Transcript 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
r
• s= arc length
• r = radius
Angular Motion
• 360o = 2rad
• 180o = rad
(rad) 

180
(deg)
Angular displacement
• Angular dispacement: the angle through which a point line, or
body is rotated in a specified direction and about a specified
axis
 
s
r
• 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

avg 
t
• Angular acceleration: The time rate of change of
angular speed, expressed in radians per second per
second
2  1 
avg 
t2  t1

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
Tangential Acceleration
• The instantaneous linear acceleration
of an object directed along the tangent
to the object’s circular path
v t
t
r

t
a t  r
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
ac 
vt 2
r
a c  r2
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
Fc  ma c
or:
Fc 
mv t 2
r
Fc  mr
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
F
P
A
• 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
Location
Some pressures
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
Atmosphere at 10 km above sea level
1.01 x
105
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
container
F1 F2
A2
Pinc 
A1

A2
F2 
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?
F much
620N weight can the wide piston lift?
• PHow


 3.1  10 Pa
3
A
F2 
A2
A1
0.20m
F1 
2
2.0m
2
0.20m
2
3
620N  6.2  10 N
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
m  V
V  Ah
P
F mg Vg Ahg



 hg
A
A
A
A
Fluid Pressure
• Gauge Pressure
P
F mg Vg Ahg



 hg
A
A
A
A
– does not take the pressure of the atmosphere into
consideration
• Fluid Pressure as a function of depth
P  P0  gh
– 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
Pnet  Pbottom  Ptop  (P0  gh2)  (P0  gh1)
 g(h2  h1)  gL
Fnet  Pnet A  gLA  gV  m f g
• 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 > 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
5 Pa
10
P  P  hg 
0
5
3
3
2
1.01  10 Pa  1.29kg / m (1.0  10 m)(9.81m / s )
 8.8  104 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
m1  m2
1 V1  2 V2
1 A1x1  2 A2x2
1 A1 v1t  2 A2 v2t
A1 v1  A2 v2
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 crosssection and elevation
• Velocity changes, so kinetic
energy changes
• This can be compensated
for by a change in
gravitational potential
energy or pressure
1 2
P  v  gh  cons tan t
2
Bernoulli’s Equation
1 2
P  v  gh  cons tan t
2
Bernoulli’s Principle: A
Special Case
• In a horizontal pipe
1
1
2
2
P1  1 v  P2  2 v
2
2
The Ideal Gas Law
PV  NkB T
• kB is a constant called the Boltzmann’s
constant and has been experimentally
determined to be 1.38 x 10-23 J/K
Ideal Gas Law Cont’d
• If the number of particles is constant then:
P1 V1
T1

P2 V2
T2
• Alternate Form:
ˆ˜ kB T kB T
˜˜
P

˜˜
mV
m
¯ m
– m=mass of each particle, M=N x m Total Mass of the gas
MK BT
Ê
ÁM
Á
Á
Á
ÁV
Ë
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
k
m
Simple Harmonic Motion
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
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
Hooke’s Law:
Felastic  kx
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
Fel
x = -0.02 m
Solution:
Fg
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
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
L
g
The Period of a massspring system
• Restoring force
Felastic  kx
– Not affected by mass
• Increasing mass increases inertia, but
not restoring force --> smaller
acceleration
The Period of a massspring 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
m
T  2
k
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

1
v
f
T
T
v  f
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, 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 than
observer B because of the relative motion of the car
The frequency heard by observer A is higher than
the 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
E / t
P
intensity 

area
area
intensity 
P
4 r
2

(power)
(4 )(distance from the source)
2
Sound Intensity
intensity 
P
4 r
2

(power)
(4 )(distance from the source)
2
• 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 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 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
v
v
fundamental frequency  f 1 

1 2L
Second harmonic
Wavelength is equal to the string length
fn  n
v
2L
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
v
fn  n
2L
n  1, 2, 3, . . .
The Fundamental frequency can be changed by
changing the vibrating air column
Standing Waves in an Air
Column
Only odd harmonics will be present
v
fn  n
2L
n  1, 3, 5, . . .
Standing Waves in an Air
Column
• 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
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