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A test tube filled with water is being spun around in an ultracentrifuge with angular velocity $ω$ . The test tube is lying along a radius and the free surface of the water is at radius $r_{0}$ (Figure). The length of the test tube is $ℓ$ and $ρ$ is the density of the water. Ignore gravity and ignore atmospheric pressure. Choose the correct statement.

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Answer:

A

Solution:

For the section of length dx , $dT = ρ Adx ω^{2} x$

∴ T at centre $= \int_{r_{0}}^{(\frac{ℓ}{2} + r_{0})} ρ A ω^{2} xdx = \frac{1}{2} ρ A ω^{2} [(\frac{ℓ}{2} + r_{0})^{2} - r_{0}^{2}]$ $= \frac{1}{2} ρ A ω^{2} [\frac{ℓ ^{2}}{4} + r_{0}^{2} + ℓ r_{0} - r_{0}^{2}] = \frac{1}{2} ρ A ω^{2} [\frac{ℓ ^{2} + 4 ℓ r _{0}}{4}] = \frac{1}{8} ρ A ω^{2} ℓ [ℓ + 4 r_{0}]$

∴ Pressure $= \frac{T}{A} = \frac{1}{8} ρ ω^{2} ℓ [ℓ + 4 r_{0}]$

Stream:JEESubject:PhysicsTopic:FluidSubtopic:Pressure

⏱ 2mℹ️ Source: QB

Physics - Fluid Question with Solution | TestHub

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