1 / 127

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

Angular Motion

- 360o 2?rad
- 180o ?rad

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?

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

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 objects

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

Tangential Acceleration

- The instantaneous linear acceleration of an

object directed along the tangent to the objects

circular path

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.

Centripetal Acceleration

- 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

Centripetal Acceleration

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

Whats 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 objects 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 Newtons second law
- or

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

Some Pressures

Table 9-2 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 - Pascals 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 container

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?

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 area

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)

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

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 -gt 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 P01.01 x 105 Pa

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

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

Bernoullis Principle

- The pressure in a fluid decreases as the fluids

velocity increases

Bernoullis 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

Bernoullis Equation

Bernoullis Principle A Special Case

- In a horizontal pipe

The Ideal Gas Law

- kB is a constant called the Boltzmanns constant

and has been experimentally determined to be 1.38

x 10-23 J/K

Ideal Gas Law Contd

- If the number of particles is constant then
- Alternate Form
- mmass of each particle, MN x m Total Mass of

the gas

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

Simple Harmonic Motion

Simple harmonic motion

- At the unstretched position, the spring is at

equilibrium (x0) - The spring force increases as the spring is

stretched away from equilibrium - As the mass moves towards equilibrium, force (and

acceleration) decreases

Simple harmonic motion

- Momentum causes mass to overshoot equilibrium
- Elastic force increases (in the opposite

direction)

Simple harmonic motion

- 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

Hookes Law

Spring force - (spring constant x displacement)

Hookes Law Example

Example 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 kg x -0.02

m g -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

Fel

Fg

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 bobs weight - When the angle of displacement is gt15o, 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

The Period of a mass-spring system

- Restoring force
- Not affected by mass
- Increasing mass increases inertia, but not

restoring force --gt smaller acceleration

The Period of a mass-spring system

- A heavier mass will take more time to complete a

cycle --gt Period increases - The greater the spring constant, the greater the

force, the greater the acceleration, which causes

a decrease in period

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

Note The distance between the adjacent crests

and troughs are the same

- Longitudinal wave A wave whose particles vibrate

parallel to the direction of wave motion. i.e.

sound

Period, Frequency, and Wave speed

- Period is the amount of time it takes for a

complete wavelength to pass a given point

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

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)

The Production of Sound Waves

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, wavelengthv/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

Sound Intensity

- Intensity decreases as the distance from the

source (r) increases - Same amount of energy spread over a larger area

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 guitars 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

natural Frequencies 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 harmonic Wavelength

is equal to twice the string length

Second harmonic Wavelength is equal to the string

length

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

The Fundamental frequency can be changed by

changing the vibrating air column

Standing Waves in an Air Column

Only odd harmonics will be present

Standing Waves in an Air Column

- Trumpets, saxophones and clarinets are similar to

a pipe closed at one end - Trumpets Players 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

Harmonics account for sound quality, or timbre

Harmonics account for sound quality, or timbre

- 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