Question

What gauge pressure must a machine produce in order to suck mud of density $1800kg/{m^3}$ up a tube by a height of $1.5m$ ?

Answer 1

In order to answer this question to know about the gauge pressure a machine must have in order to suck the mud, we will use the concept of Bernoulli’s equation. The Bernoulli equation is a crucial statement that connects a fluid's pressure, height, and velocity at a single point in its flow.

Formula used:
${p_g} = p - {p_ \circ } = \rho gh$
Here, $\rho$ = density of the mud.
$g$= gravitational force of the earth taken as $\left( {9.8m/{s^2}} \right)$
And $h$ = height of the tube

Complete step-by-step solution:
Gauge pressure is the pressure measured in relation to atmospheric pressure; it is positive for pressures greater than atmospheric pressure and negative for pressures less than atmospheric pressure. The pressure of any fluid that is not enclosed is increased by the atmospheric pressure.
In this case, Bernoulli's equation reduces to
${p_g} = p - {p_ \circ } = \rho gh$
Thus,
${P_g} = \rho g\left( { - h} \right) \\\ = - \left( {1800kg/{m^3}} \right)\left( {9.8m/{s^2}} \right)\left( {1.5m} \right) \\\ = - 2.6 \times {10^4}Pa \\\$
Therefore, gauge pressure of the machine is $- 2.6 \times {10^4}Pa$

Additional Information:
Bernoulli's equation can be thought of as a law of energy conservation for a moving fluid. Bernoulli's equation was derived from the notion that any extra kinetic or potential energy gained by a fluid system is due to external work performed on the system by another non-viscous fluid.

Note: Bernoulli's equation only applicable to inviscid and incompressible flow because inviscid flow has no viscosity and thus no viscous forces acting on the body, and incompressible flow has a constant density.