Fluid mechanics

Compute the gauge pressure and the absolute pressure at a depth of $12.0\mathrm{m}$ in a freshwater lake, with atmospheric pressure $p_0=1.013\cdot 10^5\mathrm{Pa}$.

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A tub contains water $40\mathrm{cm}$ deep. In the bottom there is a circular plug of diameter $4.0\mathrm{cm}$, below which there is air at atmospheric pressure. Compute the resultant force that the water and the air exert on the plug.

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Into one arm of a U-tube containing water a liquid that does not mix with water is poured. When the column of the liquid is $15.0\mathrm{cm}$ high, in the other arm the water stands $12.0\mathrm{cm}$ above the level of the interface. Compute the density of the liquid.

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At a mountain location a mercury barometer ($\rho =13.6\cdot 10^{3}kg/{\mathrm{m}}^3$) reads $640\mathrm{mm}$. Compute the atmospheric pressure in pascals.

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In a hydraulic press the pistons, at the same height, have radii $1.5\mathrm{cm}$ and $18\mathrm{cm}$. A force of $150\mathrm{N}$ is applied to the small piston. Compute the maximum mass that the large piston can support, neglecting the masses of the pistons.

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A homogeneous plastic cylinder floats in water with three quarters of its volume immersed. Compute the density of the plastic and the fraction of the volume that would remain immersed if the cylinder floated in an oil of density $900kg/{\mathrm{m}}^3$.

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A stone hanging from a spring scale weighs $12.0\mathrm{N}$ in air and $7.5\mathrm{N}$ when completely immersed in water. Compute the volume and the density of the stone, neglecting the buoyancy of the air.

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A body hanging from a spring scale weighs $3.00\mathrm{N}$ in air, $2.00\mathrm{N}$ completely immersed in water and $2.20\mathrm{N}$ completely immersed in an unknown liquid. Compute the density of the liquid.

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A garden hose of diameter $2.0\mathrm{cm}$ carries water with flow rate $0.30L/s$ and ends with a nozzle of diameter $0.50\mathrm{cm}$. Compute the speed of the water in the hose and at the outlet of the nozzle.

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In a horizontal duct water flows with speed $2.0m/s$ and pressure $1.80\cdot 10^5\mathrm{Pa}$. Further along, the cross section of the duct is reduced to half. Treating water as an ideal fluid, compute the speed and the pressure in the narrow section.

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A very wide open tank has a small hole $5.0\mathrm{m}$ below the free surface of the water. Derive the formula for the exit speed and compute its value.

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A horizontal Venturi tube through which water flows has cross section $12\mathrm{c}{\mathrm{m}}^2$ in the wide section and $4.0\mathrm{c}{\mathrm{m}}^2$ in the constriction; the pressure difference between the two sections is $2.4\cdot 10^3\mathrm{Pa}$. Compute the velocity in the wide section and the flow rate.

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A horizontal pipe $2.0\mathrm{m}$ long and of radius $1.0\mathrm{mm}$ carries water ($\eta =1.0\cdot 10^{-3}\mathrm{Pa}\cdot \mathrm{s}$) with a pressure difference of $4.0\cdot 10^3\mathrm{Pa}$ between its ends. Compute the flow rate with Poiseuille’s law and verify with the Reynolds number that the flow is laminar.

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