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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

- Define density.
- Calculate the mass of a reservoir from its density.
- Compare and contrast the densities of various substances.

Which weighs more, a ton of feathers or a ton of bricks? This old riddle plays with the distinction between mass and density. A ton is a ton, of course; but bricks have much greater density than feathers, and so we are tempted to think of them as heavier. (See Figure 1.)

$\rho =\frac{m}{V}\\$

,
where the Greek letter
Density is mass per unit volume.

*ρ* is the symbol for density, *m* is the mass, and *V* is the volume occupied by the substance.

$\rho =\frac{m}{V}\\$

,
where Substance | $\rho \left({\text{10}}^{3}{\text{kg/m}}^{3}\text{or}\text{g/mL}\right)\\$ |
Substance | $\rho \left({\text{10}}^{3}{\text{kg/m}}^{3}\text{or}\text{g/mL}\right)\\$ |
Substance | $\rho \left({\text{10}}^{3}{\text{kg/m}}^{3}\text{or}\text{g/mL}\right)\\$ |
---|---|---|---|---|---|

Solids |
Liquids |
Gases |
|||

Aluminum | 2.7 | Water (4ºC) | 1.000 | Air | 1.29 × 10^{−3 } |

Brass | 8.44 | Blood | 1.05 | Carbon dioxide | 1.98 × 10^{−3 } |

Copper (average) | 8.8 | Sea water | 1.025 | Carbon monoxide | 1.25 × 10^{−3 } |

Gold | 19.32 | Mercury | 13.6 | Hydrogen | 0.090 × 10^{−3 } |

Iron or steel | 7.8 | Ethyl alcohol | 0.79 | Helium | 0.18 × 10^{−3 } |

Lead | 11.3 | Petrol | 0.68 | Methane | 0.72 × 10^{−3 } |

Polystyrene | 0.10 | Glycerin | 1.26 | Nitrogen | 1.25 × 10^{−3 } |

Tungsten | 19.30 | Olive oil | 0.92 | Nitrous oxide | 1.98 × 10^{−3 } |

Uranium | 18.70 | Oxygen | 1.43 × 10^{−3 } |
||

Concrete | 2.30–3.0 | Steam (100º C) | 0.60 × 10^{−3 } |
||

Cork | 0.24 | ||||

Glass, common (average) | 2.6 | ||||

Granite | 2.7 | ||||

Earth’s crust | 3.3 | ||||

Wood | 0.3–0.9 | ||||

Ice (0°C) | 0.917 | ||||

Bone | 1.7–2.0 |

A pile of sugar and a pile of salt look pretty similar, but which weighs more? If the volumes of both piles are the same, any difference in mass is due to their different densities (including the air space between crystals). Which do you think has the greater density? What values did you find? What method did you use to determine these values?

A reservoir has a surface area of 50.0 km^{2} and an average depth of 40.0 m. What mass of water is held behind the dam? (See Figure 2 for a view of a large reservoir—the Three Gorges Dam site on the Yangtze River in central China.)

**Strategy **

We can calculate the volume *V* of the reservoir from its dimensions, and find the density of water *ρ* in Table 1. Then the mass *m* can be found from the definition of density

**Solution**

Solving equation *ρ *= *m*/*V *for *m* gives *m* = *ρV*. The volume *V* of the reservoir is its surface area *A* times its average depth *h*:

*ρ* from Table 1 is 1.000 × 10^{3}. Substituting *V* and *ρ* into the expression for mass gives

**Discussion**

A large reservoir contains a very large mass of water. In this example, the weight of the water in the reservoir is *mg* = 1.96 × 10^{13} N, where *g* is the acceleration due to the Earth’s gravity (about 9.80 m/s^{2}). It is reasonable to ask whether the dam must supply a force equal to this tremendous weight. The answer is no. As we shall see in the following sections, the force the dam must supply can be much smaller than the weight of the water it holds back.

$\rho =\frac{m}{V}\\$

.
$\begin{array}{lll}V& =& {Ah}=\left(\text{50.0}{\text{km}}^{2}\right)\left(\text{40.0}\text{m}\right)\\ & =& \left[\left(\text{50.0 k}{\text{m}}^{2}\right){\left(\frac{{\text{10}}^{3}\text{m}}{1\text{km}}\right)}^{2}\right]\left(\text{40.0 m}\right)=2\text{.}\text{00}\times {\text{10}}^{9}{\text{m}}^{3}\end{array}\\$

The density of water $\begin{array}{lll}m& =& \left(1\text{.}\text{00}\times {\text{10}}^{3}{\text{ kg/m}}^{3}\right)\left(2\text{.}\text{00}\times {\text{10}}^{9}{\text{m}}^{3}\right)\\ & =& 2.00\times {\text{10}}^{\text{12}}\text{ kg}\end{array}\\$

.
- Density is the mass per unit volume of a substance or object. In equation form, density is defined as

$\rho =\frac{m}{V}\\$. - The SI unit of density is kg/m
^{3}.

1. Approximately how does the density of air vary with altitude?

2. Give an example in which density is used to identify the substance composing an object. Would information in addition to average density be needed to identify the substances in an object composed of more than one material?

3. Figure 3 shows a glass of ice water filled to the brim. Will the water overflow when the ice melts? Explain your answer.

2. Give an example in which density is used to identify the substance composing an object. Would information in addition to average density be needed to identify the substances in an object composed of more than one material?

3. Figure 3 shows a glass of ice water filled to the brim. Will the water overflow when the ice melts? Explain your answer.

2. Mercury is commonly supplied in flasks containing 34.5 kg (about 76 lb). What is the volume in liters of this much mercury?

3. (a) What is the mass of a deep breath of air having a volume of 2.00 L? (b) Discuss the effect taking such a breath has on your body’s volume and density.

4, A straightforward method of finding the density of an object is to measure its mass and then measure its volume by submerging it in a graduated cylinder. What is the density of a 240-g rock that displaces 89.0 cm^{3} of water? (Note that the accuracy and practical applications of this technique are more limited than a variety of others that are based on Archimedes’ principle.)

5. Suppose you have a coffee mug with a circular cross section and vertical sides (uniform radius). What is its inside radius if it holds 375 g of coffee when filled to a depth of 7.50 cm? Assume coffee has the same density as water.

6. (a) A rectangular gasoline tank can hold 50.0 kg of gasoline when full. What is the depth of the tank if it is 0.500-m wide by 0.900-m long? (b) Discuss whether this gas tank has a reasonable volume for a passenger car.

7. A trash compactor can reduce the volume of its contents to 0.350 their original value. Neglecting the mass of air expelled, by what factor is the density of the rubbish increased?

8. A 2.50-kg steel gasoline can holds 20.0 L of gasoline when full. What is the average density of the full gas can, taking into account the volume occupied by steel as well as by gasoline?

9. What is the density of 18.0-karat gold that is a mixture of 18 parts gold, 5 parts silver, and 1 part copper? (These values are parts by mass, not volume.) Assume that this is a simple mixture having an average density equal to the weighted densities of its constituents.

10. There is relatively little empty space between atoms in solids and liquids, so that the average density of an atom is about the same as matter on a macroscopic scale—approximately 10^{3} kg/m^{3}. The nucleus of an atom has a radius about 10^{-5} that of the atom and contains nearly all the mass of the entire atom. (a) What is the approximate density of a nucleus? (b) One remnant of a supernova, called a neutron star, can have the density of a nucleus. What would be the radius of a neutron star with a mass 10 times that of our Sun (the radius of the Sun is 7 × 10^{8})?

- density:
- the mass per unit volume of a substance or object

3. (a) 2.58 g (b) The volume of your body increases by the volume of air you inhale. The average density of your body decreases when you take a deep breath, because the density of air is substantially smaller than the average density of the body before you took the deep breath.

4. 2.70 g/cm

6. (a) 0.163 m (b) Equivalent to 19.4 gallons, which is reasonable

8. 7.9 × 10

9. 15.6 g/cm

10. (a) 10