Thus pressure
${p}_{2}$ over the second opening is reduced by
$\frac{1}{2}\rho {v}_{2}^{2}$ , so the fluid in the manometer rises by
h on the side connected to the second opening, where
$h\propto \frac{1}{2}\rho {v}_{2}^{2}.$
(Recall that the symbol
$\propto $ means “proportional to.”) Solving for
${v}_{2}$ , we see that
${v}_{2}\propto \sqrt{h}.$
Part (b) shows a version of this device that is in common use for measuring various fluid velocities; such devices are frequently used as air-speed indicators in aircraft.
A fire hose
All preceding applications of Bernoulli’s equation involved simplifying conditions, such as constant height or constant pressure. The next example is a more general application of Bernoulli’s equation in which pressure, velocity, and height all change.
Calculating pressure: a fire hose nozzle
Fire hoses used in major structural fires have an inside diameter of 6.40 cm (
[link] ). Suppose such a hose carries a flow of 40.0 L/s, starting at a gauge pressure of
$1.62\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{6}{\text{N/m}}^{2}$ . The hose rises up 10.0 m along a ladder to a nozzle having an inside diameter of 3.00 cm. What is the pressure in the nozzle?
Strategy
We must use Bernoulli’s equation to solve for the pressure, since depth is not constant.
where subscripts 1 and 2 refer to the initial conditions at ground level and the final conditions inside the nozzle, respectively. We must first find the speeds
${v}_{1}$ and
${v}_{2}$ . Since
$Q={A}_{1}{v}_{1}$ , we get
This value is a gauge pressure, since the initial pressure was given as a gauge pressure. Thus, the nozzle pressure equals atmospheric pressure as it must, because the water exits into the atmosphere without changes in its conditions.
Bernoulli’s equation states that the sum on each side of the following equation is constant, or the same at any two points in an incompressible frictionless fluid:
Bernoulli’s principle is Bernoulli’s equation applied to situations in which the height of the fluid is constant. The terms involving depth (or height
h ) subtract out, yielding
a postmodernist would say that he did not discover them, he made them up and they're not actually a reality in itself, but a mere construct by which we decided to observe the word around us
the velocity of a boat related to water is 3i+4j and that of water related to earth is i-3j. what is the velocity of the boat relative to earth.If unit vector i and j represent 1km/hour east and north respectively