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Oscillations & Waves
IB Physics
WAVES
Forced Oscillations &
Resonance
Simple Harmonic
Motion
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Make sure to read page 99
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Simple Harmonic Motion
• Oscillation
4. Physics. a. an effect expressible as a quantity that repeatedly and regularly fluctuates above and below some mean value, as the pressure of a sound wave or the voltage of an alternating current. b. a single fluctuation between maximum and minimum values in such an effect.
From: http://dictionary.reference.com
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Simple Harmonic Motion
• Terms– Displacement(x,Θ)– Amplitude (xo,Θo)– Period (T)– Frequency (f)– Phase Difference
{There’s a nice succinct explanation of the Radian on p.101. Check it out.}
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Table 13-1Typical Periods and Frequencies
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Simple Harmonic Motion
• Definition– Oscillators that are perfectly isochronous & whose
amplitude does not change in time
• Real World Approximations– Pendulum (Θ0 < 40o)
– Weight on a spring (limited Amplitude)
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Simple Harmonic Motion
• Angular Frequency– In terms of linear frequency:
ω = 2πf
• There is a connection between angular frequency and angular speed of a particle moving in a circle with a constant speed.
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13-3 Connections between Uniform Circular Motion and Simple Harmonic Motion
An object in simple harmonic motion has the same motion as one component of an object in uniform circular motion:
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13-3 Connections between Uniform Circular Motion and Simple Harmonic Motion
Here, the object in circular motion has an angular speed of
where T is the period of motion of the object in simple harmonic motion.
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Figure 13-5Position versus time in simple harmonic motion
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Figure 13-6Velocity versus time in simple harmonic motion
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Figure 13-7Acceleration versus time in simple harmonic motion
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Figure 13-2Displaying position versus time for simple harmonic motion
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Figure 13-3Simple harmonic motion as a sine or a cosine
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Simple Harmonic Motion
• Mathematical Definition
a is directly proportional to x a = - ω2 x
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Simple Harmonic Motion
• What does this mean about force?
F = - k x• Apply 2nd Law
ma = - k x
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Simple Harmonic Motion
• Acceleration not constant– Force-accel relation: 2nd order diff eq
x = P cos ω t + Q sin ω t• P & Q constants• ω = √(k/m)
• Compare T calculation for spring vs. pendulum
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13-4 The Period of a Mass on a Spring
Therefore, the period is
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13-6 The PendulumA simple pendulum consists of a mass m (of negligible size) suspended by a string or rod of length L (and negligible mass).
The angle it makes with the vertical varies with time as a sine or cosine.
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13-6 The Pendulum
Looking at the forces on the pendulum bob, we see that the restoring force is proportional to sin θ, whereas the restoring force for a spring is proportional to the displacement (which is θ in this case).
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13-6 The PendulumHowever, for small angles, sin θ and θ are approximately equal.
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13-6 The Pendulum
Substituting θ for sin θ allows us to treat the pendulum in a mathematically identical way to the mass on a spring. Therefore, we find that the period of a pendulum depends only on the length of the string:
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Solutionsof the
SHM equation
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SHM Equation Solutions
x = xocosωt
x = xosinωt
v = vocosωt
v = -vosinωt
v = ±ω√(xo2 - x2)
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Boundary Conditions
• x = xo when t=0
• x = 0 when t=0
– Solutions differ in phase by π 2
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Energy Changes
• Kinetic• Potential• Total
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Figure 13-10Energy as a function of position in simple harmonic motion
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Figure 13-11Energy as a function of time in simple harmonic motion
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Forced Oscillations & Resonance
• Damped Oscillations – decr w/ time– Heavily – decr very quickly– Critically – no/barely
• Damping Force– Opposite in direction to motion of oscillating
particle– Dissipative
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Forced Oscillations & Resonance
• Natural Frequency– Frequency at which system oscillates when not
being driven• Forced (driven) Oscillations
– Added energy to prevent damping
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Forced Oscillations & Resonance
• Driver Frequency = Natural Frequency– Max E from driver when @ max amplitude– Max amplitude of oscillation– Resonance– re: A vs. f graphs
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WAVES
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Waves• A means by which energy is transferred
between two points in a medium• No net transfer of the medium• Single: “pulse”• Continuous: “wave train”• Mechanical waves need a medium.
– Example: sound & water• Radiant energy does not need a medium.
– Example: light
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Transverse
• Vibratory is perpendicular to the direction of energy transfer.
• Examples: water & light
Crest
Trough
Height
Amplitude
Wavelength
Equilibrium
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Longitudinal
• Vibratory motion is parallel to the direction of energy transfer.
• Compressional or Pressure wave
////// / / / / / ////// / / / / /
• Example: sound
Compression
Rarefaction
Wavelength
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Waves in 2 Dimensions• Previous representations are cut-aways,
showing length & amplitude• Wave Fronts use parallel lines to represent
crests, showing width & length– Rays are often drawn perpendicular to fronts to
indicate the direction of travel of the waveλ
RAY
Fronts
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Wave CharacteristicsCrest – highest point
- (max displacement )Trough – lowest point
- (max displacement )Compression – particles are closest
- (max displacement )Rarefaction – particles are farthest apart
- (max displacement)
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Wave CharacteristicsAmplitude (A,a) – maximum displacement from
equilibrium positionPeriod (T) – time for one complete oscillationFrequency (f) – oscillations per secondWavelength (λ) – distance between two
successive particles that have the same displacement
Wave Speed (v,c) – speed energy moved through medium by the wave
Intensity (I) – energy per unit time transported across a unit area of medium
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Wave Characteristics• Wave Speed depends on nature & properties
of medium– Water waves travel faster in deep water
• Frequency of wave depends upon frequency of source– Will not change if wave enters a different medium
or the properties of the medium change
• intensity ∞ amplitude2
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Wave Characteristics
• Relationships:f = 1
Tv = fλ
• Waves are periodic in both time and space.
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Wave Graphs
Equilibrium
Position
Distance
Displacem
ent
Displacement-Position Graph
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Wave Graphs
Time
Displacem
ent
Displacement-Time Graph
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Electromagnetic Waves
• Electric & Magnetic Fields oscillating at right angles to each other
• Same speed in free space• Know spectrum p.117