ) = ρ 0 ​ ( 1 + αx ) . Find the net force exerted by the liquid on the axis of rotation. (Take the cylinder to be massless and A = cross sectional area of cylinder)

TestHub · Accepted Answer

Take a liquid element at distance x from the axis.
$d m = ρ (x) A d x$
$d m = ρ_{0} (1 + αx) A d x$
$d F = ω^{2} x d m$
$F = ρ_{0} A ω^{2} \int_{0}^{ℓ} (x + α x^{2}) d x$
$F = ρ_{0} A ω^{2} (\frac{ℓ ^{2}}{2} + \frac{α ℓ ^{3}}{3})$
$F = ρ_{0} A ω^{2} ℓ^{2} (\frac{1}{2} + \frac{α ℓ}{3})$
Option A.

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