Oscillations & Waves

Oscillations & Waves

IB Physics

WAVES

Forced Oscillations &

Resonance

Simple Harmonic

Motion

Make sure to read page 99

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


• 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.}

Table 13-1Typical Periods and Frequencies


• 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)


• 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.

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:

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.

Figure 13-5Position versus time in simple harmonic motion

Figure 13-6Velocity versus time in simple harmonic motion

Figure 13-7Acceleration versus time in simple harmonic motion

Figure 13-2Displaying position versus time for simple harmonic motion

Figure 13-3Simple harmonic motion as a sine or a cosine


• Mathematical Definition

a is directly proportional to x a = - ω2 x


• What does this mean about force?

F = - k x• Apply 2nd Law

ma = - k x


• 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

13-4 The Period of a Mass on a Spring

Therefore, the period is

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.

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).

13-6 The PendulumHowever, for small angles, sin θ and θ are approximately equal.

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:

Solutionsof the

SHM equation

SHM Equation Solutions

x = xocosωt

x = xosinωt

v = vocosωt

v = -vosinωt

v = ±ω√(xo2 - x2)

Boundary Conditions

• x = xo when t=0

• x = 0 when t=0

– Solutions differ in phase by π 2

Energy Changes

• Kinetic• Potential• Total

Figure 13-10Energy as a function of position in simple harmonic motion

Figure 13-11Energy as a function of time in simple harmonic motion

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


• Natural Frequency– Frequency at which system oscillates when not

being driven• Forced (driven) Oscillations

– Added energy to prevent damping


• Driver Frequency = Natural Frequency– Max E from driver when @ max amplitude– Max amplitude of oscillation– Resonance– re: A vs. f graphs

WAVES

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

Transverse

• Vibratory is perpendicular to the direction of energy transfer.

• Examples: water & light

Crest

Trough

Height

Amplitude

Wavelength

Equilibrium

Longitudinal

• Vibratory motion is parallel to the direction of energy transfer.

• Compressional or Pressure wave

////// / / / / / ////// / / / / /

• Example: sound

Compression

Rarefaction

Wavelength

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

Wave CharacteristicsCrest – highest point

- (max displacement )Trough – lowest point

- (max displacement )Compression – particles are closest

- (max displacement )Rarefaction – particles are farthest apart

- (max displacement)

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

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

Wave Characteristics

• Relationships:f = 1

Tv = fλ

• Waves are periodic in both time and space.

Wave Graphs

Equilibrium

Position

Distance

Displacement

Displacement-Position Graph

Wave Graphs

Time

Displacement

Displacement-Time Graph

Electromagnetic Waves

• Electric & Magnetic Fields oscillating at right angles to each other

• Same speed in free space• Know spectrum p.117

Oscillations & Waves

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