A barrel will rupture when the gauge pressure within it...

A barrel will rupture when the gauge pressure within it reaches $350 \mathrm{kPa}$. It is attached to the lower end of a vertical pipe, with the pipe and barrel filled with oil $\left(\rho=890 \mathrm{~kg} / \mathrm{m}^{3}\right)$. How long can the pipe be if the barrel is not to rupture? From $P=\rho g h$ we have $$ h=\frac{P}{\rho g}=\frac{350 \times 10^{3} \mathrm{~N} / \mathrm{m}^{2}}{\left(9.81 \mathrm{~m} / \mathrm{s}^{2}\right)\left(890 \mathrm{~kg} / \mathrm{m}^{3}\right)}=40.1 \mathrm{~m} $$

A barrel will rupture when the gauge pressure within it reaches $350 \mathrm{kPa}$. It is attached to the lower end of a vertical pipe, with the pipe and barrel filled with oil $\left(\rho=890 \mathrm{~kg} / \mathrm{m}^{3}\right)$. How long can the pipe be if the barrel is not to rupture?
From $P=\rho g h$ we have
$$
h=\frac{P}{\rho g}=\frac{350 \times 10^{3} \mathrm{~N} / \mathrm{m}^{2}}{\left(9.81 \mathrm{~m} / \mathrm{s}^{2}\right)\left(890 \mathrm{~kg} / \mathrm{m}^{3}\right)}=40.1 \mathrm{~m}
$$

Key Concepts

Hydrostatic Pressure

This is the pressure exerted by a fluid due to the force of gravity acting on its mass. It increases with depth and is calculated using the relation P = ?gh, where ? is the density of the fluid, g is the gravitational acceleration, and h is the vertical height of the fluid column. This principle is fundamental in understanding pressure variations in fluid statics.

Gauge Pressure

Gauge pressure is the pressure measured relative to the ambient atmospheric pressure. It is the pressure read by instruments that do not account for atmospheric pressure, focusing instead on the excess pressure within a system. Recognizing the distinction between gauge and absolute pressure is crucial for engineering applications involving pressure containment.

Density

Density represents the mass per unit volume of a substance, generally expressed in kilograms per cubic meter (kg/m³) for fluids. In hydrostatic pressure calculations, density is a key factor as it influences how much mass is present in a given volume of fluid, thereby affecting the overall pressure exerted by a fluid column.

Gravitational Acceleration

Gravitational acceleration is the acceleration due to gravity, typically approximated as 9.81 m/s² near the Earth's surface. It determines the weight of a fluid column and, when used in the hydrostatic pressure formula (P = ?gh), directly impacts the resulting pressure. This constant is essential in calculations involving forces exerted by fluids in gravitational fields.

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