- Accounting
- Aerospace Engineering
- Anatomy
- Anthropology
- Arts & Humanities
- Astronomy
- Biology
- Business
- Chemistry
- Civil Engineering
- Computer Science
- Communications
- Economics
- Electrical Engineering
- English
- Finance
- Geography
- Geology
- Health Science
- History
- Industrial Engineering
- Information Systems
- Law
- Linguistics
- Management
- Marketing
- Material Science
- Mathematics
- Mechanical Engineering
- Medicine
- Nursing
- Philosophy
- Physics
- Political Science
- Psychology
- Religion
- Sociology
- Statistics

HomeStudy GuidesPhysics

Menu

The Nature of Science and Physics

Kinematics

Introduction to One-Dimensional KinematicsDisplacementVectors, Scalars, and Coordinate SystemsTime, Velocity, and SpeedVideo: One-Dimensional KinematicsAccelerationMotion Equations for Constant Acceleration in One DimensionProblem-Solving Basics for One-Dimensional KinematicsFalling ObjectsGraphical Analysis of One-Dimensional Motion

Two-Dimensional Kinematics

Dynamics: Force and Newton's Laws of Motion

Introduction to Dynamics: Newton's Laws of MotionDevelopment of Force ConceptNewton's First Law of Motion: InertiaNewton's Second Law of Motion: Concept of a SystemNewton's Third Law of Motion: Symmetry in ForcesVideo: Newton's LawsNormal, Tension, and Other Examples of ForcesProblem-Solving StrategiesFurther Applications of Newton's Laws of MotionExtended Topic: The Four Basic Forces—An Introduction

Further Applications of Newton's Laws: Friction, Drag, and Elasticity

Uniform Circular Motion and Gravitation

Introduction to Uniform Circular Motion and GravitationRotation Angle and Angular VelocityCentripetal AccelerationCentripetal ForceFictitious Forces and Non-inertial Frames: The Coriolis ForceNewton's Universal Law of GravitationVideo: GravitationSatellites and Kepler's Laws: An Argument for Simplicity

Work, Energy, and Energy Resources

Introduction to Work, Energy, and Energy ResourcesWork: The Scientific DefinitionKinetic Energy and the Work-Energy TheoremGravitational Potential EnergyVideo: Potential and Kinetic EnergyConservative Forces and Potential EnergyNonconservative ForcesConservation of EnergyPowerWork, Energy, and Power in HumansWorld Energy Use

Linear Momentum and Collisions

Rotational Motion and Angular Momentum

Introduction to Rotational Motion and Angular MomentumAngular AccelerationKinematics of Rotational MotionVideo: Rotational MotionDynamics of Rotational Motion: Rotational InertiaRotational Kinetic Energy: Work and Energy RevisitedAngular Momentum and Its ConservationVideo: Angular MomentumCollisions of Extended Bodies in Two DimensionsGyroscopic Effects: Vector Aspects of Angular Momentum

Statics and Torque

Fluid Statics

Introduction to Fluid StaticsWhat Is a Fluid?DensityPressureVariation of Pressure with Depth in a FluidPascal's PrincipleGauge Pressure, Absolute Pressure, and Pressure MeasurementArchimedes' PrincipleVideo: BuoyancyCohesion and Adhesion in Liquids: Surface Tension and Capillary ActionPressures in the Body

Fluid Dynamics and Its Biological and Medical Applications

Introduction to Fluid Dynamics and Biological and Medical ApplicationsFlow Rate and Its Relation to VelocityBernoulli's EquationVideo: Fluid FlowThe Most General Applications of Bernoulli's EquationViscosity and Laminar Flow; Poiseuille's LawThe Onset of TurbulenceMotion of an Object in a Viscous FluidMolecular Transport Phenomena: Diffusion, Osmosis, and Related Processes

Temperature, Kinetic Theory, and the Gas Laws

Heat and Heat Transfer Methods

Thermodynamics

Introduction to ThermodynamicsThe First Law of ThermodynamicsThe First Law of Thermodynamics and Some Simple ProcessesIntroduction to the Second Law of Thermodynamics: Heat Engines and Their EfficiencyCarnot's Perfect Heat Engine: The Second Law of Thermodynamics RestatedApplications of Thermodynamics: Heat Pumps and RefrigeratorsEntropy and the Second Law of Thermodynamics: Disorder and the Unavailability of EnergyStatistical Interpretation of Entropy and the Second Law of Thermodynamics: The Underlying Explanation

Oscillatory Motion and Waves

Introduction to Oscillatory Motion and WavesHooke's Law: Stress and Strain RevisitedPeriod and Frequency in OscillationsSimple Harmonic Motion: A Special Periodic MotionVideo: Harmonic MotionThe Simple PendulumEnergy and the Simple Harmonic OscillatorUniform Circular Motion and Simple Harmonic MotionDamped Harmonic MotionForced Oscillations and ResonanceWavesSuperposition and InterferenceEnergy in Waves: Intensity

Physics of Hearing

- Calculate using Torricelli’s theorem.
- Calculate power in fluid flow.

${P}_{1}+\frac{1}{2}{{\rho v}_{1}}^{2}+\rho gh_{1}={P}_{2}+\frac{1}{2}{{\rho v}_{2}}^{2}+\rho gh_{2}\\$

.
Both $\frac{1}{2}{{\rho v}_{1}}^{2}+\rho gh_{1}=\frac{1}{2}{{\rho v}_{2}}^{2}+\rho gh_{2}\\$

.
Solving this equation for ${{v}_{2}}^{2}$

, noting that the density ${{v}_{2}}^{2}={{v}_{1}}^{2}+2g\left({h}_{1}-{h}_{2}\right)\\$

.
We let ${{v}_{2}}^{2}={{v}_{1}}^{2}+2gh\\$

where
Fire hoses used in major structure fires have inside diameters of 6.40 cm. Suppose such a hose carries a flow of 40.0 L/s starting at a gauge pressure of 1.62 × 10^{6} N/m^{2}. The hose goes 10.0 m up a ladder to a nozzle having an inside diameter of 3.00 cm. Assuming negligible resistance, what is the pressure in the nozzle?

**Strategy**

Here we must use Bernoulli’s equation to solve for the pressure, since depth is not constant.

**Solution**

Bernoulli’s equation states

*v*_{1} and *v*_{2}. Since *Q* = *A*_{ 1 }*v*_{ 1 } , we get

*h*_{1} to be zero, we solve Bernoulli’s equation for *P*_{2}:

**Discussion**

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.

${P}_{1}+\frac{1}{2}{{\rho v}_{1}}^{2}+\rho gh_{1}={P}_{2}+\frac{1}{2}{{\rho v}_{2}}^{2}+\rho gh_{2}\\$

.
where the 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 ${v}_{1}=\frac{Q}{{A}_{1}}=\frac{{40.0} \times {10}^{-3}\text{ m}^{3}\text{s}}{\pi \left({3.20} \times {10}^{-2}\text{ m}\right)^{2}}=12.4 \text{ m/s}\\$

.
Similarly, we find*v*_{2 }= 56.6 m/s.

${P}_{2}={P}_{1}+\frac{1}{2}\rho \left({{v}_{1}}^{2}-{{v}_{2}}^{2}\right)-\rho gh_{2}\\$

.
Substituting known values yields$\begin{array}{c}{P}_{2}=1.62\times{10}^{6}\text{ N/m}^{2}+\frac{1}{2}\left({1000}\text{ kg/m}^{3}\right)\left[\left({12.4}\text{ m/s}\right)^{2}-\left(56.6\text{ m/s}\right)^{2}\right] \\ -\left({1000}\text{ kg/m}^{3}\right)\left(9.80 \text{ m/s}^{2}\right)\left({10.0}\text{ m}\right)=0\end{array}\\$

$P+\frac{1}{2}{\rho v}^{2}+\rho{gh}=\text{ constant}\\$

All three terms have units of energy per unit volume, as discussed in the previous section. Now, considering units, if we multiply energy per unit volume by flow rate (volume per unit time), we get units of power. That is, ($\left(P+\frac{1}{2}{\rho v}^{2}+\rho gh\right)Q=\text{ power}\\$

.
Each term has a clear physical meaning. For example, $\frac{1}{2}{\rho v}^{2}Q\\$

is the power supplied to a fluid to give it its kinetic energy. And Power is defined as the rate of energy transferred, or *E*/*t*. Fluid flow involves several types of power. Each type of power is identified with a specific type of energy being expended or changed in form.

Suppose the fire hose in the previous example is fed by a pump that receives water through a hose with a 6.40-cm diameter coming from a hydrant with a pressure of 0.700 × 10^{6} N/m^{2}. What power does the pump supply to the water?

**Strategy**

Here we must consider energy forms as well as how they relate to fluid flow. Since the input and output hoses have the same diameters and are at the same height, the pump does not change the speed of the water nor its height, and so the water’s kinetic energy and gravitational potential energy are unchanged. That means the pump only supplies power to increase water pressure by 0.92 × 10^{6} N/m^{2} (from 0.700 × 10^{6} N/m^{2} to 1.62 × 10^{6} N/m^{2}).

**Solution**

As discussed above, the power associated with pressure is

**Discussion**

Such a substantial amount of power requires a large pump, such as is found on some fire trucks. (This kilowatt value converts to about 50 hp.) The pump in this example increases only the water’s pressure. If a pump—such as the heart—directly increases velocity and height as well as pressure, we would have to calculate all three terms to find the power it supplies.

$\begin{array}{lll}\text{power}& =& PQ\\ & =& \left(\text{0.920}\times {\text{10}}^{6}{\text{ N/m}}^{2}\right)\left(40.0\times {10}^{-3}{\text{ m}}^{3}\text{/s}\right)\text{.}\\ \text{}& =& 3.68\times {10}^{4}\text{ W}=36.8\text{ kW}\end{array}\\$

- Power in fluid flow is given by the equation $\left({P}_{1}+\frac{1}{2}{\rho v}^{2}+\rho gh\right)Q=\text{power}\\$, where the first term is power associated with pressure, the second is power associated with velocity, and the third is power associated with height.

1. 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.)

2. Water that has emerged from a hose into the atmosphere has a gauge pressure of zero. Why? When you put your hand in front of the emerging stream you feel a force, yet the water’s gauge pressure is zero. Explain where the force comes from in terms of energy.

3. The old rubber boot shown in Figure 3 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.

4. 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.

1. Hoover Dam on the Colorado River is the highest dam in the United States at 221 m, with an output of 1300 MW. The dam generates electricity with water taken from a depth of 150 m and an average flow rate of 650 m^{3}/s. (a) Calculate the power in this flow. (b) What is the ratio of this power to the facility’s average of 680 MW?

2. A frequently quoted rule of thumb in aircraft design is that wings should produce about 1000 N of lift per square meter of wing. (The fact that a wing has a top and bottom surface does not double its area.) (a) At takeoff, an aircraft travels at 60.0 m/s, so that the air speed relative to the bottom of the wing is 60.0 m/s. Given the sea level density of air to be 1.29 kg/m^{3}, how fast must it move over the upper surface to create the ideal lift? (b) How fast must air move over the upper surface at a cruising speed of 245 m/s and at an altitude where air density is one-fourth that at sea level? (Note that this is not all of the aircraft’s lift—some comes from the body of the plane, some from engine thrust, and so on. Furthermore, Bernoulli’s principle gives an approximate answer because flow over the wing creates turbulence.)

3. The left ventricle of a resting adult’s heart pumps blood at a flow rate of 83.0 cm^{3}/s, increasing its pressure by 110 mm Hg, its speed from zero to 30.0 cm/s, and its height by 5.00 cm. (All numbers are averaged over the entire heartbeat.) Calculate the total power output of the left ventricle. Note that most of the power is used to increase blood pressure.

4. A sump pump (used to drain water from the basement of houses built below the water table) is draining a flooded basement at the rate of 0.750 L/s, with an output pressure of 3.00 × 10^{5} N/m^{2}. (a) The water enters a hose with a 3.00-cm inside diameter and rises 2.50 m above the pump. What is its pressure at this point? (b) The hose goes over the foundation wall, losing 0.500 m in height, and widens to 4.00 cm in diameter. What is the pressure now? You may neglect frictional losses in both parts of the problem.

3. 1.26 W