Chapter 7

Chapter 7

7.1 Measuring rotational motion

Rotational Quantities

• Rotational motion: motion of a body that spins about an axis– Axis of rotation: the line

about which the rotation occurs

• Circular motion: motion of a point on a rotating object

Rotational Quantities

• Circular Motion– Direction is constantly changing– Described as an angle– All points (except points on the axis) move through

the same angle during any time interval

Circular Motion

• Useful to set a reference line

• Angles are measured in radians

• s= arc length• r = radius

θ =

s

r

Angular Motion

• 360o = 2rad• 180o = rad

θ ( r a d ) =

Π

1 8 0

θ ( d e g )

Angular displacement

• Angular dispacement: the angle through which a point line, or body is rotated in a specified direction and about a specified axis

• Practice:– Earth has an equatorial radius of approximately 6380km and

rotates 360o every 24 h.• What is the angular displacement (in degrees) of a person standing at

the equator for 1.0 h?• Convert this angular displacement to radians• What is the arc length traveled by this person?

Δ θ =

Δ s

r

Angular speed and acceleration

• Angular speed: The rate at which a body rotates about an axis, usually expressed in radians per second

• Angular acceleration: The time rate of change of angular speed, expressed in radians per second per second

ω a v g =

Δ θ

Δ t

α a v g =

ω 2 − ω 1

t 2 − t 1

=

Δ ω

Δ t

Angular speed and acceleration

ALL POINTS ON A ROTATING RIGID OBJECT HAVE THE SAME ANGULAR

SPEED AND ANGULAR ACCELERATION

Rotational kinematic equations

Angular kinematics

• Practice– A barrel is given a downhill rolling start of

1.5 rad/s at the top of a hill. Assume a constant angular acceleration of 2.9 rad/s

• If the barrel takes 11.5 s to get to the bottom of the hill, what is the final angular speed of the barrel?

• What angular displacement does the barrel experience during the 11.5 s ride?

Homework Assignment

• Page 269: 5 - 12

Chapter 7

7.2 Tangential and Centripetal Acceleration

Tangential Speed

• Let us look at the relationship between angular and linear quantities.

• The instantaneous linear speed of an object directed along the tangent to the object’s circular path

• Tangent: the line that touches the circle at one and only one point.

Tangential Speed

• In order for two points at different distances to have the same angular displacement, they must travel different distances

• The object with the larger radius must have a greater tangential speed

Tangential Speed

v t = r ω

Tangential Acceleration

• The instantaneous linear acceleration of an object directed along the tangent to the object’s circular path

Δ v t

Δ t

= r

Δ ω

Δ t

a t = r α

Lets do a problem

• A yo-yo has a tangential acceleration of 0.98m/s2 when it is released. The string is wound around a central shaft of radius 0.35cm. What is the angular acceleration of the yo-yo?

Centripetal Acceleration

• Acceleration directed toward the center of a circular path

• Although an object is moving at a constant speed, it can still have an acceleration.

• Velocity is a vector, which has both magnitude and DIRECTION.

• In circular motion, velocity is constantly changing direction.


• vi and vf in the figure to the right differ only in direction, not magnitude

• When the time interval is very small, vf and vi will be almost parallel to each other and acceleration is directed towards the center


a c = r ω

2

ac =vt2

r

Tangential and centripetal accelerations

• Summary:– The tangential component of

acceleration is due to changing speed; the centripetal component of acceleration is due to changing direction

• Pythagorean theorem can be used to find total acceleration and the inverse tangent function can be used to find direction

What’s coming up

• HW: Pg 270, problems 21 - 26

• Monday: Section 7.3

• Wednesday: Review

• Friday: TEST over Chapter 7

Chapter 7

7.3: Causes of Circular Motion

Causes of circular motion

• When an object is in motion, the inertia of the object tends to maintain the object’s motion in a straight-line path.

• In circular motion (I.e. a weight attached to a string), the string counteracts this tendency by exerting a force

• This force is directed along the length of the string towards the center of the circle

Force that maintains circular motion

• According to Newton’s second law

or:

F c = m a c

F c =

m v t

2

r F c = m r ω

2

Force that maintains circular motion

• REMEMBER: The force that maintains circular motion acts at right angles to the motion.

• What happens to a person in a car(in terms of forces) when the car makes a sharp turn.

Chapter 9

9.2 - Fluid pressure and temperature

Pressure

• What happens to your ears when you ride in an airplane?

• What happens if a submarine goes too deep into the ocean?

What is Pressure?

• Pressure is defined as the measure of how much force is applied over a given area

• The SI unit of pressure is the pascal (PA), which is equal to N/m2

• 105Pa is equal to 1 atm

P =FA

Some Pressures

Table 9-2 Some pressures

Location P(Pa)Center of the sun 2 x 1016

Center of Earth 4 x 1011

Bottom of the Pacific Ocean 6 x 107

Atmosphere at sea level 1.01 x 105

Atmosphere at 10 km above sea level 2.8 x 104

Best vacuum in a laboratory 1 x 10-12

Pressure applied to a fluid

• When you inflate a balloon/tire etc, pressure increases

• Pascal’s Principle– Pressure applied to a fluid in a closed

container is transmitted equally to every point of the fluid and to the walls of a containerPinc =

F1

A1

=F2

A2

F2 =A2

A1

F1

Lets do a problem

• In a hydraulic lift, a 620 N force is exerted on a 0.20 m2 piston in order to support a weight that is placed on a 2.0 m2 piston.

• How much pressure is exerted on the narrow piston?

• How much weight can the wide piston lift?P =FA

=620N

0.20m2=3.1 ×103Pa

F2 =A2

A1

F1 =2.0m2

0.20m2620N=6.2 ×103N

Pressure varies with depth in a fluid

• Water pressure increases with depth. WHY?

• At a given depth, the water must support the weight of the water above it

• The deeper you are, the more water there is to support

• A submarine can only go so deep an withstand the increased pressure

The example of a submarine

• Lets take a small area on the hull of the submarine

• The weight of the entire column of water above that area exerts a force on that areaV =Ah m =ρV

P =FA

=mgA

=ρVgA

=ρAhgA

=ρhg

Fluid Pressure

• Gauge Pressure

– does not take the pressure of the atmosphere into consideration

• Fluid Pressure as a function of depth

– Absolute pressure = atmospheric pressure + (density x free-fall acceleration x depth)

P =FA

=mgA

=ρVgA

=ρAhgA

=ρhg

P =P0 +ρgh

Point to remember

These equations are valid ONLY if the density is the same throughout the

fluid

The Relationship between Fluid pressure

and buoyant forces

• Buoyant forces arise from the differences in fluid pressure between the top and bottom of an immersed object

Pnet =Pbottom+Ptop =(P0 +ρgh2) −(P0 +ρgh1)

=ρg(h2 −h1) =ρgL

Fnet =PnetA=ρgLA=ρgV=mfg

Atmospheric Pressure

• Pressure from the air above• The force it exerts on our body is

200 000N (40 000 lb)• Why are we still alive??• Our body cavities are permeated

with fluids and gases that are pushing outward with a pressure equal to that of the atmosphere -> Our bodies are in equilibrium

Atmospheric

• A mercury barometer is commonly used to measure atmospheric pressure

Kinetic Theory of Gases

• Gas contains particles that constantly collide with each other and surfaces

• When they collide with surfaces, they transfer momentum

• The rate of transfer is equal to the force exerted by the gas on the surface

• Force per unit time is the gas pressure

Lets do a Problem

• Find the atmospheric pressure at an altitude of 1.0 x 103 m if the air density is constant. Assume that the air density is uniformly 1.29 kg/m3 and P0=1.01 x 105 PaP =P0 +ρhg=

1.01 ×105Pa+ 1.29kg/m3(−1.0 ×103m)(9.81m/ s2)=8.8 ×104Pa

Temperature in a gas

• Temperature is the a measure of the average kinetic energy of the particles in a substance

• The higher the temperature, the faster the particles move

• The faster the particles move, the higher the rate of collisions against a given surface

• This results in increased pressure

HW Assignment

• Page 330: Practice 9C, page 331: Section Review

Chapter 9

9.3 - Fluids in Motion

Fluid Flow

• Fluid in motion can be characterized in two ways:– Laminar: Every particle passes a particular

point along the same smooth path (streamline) traveled by the particles that passed that point earlier

– Turbulent: Abrupt changes in velocity• Eddy currents: Irregular motion of the fluid

Ideal Fluid

• A fluid that has no internal friction or viscosity and is incompressible– Viscosity: The amount of internal friction

within a fluid– Viscous fluids loose kinetic energy

because it is transformed into internal energy because of internal friction.

Ideal Fluid

• Characterized by Steady flow– Velocity, density and pressure are constant at

each point in the fluid– Nonturbulent

• There is no such thing as a perfectly ideal fluid, but the concept does allow us to understand fluid flow better

• In this class, we will assume that fluids are ideal fluids unless otherwise stated

Principles of Fluid Flow

• If a fluid is flowing through a pipe, the mass flowing into the pipe is equal to the mass flowing out of the pipe

m1 =m2

ρ1V1 = ρ2V2

ρ1A1Δx1 = ρ2A2Δx2

ρ1A1v1Δt = ρ2A2v2Δt

A1v1 =A2v2

Pressure and Speed of Flow

• In the Pipe shown to the right, water will move faster through the narrow part

• There will be an acceleration

• This acceleration is due to an unbalanced force

• The water pressure will be lower, where the velocity is higher

Bernoulli’s Principle

• The pressure in a fluid decreases as the fluid’s velocity increases

Bernoulli’s Equation

• Pressure is moving through a pipe with varying cross-section and elevation

• Velocity changes, so kinetic energy changes

• This can be compensated for by a change in gravitational potential energy or pressure

P +12ρv2 +ρgh=constant

Bernoulli’s Equation

P +12ρv2 +ρgh=constant

Bernoulli’s Principle: A Special Case

• In a horizontal pipe

P1 +12ρ1v

2 =P2 +12ρ2v

2

The Ideal Gas Law

• kB is a constant called the Boltzmann’s constant and has been experimentally determined to be 1.38 x 10-23 J/K

PV =NkBT

Ideal Gas Law Cont’d

• If the number of particles is constant then:

• Alternate Form:

– m=mass of each particle, M=N x m Total Mass of the gas

P1V1

T1

=P2V2

T2

P =MKBT

mV=

MV

Ê

Ë

ÁÁÁÁÁÁˆ

¯

˜̃̃˜̃̃kBT

m=ρkBT

m

Real Gas

• An ideal gas can be described by the ideal gas law

• Real gases depart from ideal gas behavior at high pressures and low temperatures.

Chapter 12: Vibration and Waves

12.1 Simple Harmonic Motion

Simple harmonic motion

• Periodic motion: Back and forth motion over the same path– E.g. Mass attached to a spring

km

Simple Harmonic Motion


• At the unstretched position, the spring is at equilibrium (x=0)

• The spring force increases as the spring is stretched away from equilibrium

• As the mass moves towards equilibrium, force (and acceleration) decreases


• Momentum causes mass to overshoot equilibrium

• Elastic force increases (in the opposite direction)


• Defined as a vibration about an equilibrium position in which a restoring force is proportional to the displacement from equilibrium

• The force that pushes or pulls the mass back to its original equilibrium position is called the restoring force

Felastic =−kxSpring force = - (spring constant x displacement)

Hooke’s Law:

Hookes Law ExampleExample 1: If a mass of 0.55kg attached to a vertical spring stretches the spring 2 cm from its equilibrium position, what is the spring constant?

Given: m = 0.55 kgx = -0.02 mg = -9.8 m/s2

Solution:Fnet = 0 = Felastic + Fg

0 = - kx + mg or, kx = mg k = mg/x = (0.55 g)(-9.8 m/s2)/(-0.02 m) =

270 N/m

x = -0.02 m

Fg

Fel

Energy

• What kind of energy does a springs has when it is stretched or compressed?– Elastic Potential energy

• Elastic Potential energy can be converted into other forms of energy– i.e. Bow and Arrow

The Simple Pendulum

• Consists of a mass, which is called a bob, which is attached to a fixed string

• Assumptions:– Mass of the string is

negligible– Disregard friction

The Simple Pendulum

• The restoring force is proportional to the displacement

• The restoring force is equal to the x component of the bob’s weight

• When the angle of displacement is >15o, a pendulums motion is simple harmonic

The Simple Pendulum

• In the absence of friction, Mechanical energy is conserved

Simple Harmonic motion

Chapter 12: Vibration and Waves

12.2 Measuring simple harmonic motion

Amplitude, Period and Frequency

• Amplitude: The maximum displacement from the equilibrium position

• Period (T): The time it takes to execute a complete cycle of motion

• Frequency (f): the number of cycles/vibrations per unit time

Period and Frequency

• If the time it takes to complete one cycle is 20 seconds:– The Period is said to be 20s– The frequency is 1/20 cycles/s or 0.05

cycles/s– SI unit for frequency is s-1 a.k.a hertz (Hz)

Measures of simple harmonic motion

The period of a simple pendulum

• Changing mass does not change the period– Has larger restoring force, but needs larger force

to get the same acceleration

• Changing the amplitude also does not change the period (for small amplitudes)– Restoring force increases, acceleration is greater,

but distance also increases

The Period of a simple pendulum

• LENGTH of a pendulum does affect its period– Shorter pendulums have a smaller arc to travel

through, while acceleration is the same

• Free-fall acceleration also affects the period of a pendulum

T =2 Lg

The Period of a mass-spring system

• Restoring force

– Not affected by mass

• Increasing mass increases inertia, but not restoring force --> smaller acceleration

Felastic =−kx

The Period of a mass-spring system

• A heavier mass will take more time to complete a cycle --> Period increases

• The greater the spring constant, the greater the force, the greater the acceleration, which causes a decrease in period

T =2 mk

Chapter 12

12.3 Properties of Waves

Wave Motion

• Lets say we drop a pebble into water– Waves travel away from

disturbance

– If there is an object floating in the water, it will move up and down, back and forth about its original position

– Indicates that the water particles move up and down

Wave Motion

• Water is the medium– Material through which the

disturbance travels

• Mechanical wave– A wave that propagates through a

deformable, elastic medium• i.e. sound - cannot travel through

outer space

• Electromagnetic wave– Does not require a medium

• i.e. visible light, radio waves, microwaves, x rays

Types of Waves• Pulse Wave: Single nonperiodic disturbance• Periodic Wave: A wave whose source is some form of periodic motion• Sine Wave: A wave whose source vibrates with simple harmonic

motion– Every point vibrates up and down

Types of Waves

• Transverse wave: A wave whose particles vibrate perpendicularly to the direction of wave motion

• Longitudinal wave: A wave whose particles vibrate parallel to the direction of wave motion. i.e. sound

Note: The distance between the adjacent crests and troughs are the same

Period, Frequency, and Wave speed

• Period is the amount of time it takes for a complete wavelength to pass a given point

v =λT

f =1T

v =fλ

Waves and Energy

• Waves carry a certain amount of energy

• Energy transfers from one place to another

• Medium remains essentially in the same place

• The greater the amplitude of the wave, the more energy transfered

Chapter 12

12.4 Wave Interactions

Wave Interference

• Waves are not matter, but displacements of matter– Two waves can occupy the same space at the

same time– Forms an interference pattern

• Superposition: Combination of two overlapping waves

Constructive interference

• Individual displacements on the same side of the equilibrium position are added together to form a resultant wave

Destructive Interference

• Individual displacements on opposite sides of the equilibrium position are added together to form the resultant wave

Reflection

• When a wave encounters a boundary, it is reflected– If it is a free boundary/reflective surface the wave

is reflected unchanged– If it is a fixed boundary, the wave is reflected and

inverted

Standing Waves

• A wave pattern that results when two waves of the same frequency travel in opposite directions and interfere– Nodes: point in standing wave that always

undergoes complete destructive interference and is stationary

– Antinode: Point in standing wave, halfway between two nodes, with largest amplitude

Chapter 13 - Sound

13.1 Sound Waves

The Production of Sound Waves


• Compression: the region of a longitudinal wave in which the density and pressure are greater than normal

• Rarefaction: the region of a longitudinal wave in which the density and pressure are less than normal

• These compressions and rarefactions expand and spread out in all directions (like ripples in water)


Characteristics of Sound Waves

• The average human ear can hear frequencies between 20 and 20,000 Hz.

• Below 20Hz are called infrasonic waves• Above 20,000 Hz are called ultrasonic waves

– Can produce images (i.e. ultrasound)– f = 10 Mhz, v = 1500m/s, wavelength=v/f = 1.5mm– Reflected sound waves are converted into an electric signal,

which forms an image on a fluorescent screen.

Characteristics of Sound Waves

• Frequency determines pitch - the perceived highness or lowness of a sound.

Speed of Sound

• Depends on medium– Travels faster through solids, than through gasses. – Depends on the transfer of motion from particle to another

particle.– In Solids, molecules are closer together

• Also depends on temperature– At higher temperatures, gas particles collide more frequently– In liquids and solids, particles are close enough together that

change in speed due to temperature is less noticeable

Speed of Sound

Propagation of Sound Waves

• Sound waves spread out in all directions (in all 3 dimensions)

• Such sound waves are approximately spherical

Propagation of Sound Waves

The Doppler Effect

• When an ambulance passes with sirens on, the pitch will be higher as it approaches you and lower as it moves away

• The frequency is staying the same, but the pitch is changing

The Doppler Effect

The wave fronts reach observer A more often thanobserver B because of the relative motion of the car

The frequency heard by observer A is higher thanthe frequency heard by observer B

HW Assignment

• Section 13-1: Concept Review

Chapter 13 - Sound

13.2 - Sound intensity and resonance

Sound Intensity

• When you play the piano– Hammer strikes wire– Wire vibrates– Causes soundboard to

vibrate– Causes a force on the air

molecules– Kinetic energy is

converted to sound waves

Sound Intensity

• Sound intensity is the rate at which energy flows through a unit area of the plane wave– Power is the rate of energy transfer– Intensity can be described in terms of power– SI unit: W/m2

intensity =ΔE / Δtarea

=P

area

intensity =P

4r 2=

(power)

(4)(distance from the source)2

Sound Intensity

• Intensity decreases as the distance from the source (r) increases

• Same amount of energy spread over a larger area

intensity =P

4r 2=

(power)

(4)(distance from the source)2

Intensity and Frequency

Human Hearing depends both on frequency and intensity

Relative Intensity

• Intensity determines loudness (volume)• Volume is not directly proportional to intensity• Sensation of loudness is approximately logarithmic• The decibel level is a more direct indication of

loudness as perceived by the human ear– Relative intensity, determined by relating the intensity of a

sound wave to the intensity at the threshold of hearing

Relative Intensity

•When intensity is multiplied by 10, 10dB are added to the decibel level•10dB increase equates to sound being twice as loud

Forced Vibrations

• Vibrating strings cause bridge to vibrate

• Bridge causes the guitar’s body to vibrate

• These forced vibrations are called sympathetic vibrations

• Guitar body cause the vibration to be transferred to the air more quickly– Larger surface area

Resonance

• In Figure 13.11, if a blue pendulum is set into motion, the others will also move

• However, the other blue pendulum will oscillate with a much larger amplitude than the red and green– Because the natural frequency matches the frequency of the first

blue pendulum

• Every guitar string will vibrate at a certain frequency• If a sound is produced with the same frequency as one of the

strings, that string will also vibrate

The Human Ear

The basilar membrane has different naturalFrequencies at different positions

Chapter 13 - Sound

13.3 - Harmonics

Standing Waves on a Vibrating String

• Musical instruments, usually consist of many standing waves together, with different wavelengths and frequencies even though you hear a single pitch

• Ends of the string will always be the nodes• In the simplest vibration, the center of the

string experiences the most displacement• This frequency of this vibration is called the

fundamental frequency

The Harmonic Series

Fundamental frequency or first harmonicWavelength is equal to twice the string length

Second harmonicWavelength is equal to the string length

fundamental frequency = f 1 =vλ1

=v2L

f n =nv2L

n = 1, 2, 3, . . .

Standing Waves on a Vibrating String

• When a guitar player presses down on a string at any point, that point becomes a node

Standing Waves in an Air Column

• Harmonic series in an organ pipe depends on whether the reflecting end of the pipe is open or closed.

• If open - that end becomes and antinode

• If closed - that end becomes a node

Standing waves in an Air Column

f n =nv2L

n=1, 2, 3, . . .

The Fundamental frequency can be changed by changing the vibrating air column


Only odd harmonics will be present

f n =nv2L

n=1, 3, 5, . . .


• Trumpets, saxophones and clarinets are similar to a pipe closed at one end– Trumpets: Player’s mouth closes

one end– Saxophones and clarinets: reed

closes one end

• Fundamental frequency formula does not directly apply to these instruments– Deviations from the cylindrical

shape of a pipe affect the harmonic series

Harmonics account for sound quality, or timbre

• Each instrument has its own characteristic mixture of harmonics at varying intensities

• Tuning fork vibrates only at its fundamental, resulting in a sine wave

• Other instruments are more complex because they consist of many harmonics at different intensities



• The mixture of harmonics produces the characteristic sound of an instrument : timbre

• Fuller sound than a tuning fork

Fundamental Frequency determines pitch

• In musical instruments, the fundamental frequency determines pitch

• Other harmonics are sometimes referred to as overtones

• An frequency of the thirteenth note is twice the frequency of the first note

Fundamental Frequency determines pitch

Beats

• When two waves differ slightly in frequency, they interfere and the pattern that results is an alternation between loudness and softness - Beat

• Out of phase: complete destructive interference

• In Phase - complete constructive interference

Beats

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Chapter 7