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L 21 – Vibration and Sound [1] Resonance Tacoma Narrows Bridge Collapse clocks – pendulum...
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Transcript of L 21 – Vibration and Sound [1] Resonance Tacoma Narrows Bridge Collapse clocks – pendulum...
![Page 1: L 21 – Vibration and Sound [1] Resonance Tacoma Narrows Bridge Collapse clocks – pendulum springs harmonic motion mechanical waves sound waves musical.](https://reader035.fdocuments.net/reader035/viewer/2022081605/5a4d1ace7f8b9ab0599709b7/html5/thumbnails/1.jpg)
L 21 – Vibration and Sound [1]
• Resonance• Tacoma Narrows
Bridge Collapse• clocks – pendulum• springs• harmonic motion• mechanical waves• sound waves• musical instruments
November 7, 1940
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Flow past an object
wind
vortex street - exerts aperiodic force on the object
object
an example of resonance in mechanical systems
vorticies
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Vortex street behind Selkirk Islandin the South Pacific
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The earth is shaking
Earthquakes http://www.classzone.com/books/earth_science/terc/content/visualizations/es1005/es1005page01.cfm?chapter_no=visualization
http://www.geo.mtu.edu/UPSeis/waves.html
S waves
P waves
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Earthquakes andTsunamis
Plate tectonics
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Keeping time Clocks
Length candle burns
hourglass sundial
1800 Clockswith mainspring
Digital clock
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length of a shadow
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Clocks based on repetitive motion
• based on an object whose motion repeats itself at regular intervals
• pendulum clock• first used by Galileo to measure time
(Before this, Galileo, who was trained as a physician, used his own pulse as a clock.)
• based on harmonic oscillators – objects that vibrate back and forth
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The pendulum- a closer look
• The pendulum is driven by gravity – the mass is falling from point A to point B then rises from B to C
• the tension in the string T provides the centripetal force to keep m moving in a circle
• one component of mg is along the circular arc – always pointing toward point B on either side. At point B this blue force vanishes.
mg
T
L
mg
T
AB
C
mg
T
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The “restoring” force
• To start the pendulum, you displace it from point B to point A and let it go!
• point B is the equilibrium position of the pendulum
• on either side of B the blue force always act to bring (restore) the pendulum back to equilibrium, point B
• this is a “restoring” forcemg
T
L
mg
T
AB
C
mg
T
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the role of the restoring force
• the restoring force is the key to understanding all systems that oscillate or repeat a motion over and over.
• the restoring force always points in the direction to bring the object back to equilibrium (for a pendulum at the bottom)
• from A to B the restoring force accelerates the pendulum down
• from B to C it slows the pendulum down so that at point C it can turn around
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Repeating motions
• if there are no forces (friction or air resistance) to interfere with the motion, the motion repeats itself forever it is a harmonic oscillator
• harmonic – repeats at the same intervals• notice that at the very bottom of the
pendulum’s swing (at B ) the restoring force is ZERO, so what keeps it going?
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it’s the INERTIA !
• even though the restoring force is zero at the bottom of the pendulum swing, the ball is moving and since it has inertia it keeps moving to the left.
• as it moves from B to C, gravity slows it down (as it would any object that is moving up), until at C it momentarily comes to rest.
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Energy in a pendulum • to start the pendulum, we move it from B to A. A t
point A it has only gravitational potential energy (GPE) due to gravity
• from A to B, its GPE is converted to kinetic energy, which is maximum at B (its speed is maximum at B too)
• from B to C, it uses its kinetic energy to climb up the hill, converting its KE back to GPE
• at C it has just as much GPE as it did at A• large pendulum demo
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Some terminology
• the maximum displacement of an object from equilibrium is called the AMPLITUDE
• the time that it takes to complete one full cycle (A B C B A ) is called the PERIOD of the motion
• if we count the number of full cycles the oscillator completes in a given time, that is called the FREQUENCY of the oscillator
AA
0
The mass/spring oscillatoris the simplest example
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period and frequency
• The period T and frequency f are related to each other.
• if it takes ½ second for an oscillator to go through one cycle, its period is T = 0.5 s.
• in one second, then the oscillator would complete exactly 2 cycles ( f = 2 per second or 2 Hertz, Hz)
• 1 Hz = 1 cycle per second.• thus the frequency is: f = 1/T and, T = 1/f
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Mass hanging from a spring
• a mass hanging from a spring also executes harmonic motion up and down.
• to understand this motion we have to first understand how springs work.
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springs are amazing devices!
the harder I pull on a spring, the harder it pulls back
the harder I push ona spring, the harder it
pushes back
stretching
compression
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Springs obey Hooke’s Lawsp
ring
forc
e (N
)
amount of stretching or compressing in meters
• the strength of a spring is measured by how much force it provides for a given amount of stretch • we call this quantity k, the spring constant in N/m• magnitude of spring force = k amount of stretch
elastic limit of the spring
1 2
1
2
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springs are useful !
Springs help you sleep more comfortably!
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the mass/spring oscillator• as the mass falls down it
stretches the spring, which makes the spring force bigger, thus slowing the mass down
• after the mass has come momentarily to rest at the bottom, the spring pulls it back up
• at the top, the mass starts falling again and the process continues – oscillation!
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the mass spring oscillator does not need gravity
the time to complete an oscillation does notdepend on where the mass starts!
frictionlesssurface
spring that can be stretched or
compressed
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2pendulum
LTg
2mass spring
mTk
• L = length (m)• g = 10 m/s2
• does not depend
on mass
• m = mass in kg• k = spring constant
in N/m
T = period, time for one complete cycle
Pendulum Mass-spring