L 21 – Vibrations and Waves [1]
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Transcript of L 21 – Vibrations and Waves [1]
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L 21 – Vibrations and Waves [1]L 21 – Vibrations and Waves [1]
resonanceresonance Tacoma Narrows Tacoma Narrows Bridge CollapseBridge Collapse
clocks – pendulumclocks – pendulum springssprings harmonic motionharmonic motion mechanical wavesmechanical waves sound wavessound waves musical instrumentsmusical instruments
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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 from Madeira IslandVortex street from Madeira Island((off the NW coast of Africa)off the NW coast of Africa)
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The earth is shaking
earthquakes
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Earthquakes in the Midwest
New Madrid, MO
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Clocks
1800
Length candle burns hourglass
sundial Clock with mainspring
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water clock- Greeks
As the water leaks out,level goes down.
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length of a shadow
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Non-repetitive Clocks• measures a single interval of time
• hourglass• shadows-sundials• candles• water clocks
• poorly suited to subdividing a day
• requires frequent operator intervention
• not portable
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Repetitive Clocks
• based on an object whose motion repeats itself at regular intervals
• does not require frequent operator intervention
pendulum clock
• first used by Galileo to measure time
• based on harmonic oscillators – objects that vibrate
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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 forces it to move 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 just the right direction to bring the object back to equilibrium
• 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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let’s look at energy • 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
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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 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 Law
sprin
g fo
rce
(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
elastic limit of the spring
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springs are useful !
springs help make a bumpy road
seem less bumpy
springs help yousleep morecomfortably!
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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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Period of the mass-spring system
2m
Tk
• This formula is good for a mass on a hanging spring or for a horizontal spring on the air track
• If the mass is quadrupled, the period is doubled