raymond a. serway chris vuille phys 101 spring 2014 review of ch 8, 9, 10, 11, 13 and 14

Raymond A. SerwayChris Vuille

PHYS 101 SPRING 2014Review of CH 8, 9, 10, 11, 13 and 14

Ch 8:Rotational Equilibrium andRotational Dynamics

Section 9.2

• Torque, , is the tendency of a force to rotate an object about some axis– = r Fsin

• is the torque– Symbol is the Greek tau

• r is the length of the position vector• F is the tangential force

• SI unit is Newton . meter (N.m)


Section 9.2

Only the tangential component of force causes a torque:


Section 9.2

• The net torque is the sum of all the torques produced by all the forces– Remember to account for the direction of the

tendency for rotation• Counterclockwise torques are positive• Clockwise torques are negative


Section 9.2

• First Condition of Equilibrium– The net external force must be zero

– This is a statement of translational equilibrium• The Second Condition of Equilibrium states

– The net external torque must be zero

– This is a statement of rotational equilibrium


Section 9.2

• When a rigid object is subject to a net torque (Στ ≠ 0), it undergoes an angular acceleration

Where I is the moment of inertia:


Section 9.2

Conservation of Mechanical Energy

– Remember, this is for conservative forces, no dissipative forces such as friction can be present

– Potential energies of any other conservative forces could be added



Work-Energy Theorem• In the case where there are dissipative forces

such as friction, use the generalized Work-Energy Theorem instead of Conservation of Energy

• Wnc = E =KEt + KEr + PE


Angular Momentum L and Conservation of L

Angular momentum is defined as : L = I ω

Since

If = 0 then L=0 in other words ff the sum of all torques on an object is zero then angular momentum is conserved:

Li = Lf Ii ωi=If ωf

Ch 9: Density

• The density of a substance of uniform composition is defined as its mass per unit volume:

• SI unit: kg/m3 (SI) – Often see g/cm3 (cgs)

• 1 g/cm3 = 1000 kg/m3

Section 9.2

Ch 9: Pressure

• The average pressure P is the force divided by the area

Section 9.2

Ch 9: Young’s Modulus

• stress = elastic modulus x strain

In the case of stretching a bar we have tensile stress F/A and tensile strain L/L0 and the elastic modulus is called Young’s modulus

Section 9.3

CH 9: Shear Modulus

Section 9.3

In the case of applying a shear stress F/A on an object we have shear strain x/h and the elastic modulus is called the shear modulus

CH 9: Bulk Modulus

Section 9.3

In the case of applying a volume stress P on an object we have volume strain V/V and the elastic modulus is called the bulk modulus

Ch 9: Pressure and Depth equation

• • Po is normal atmospheric

pressure– 1.013 x 105 Pa = 14.7 lb/in.2

• The pressure does not depend upon the shape of the container

Section 9.4

• One atmosphere (1 atm) = 1.013 x 105 Pa

Ch 9: Pascal’s Principle

• A change in pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and to the walls of the container.

Section 9.4

Ch 9: Archimedes' Principle

• Any object completely or partially submerged in a fluid is buoyed up by a force whose magnitude is equal to the weight of the fluid displaced by the object

Section 9.6

Ch 9: Buoyant Force, cont.

• The magnitude of the buoyant force always equals the weight of the displaced fluid

Section 9.6

Ch9: Archimedes’ Principle:Totally Submerged Object

• The upward buoyant force is B=ρfluidVobjg

• The downward gravitational force is w=mg=ρobjVobjg

• The net force is B-w=(ρfluid-ρobj)Vobjg

Section 9.6

Ch 9: Archimedes’ Principle:Floating Object

• The object is in static equilibrium• The upward buoyant force is balanced by the

downward force of gravity• Volume of the fluid displaced corresponds to

the volume of the object beneath the fluid level

Section 9.6

Ch 9: Equation of Continuity

• For a steady flow the conservation of mass leads to:

1A1v1 = 2A2v2 For an incompressible fluid:

A1v1 = A2v2

• The product of the cross-sectional area of a pipe and the fluid speed is a constant

• The product Av is called the flow rate, Av=V/t Section 9.7

Ch 9: Bernoulli’s Equation

• States that the sum of the pressure, kinetic energy per unit volume, and the potential energy per unit volume has the same value at all points along a streamline

Section 9.7

Ch 9: Applications of Bernoulli’s Principle: Measuring Speed

• Shows fluid flowing through a horizontal constricted pipe

• Speed changes as diameter changes

• Can be used to measure the speed of the fluid flow

• Swiftly moving fluids exert less pressure than do slowly moving fluids

Section 9.7

Ch 9: Application – Airplane Wing

• The air speed above the wing is greater than the speed below

• The air pressure above the wing is less than the air pressure below

• There is a net upward force– Called lift

• Other factors are also involved

Ch 10: Linear Thermal Expansion

• For small changes in temperature

• α, the coefficient of linear expansion, depends on the material

Section 10.3

Ch 10: Area Thermal Expansion

• Two dimensions expand according to

Section 10.3

€

A = A − A0

= γA0ΔT

where γ is the coefficient of area

expansion and γ = 2α

Ch 10: Volume Thermal Expansion

• Three dimensions expand

Section 10.3

€

V =V −V0

= βV0ΔT

where β is the coefficient of volume

expansion and β = 3α

CH 10: Ideal Gas• If a gas is placed in a container the pressure, volume,

temperature and amount of gas are related to each other by an equation of state

• An ideal gas is one that is dilute enough, and far away enough from condensing, that the interactions between molecules can be ignored.

• Most gases at room temperature and pressure behave approximately as an ideal gas

Section 10.4

Ch 10: Moles

• One mole is the amount of the substance that contains as many particles as there are atoms in 12 g of carbon-12

• Molar mass is the mass of 1 mole of a substance.

Section 10.4

Avogadro’s Number NA

The number of elementary entities (atoms or molecules) in a mole is given by Avogadro’s number:

Therefore, n moles of gas will contain molecules.

Ch 10: Ideal Gas Law

• PV = n R T– R is the Universal Gas Constant– n is the number of moles– R = 8.31 J / mol.K– R = 0.0821 L. atm / mol.K– Is the equation of state for an ideal gas– Temperatures used in the ideal gas law must be in

kelvins

Section 10.4

Ch 10: Ideal Gas Law, Alternative Version

• P V = N kB T– kB is Boltzmann’s Constant

– kB = R / NA = 1.38 x 10-23 J/ K

– N is the total number of molecules

• n = N / NA

– n is the number of moles– N is the number of molecules

Section 10.4

Ch 10: Pressure of an Ideal Gas

• The pressure is proportional to the number of molecules per unit volume and to the average translational kinetic energy of a molecule

Section 10.5

€

P =2

3

N

V

⎛

⎝ ⎜

⎞

⎠ ⎟1

2m v 2 ⎛

⎝ ⎜

⎞

⎠ ⎟

Ch 10: Molecular Interpretation of Temperature

• Temperature is a direct measure of the average molecular kinetic energy of the gas

Section 10.5

Ch 10: Speed of the Molecules

• Expressed as the root-mean-square (rms) speed

• At a given temperature, lighter molecules move faster, on average, than heavier ones– Lighter molecules can more easily reach escape speed

from the earth

Section 10.5

CH 11: Energy Transfer

• When two objects of different temperatures are placed in thermal contact, the temperature of the warmer decreases and the temperature of the cooler increases

• The energy exchange ceases when the objects reach thermal equilibrium

• The concept of energy was broadened from just mechanical to include internal– Made Conservation of Energy a universal law of nature

Introduction

Ch 11: Units of Heat

• Calorie– A calorie is the amount of energy necessary to

raise the temperature of 1 g of water from 14.5° C to 15.5° C .

• A Calorie (food calorie) is 1000 cal

• 1 cal = 4.186 J– This is called the Mechanical Equivalent of Heat

Section 11.1

CH 11: Heat Capacity

• The heat capacity of an object is the amount of heat added to it divided by its rise in temperature:

Section 11.2

Q is positive if ΔT is positive; that is, if heat is added to a system.

Q is negative if ΔT is negative; that is, if heat is removed from a system.

Ch 11: Specific Heat

• The heat capacity of an object depends on its mass. A quantity which is a property only of the material is the specific heat:

Section 11.2

Ch 11: Calorimetry

• You can start with Qk = 0– Qk is the energy of the kth object where Qk = mk ck Tk

– Use Tf – Ti

– You don’t have to determine before using the equation which materials will gain or lose heat

Section 11.3

Ch 11: Phase Changes

• A phase change occurs when the physical characteristics of the substance change from one form to another

• Common phases changes are– Solid to liquid – melting (fusion)– Liquid to Solid – freezing (fusion)– Gas to Liquid – condensation (vaporization)– Liquid to gas – boiling (vaporization)

• Phases changes involve a change in the internal energy, but no change in temperature

Section 11.4

Section 11.4

Ch 11: Latent Heat

• The energy Q needed to change the phase of a given pure substance is– Q = ±m L

• L is the called the latent heat of the substance– Latent means hidden– L depends on the substance and the nature of the phase

change• Choose a positive sign if you are adding energy to the

system and a negative sign if energy is being removed from the system

Section 11.4

Ch 11: Latent Heat, cont.

• SI unit of latent heat are J / kg• Latent heat of fusion, Lf, is used for melting or

freezing• Latent heat of vaporization, Lv, is used for

boiling or condensing• Table 11.2 gives the latent heats for various

substances

Section 11.4

Ch 11: Graph of Ice to Steam

Section 11.4

Ch 11: Methods of Heat Transfer

• Methods of Heat Transfer include– Conduction– Convection– Radiation

Section 11.5

Ch 11: Conduction

• The transfer can be viewed on an atomic scale– It is an exchange of energy between microscopic particles

by collisions– Less energetic particles gain energy during collisions with

more energetic particles

• Rate of conduction depends upon the characteristics of the substance

Section 11.5

Ch 11: Conduction, equation

• The slab of material allows energy Q to transfer from the region of higher temperature to the region of lower temperature

• A is the cross-sectional area

Section 11.5

Ch 11: Multiple Materials, cont.

• The rate through the multiple materials will be

• TH and TC are the temperatures at the outer extremities of the compound material

Section 11.5

Ch 11: Radiation

• Radiation does not require physical contact• All objects radiate energy continuously in the

form of electromagnetic waves due to thermal vibrations of the molecules

• Rate of radiation is given by Stefan’s Law

Section 11.5

Ch 11: Radiation equation

• P = σ A e T4

– The power is the rate of energy transfer, in Watts– σ = 5.669 6 x 10-8 W/m2.K4

• Called the Stefan-Boltzmann constant

– A is the surface area of the object– e is a constant called the emissivity

• e varies from 0 to 1

– T is the temperature in Kelvins

Section 11.5

Ch 11: Energy Absorption and Emission by Radiation

• The rate at which the object at temperature T with surroundings at To radiates is– Pnet = σ A e (T4 - To

4)

– When an object is in equilibrium with its surroundings, it radiates and absorbs at the same rate

• Its temperature will not change

Section 11.5

Ch 13: Periodic Motion

Period, T: time required for one cycle of periodic motion

Frequency, f: number of oscillations per unit time

This unit is called the Hertz=1/s

€

f =1

T

SI unit : cycle/second = 1/s

Ch 13: Simple Harmonic Motion

Simple Harmonic Motion occurs when the net force along the direction of motion obeys Hooke’s Law (F=-kx), in other words the force is proportional to the displacement and always directed toward the equilibrium position.

Ch 13: Connections between Uniform Circular Motion and Simple Harmonic Motion

An object in simple harmonic motion has the same motion as one component of an object in uniform circular motion:

Ch 13: Connections between Uniform Circular Motion and Simple Harmonic Motion

The velocity as a function of time:

And the acceleration:

The position as a function of time:

€

v = -Aω sin ωt( )

€

a = -ω 2Acos ωt( ) = -ω 2x

€

x = Acos ωt( )

Ch 13: The Period of a Mass on a SpringSince the force on a mass on a spring is proportional to the displacement, and also to the acceleration, we find that

.

Substituting the time dependencies of a and x gives

€

ω 2 =k

m⇒ ω =

k

m⇒

2π

T=

k

m⇒ T = 2π

m

k

Ch 13: Energy Conservation in Oscillatory Motion

In an ideal system with no nonconservative forces, the total mechanical energy is conserved. For a mass on a spring:

Since we know the position and velocity as functions of time, we can find the maximum kinetic and potential energies:

€

E =KE + PE =1

2mv 2 +

1

2kx 2

€

PEmax =1

2kA2

KEmax =1

2mvmax

2 =1

2kA2


€

E = PE +KE =1

2kx 2 +

1

2mv 2 =

1

2kA2 cosωt( )

2+

1

2mω 2A2 sinωt( )

2=

1

2kA2 cosωt( )

2+

1

2mk

mA2 sinωt( )

2=

1

2kA2 cosωt( )

2+ sinωt( )

2[ ] =

1

2kA2

PE +KE =1

2kA2


This diagram shows how the energy transforms from potential to kinetic and back, while the total energy remains the same.

Ch 13: Velocity as a Function of Position

Section 13.2

€

E total =1

2kx 2 +

1

2mv 2 =

1

2kA2 ⇒

kx 2 +mv 2 = kA2 ⇒ v 2 =k

mA2 −

k

mx 2 =

k

mA2 − x 2

( )⇒

v =k

mA2 − x 2

( )

Ch 13: The PendulumA simple pendulum consists of a mass m (of negligible size) suspended by a string or rod of length L (and negligible mass).

The angle it makes with the vertical varies with time as a sine or cosine.

Ch 13: The Pendulum

Looking at the forces on the pendulum bob, we see that the restoring force is proportional to sin θ, whereas the restoring force for a spring is proportional to the displacement (which is θ in this case).

Ch 13: The Pendulum

Substituting θ for sin θ allows us to treat the pendulum in a mathematically identical way to the mass on a spring. Therefore, we find that the period of a pendulum depends only on the length of the string:

€

T = 2πL

g

Ch 13: Waves

A wave is a disturbance that propagates from one place to another.

In a transverse wave, the displacement of the medium is perpendicular to the direction of motion of the wave.

Ch 13: Waves

In a longitudinal wave, the displacement is along the direction of wave motion.

Ch 13: Types of WavesWavelength λ: distance over which wave repeats

Period T: time for one wavelength to pass a given point

Frequency f:

Speed of a wave:

Ch 13: Speed of a Wave on a String

• The speed of a wave on a string stretched under some tension, F

– is called the linear density• The speed depends only upon the properties

of the medium through which the disturbance travels

Section 13.9

Sound Waves

Sound waves are longitudinal waves, similar to the waves on a Slinky:

Here, the wave is a series of compressions and stretches.

Sound Waves

In a sound wave, the density and pressure of the air (or other medium carrying the sound) are the quantities that oscillate.

Speed of Sound in a Liquid

• In a fluid, the speed depends on the fluid’s compressibility and inertia

– B is the Bulk Modulus of the liquid– ρ is the density of the liquid– Compares with the equation for a transverse wave on a

string

Section 14.3

Speed of Sound in a Solid Rod

• The speed depends on the rod’s compressibility and inertial properties

– Y is the Young’s Modulus of the material– ρ is the density of the material

Section 14.3

Speed of Sound in Air

• 331 m/s is the speed of sound at 0° C• T is the absolute temperature

Section 14.3

Sound IntensityThe intensity of a sound is the amount of energy that passes through a given area in a given time.

Various Intensities of Sound

• Threshold of hearing– Faintest sound most humans can hear– About 1 x 10-12 W/m2

• Threshold of pain– Loudest sound most humans can tolerate– About 1 W/m2

• The ear is a very sensitive detector of sound waves– It can detect pressure fluctuations as small as about 3

parts in 1010

Section 14.4

Intensity Level of Sound Waves

• The sensation of loudness is logarithmic in the human ear

• β is the intensity level or the decibel level of the sound

• Io is the threshold of hearing

Section 14.4

Spherical Waves

• A spherical wave propagates radially outward from the oscillating sphere

• The energy propagates equally in all directions

• The intensity is

Section 14.5

Doppler Effect

• A Doppler effect is experienced whenever there is relative motion between a source of waves and an observer.– When the source and the observer are moving

toward each other, the observer hears a higher frequency

– When the source and the observer are moving away from each other, the observer hears a lower frequency

Section 14.6

Doppler Effect, General Case

• Both the source and the observer could be moving

• Use positive values of vo and vs if the motion is toward– Frequency appears higher

• Use negative values of vo and vs if the motion is away– Frequency appears lower

Section 14.6

Shock Waves

• A shock wave results when the source velocity exceeds the speed of the wave itself

• The circles represent the wave fronts emitted by the source

Section 14.6

Shock Waves, cont

• Tangent lines are drawn from Sn to the wave front centered on So

• The angle between one of these tangent lines and the direction of travel is given by sin θ = v / vs

• The ratio vs /v is called the Mach Number

• The conical wave front is the shock wave

Section 14.6

Interference of Sound Waves

• Sound waves interfere– Constructive interference occurs when the path

difference between two waves’ motion is zero or some integer multiple of wavelengths

• Path difference = nλ (n = 0, 1, 2, … )

– Destructive interference occurs when the path difference between two waves’ motion is an odd half wavelength

• Path difference = (n + ½)λ (n = 0, 1, 2, … )

Section 14.7

Standing Waves

A standing wave is fixed in location, but oscillates with time. These waves are found on strings with both ends fixed, such as in a musical instrument, and also in vibrating columns of air.

Standing Waves

The fundamental, or lowest, frequency on a fixed string has a wavelength twice the length of the string. Higher frequencies are called harmonics.

Standing Waves

There must be an integral number of half-wavelengths on the string; this means that only certain frequencies are possible.

Points on the string which never move are called nodes; those which have the maximum movement are called antinodes.

Standing Waves on a String, final

• The lowest frequency of vibration (b) is called the fundamental frequency (ƒ1)

• Higher harmonics are positive integer multiples of the fundamental

Section 14.8

Driven Oscillations and ResonanceAn oscillation can be driven by an oscillating driving force; the frequency of the driving force may or may not be the same as the natural frequency of the system.

Driven Oscillations and Resonance

If the driving frequency is close to the natural frequency, the amplitude can become quite large, especially if the damping is small. This is called resonance.

Driven Oscillations and Resonance

Standing Waves in Air Columns

• If one end of the air column is closed, a node must exist at this end since the movement of the air is restricted

• If the end is open, the elements of the air have complete freedom of movement and an antinode exists

Section 14.10

Tube Open at Both Ends

Section 14.10

Resonance in Air Column Open at Both Ends

• In a pipe open at both ends, the natural frequency of vibration forms a series whose harmonics are equal to integral multiples of the fundamental frequency

Section 14.10

Tube Closed at One End

Section 14.10

Resonance in an Air Column Closed at One End

• The closed end must be a node• The open end is an antinode

• There are no even multiples of the fundamental harmonic

Section 14.10

Beats

• Beats are alternations in loudness, due to interference

• Waves have slightly different frequencies and the time between constructive and destructive interference alternates

• The beat frequency equals the difference in frequency between the two sources:

Section 14.11

Beats, cont.

Section 14.11

raymond a. serway chris vuille phys 101 spring 2014 review of ch 8, 9, 10, 11, 13 and 14

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