14.6 Bernoulli’s equation  (Page 6/8)

 Page 6 / 8

David rolled down the window on his car while driving on the freeway. An empty plastic bag on the floor promptly flew out the window. Explain why.

Based on Bernoulli’s equation, what are three forms of energy in a fluid? (Note that these forms are conservative, unlike heat transfer and other dissipative forms not included in Bernoulli’s equation.)

Potential energy due to position, kinetic energy due to velocity, and the work done by a pressure difference.

The old rubber boot shown below has two leaks. To what maximum height can the water squirt from Leak 1? How does the velocity of water emerging from Leak 2 differ from that of Leak 1? Explain your responses in terms of energy.

Water pressure inside a hose nozzle can be less than atmospheric pressure due to the Bernoulli effect. Explain in terms of energy how the water can emerge from the nozzle against the opposing atmospheric pressure.

The water has kinetic energy due to its motion. This energy can be converted into work against the difference in pressure.

Problems

Verify that pressure has units of energy per unit volume.

$\begin{array}{}\\ \\ \hfill F& =\hfill & pA⇒p=\frac{F}{A},\hfill \\ \hfill \left[p\right]& =\hfill & {\text{N/m}}^{2}=\text{N}·{\text{m/m}}^{3}={\text{J/m}}^{3}=\text{energy/volume}\hfill \end{array}$

Suppose you have a wind speed gauge like the pitot tube shown in [link] . By what factor must wind speed increase to double the value of h in the manometer? Is this independent of the moving fluid and the fluid in the manometer?

If the pressure reading of your pitot tube is 15.0 mm Hg at a speed of 200 km/h, what will it be at 700 km/h at the same altitude?

−135 mm Hg

Every few years, winds in Boulder, Colorado, attain sustained speeds of 45.0 m/s (about 100 mph) when the jet stream descends during early spring. Approximately what is the force due to the Bernoulli equation on a roof having an area of $220{\text{m}}^{2}$ ? Typical air density in Boulder is $1.14{\text{kg/m}}^{3}$ , and the corresponding atmospheric pressure is $8.89\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{4}{\text{N/m}}^{2}$ . (Bernoulli’s principle as stated in the text assumes laminar flow. Using the principle here produces only an approximate result, because there is significant turbulence.)

What is the pressure drop due to the Bernoulli Effect as water goes into a 3.00-cm-diameter nozzle from a 9.00-cm-diameter fire hose while carrying a flow of 40.0 L/s? (b) To what maximum height above the nozzle can this water rise? (The actual height will be significantly smaller due to air resistance.)

a. $1.58\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{6}\phantom{\rule{0.2em}{0ex}}{\text{N/m}}^{2}$ ; b. 163 m

(a) Using Bernoulli’s equation, show that the measured fluid speed v for a pitot tube, like the one in [link] (b), is given by $v={\left(\frac{2{\rho }^{\prime }gh}{\rho }\right)}^{1\text{/}2}$ , where h is the height of the manometer fluid, ${\rho }^{\prime }$ is the density of the manometer fluid, $\rho$ is the density of the moving fluid, and g is the acceleration due to gravity. (Note that v is indeed proportional to the square root of h , as stated in the text.) (b) Calculate v for moving air if a mercury manometer’s h is 0.200 m.

A container of water has a cross-sectional area of $A=0.1\phantom{\rule{0.2em}{0ex}}{\text{m}}^{2}$ . A piston sits on top of the water (see the following figure). There is a spout located 0.15 m from the bottom of the tank, open to the atmosphere, and a stream of water exits the spout. The cross sectional area of the spout is ${A}_{\text{s}}=7.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-4}{\text{m}}^{2}$ . (a) What is the velocity of the water as it leaves the spout? (b) If the opening of the spout is located 1.5 m above the ground, how far from the spout does the water hit the floor? Ignore all friction and dissipative forces.

a. ${v}_{2}=3.28\frac{\text{m}}{\text{s}}$ ;
b. $t=0.55\phantom{\rule{0.2em}{0ex}}\text{s}$
$x=vt=1.81\phantom{\rule{0.2em}{0ex}}\text{m}$

A fluid of a constant density flows through a reduction in a pipe. Find an equation for the change in pressure, in terms of ${v}_{1},{A}_{1},{A}_{2}$ , and the density.

Hi all, love you all!!!
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