Oscillations & Waves IB Physics

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:

Simple Harmonic Motion Terms – Displacement( x, Θ ) – Amplitude ( x o, Θ o ) – Period (T) – Frequency ( f ) – Phase Difference {There’s a nice succinct explanation of the Radian on p.101. Check it out.}

Table 13-1 Typical Periods and Frequencies

Simple Harmonic Motion Definition – Oscillators that are perfectly isochronous & whose amplitude does not change in time Real World Approximations – Pendulum ( Θ 0 < 40 o ) – Weight on a spring (limited Amplitude)

Simple Harmonic Motion Angular Frequency – In terms of linear frequency: ω = 2π f 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 13-5 Position versus time in simple harmonic motion

Figure 13-6 Velocity versus time in simple harmonic motion

Figure 13-7 Acceleration versus time in simple harmonic motion

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

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

Simple Harmonic Motion Mathematical Definition a is directly proportional to x a = - ω 2 x

Simple Harmonic Motion What does this mean about force? F = - k x Apply 2 nd Law ma = - k x

Simple Harmonic Motion Acceleration not constant – Force-accel relation: 2 nd 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 Pendulum A 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 Pendulum However, 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:

SHM Equation Solutions x = x o cos ω t x = x o sin ω t v = v o cos ω t v = - v o sin ω t v = ± ω√( x o 2 - x 2 )

Boundary Conditions x = x o when t=0 x = 0 when t=0 – Solutions differ in phase by π 2

Energy Changes Kinetic Potential Total

Figure Energy as a function of position in simple harmonic motion

Figure Energy 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

Forced Oscillations & Resonance Natural Frequency – Frequency at which system oscillates when not being driven Forced (driven) Oscillations – Added energy to prevent damping

Forced Oscillations & Resonance Driver Frequency = Natural Frequency – Max E from driver max amplitude – Max amplitude of oscillation – Resonance – re: A vs. f graphs

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 wave λ RAY Fronts

Wave Characteristics Crest – highest point - (max displacement  ) Trough – lowest point - (max displacement  ) Compression – particles are closest - (max displacement  ) Rarefaction – particles are farthest apart - (max displacement  )

Wave Characteristics Amplitude (A,a) – maximum displacement from equilibrium position Period (T) – time for one complete oscillation Frequency (f) – oscillations per second Wavelength (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 ∞ amplitude 2

Wave Characteristics Relationships: f =1 T v = 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