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Examples of oscillations & waves : Earthquake – Tsunami Electric guitar – Sound wave Watch – quartz crystal Radar speed-trap Radio telescope. Part Two: Oscillations, Waves, & Fluids. Examples of fluid mechanics : Flow speed vs river width Plane flight.

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Part Two: Oscillations, Waves, & Fluids

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Examples of oscillations & waves:

Earthquake – Tsunami

Electric guitar – Sound wave

Watch – quartz crystal

Radar speed-trap

Radio telescope

Part Two: Oscillations, Waves, & Fluids

Examples of fluidmechanics:

Flow speed vs river width

Plane flight

High-speed photo: spreading circular waves on water.

13. Oscillatory Motion

Describing Oscillatory Motion

Simple Harmonic Motion

Applications of Simple Harmonic Motion

Circular & Harmonic Motion

Energy in Simple Harmonic Motion

Damped Harmonic Motion

Driven Oscillations & Resonance

Dancers from the Bandaloop Project perform on vertical surfaces, executing graceful slow-motion jumps.

What determines the duration of these jumps?

Wilberforce Pendulum

pendulum motion: rope length & g

Disturbing a system from equilibrium results in oscillatory motion.

Absent friction, oscillation continues forever.

Examples of oscillatory motion:

Microwave oven: Heats food by oscillating H2O molecules in it.

CO2 molecules in atmosphere absorb heat by vibrating  global warming.

Watch keeps time thru oscillation ( pendulum, spring-wheel, quartz crystal, …)

Earth quake induces vibrations  collapse of buildings & bridges .


13.1. Describing Oscillatory Motion

Characteristics of oscillatory motion:

  • Amplitude A = max displacement from equilibrium.

  • PeriodT = time for the motion to repeat itself.

  • Frequencyf = # of oscillations per unit time.

same period T

same amplitude A

[ f ] = hertz (Hz) = 1 cycle / s

A, T, f do not specify an oscillation completely.


Example 13.1. Oscillating Ruler

An oscillating ruler completes 28 cycles in 10 s & moves a total distance of 8.0 cm.

What are the amplitude, period, & frequency of this oscillatory motion?

Amplitude = 8.0 cm / 2 = 4.0 cm.

13.2. Simple Harmonic Motion

Simple Harmonic Motion (SHM):

2nd order diff. eq  2 integration const.


angular frequency

A, B determined by initial conditions

( t )  2

x  2A

Amplitude & Phase

C = amplitude

 = phase

Note: is independent of amplitude only for SHM.

Curve moves to the right for < 0.


Velocity & Acceleration in SHM

|x| = max at v = 0

|v| = max at a = 0

GOT IT? 13.1.

Two identical mass-springs are displaced different amounts from equilibrium & then released at different times.

Of the amplitudes, frequencies, periods, & phases of the subsequent motions, which are the same for both systems & which are different?

Same: frequencies, periods


amplitudes ( different displacement )

phases ( different release time )

Application: Swaying skyscraper

Tuned mass damper :

Damper highly damped ,

Overall oscillation overdamped.

Taipei 101 TMD:

41 steel plates,

730 ton, d = 550 cm,

87th-92nd floor.

Also used in:

  • Tall smokestacks

  • Airport control towers.

  • Power-plant cooling towers.

  • Bridges.

  • Ski lifts.


Tuned Mass Damper

Example 13.2. Tuned Mass Damper

The tuned mass damper in NY’s Citicorp Tower consists of a 373-Mg (vs 101’s 3500 Mg) concrete block that completes one cycle of oscillation in 6.80 s.

The oscillation amplitude in a high wind is 110 cm.

Determine the spring constant & the maximum speed & acceleration of the block.

13.3. Applications of Simple Harmonic Motion

  • The Vertical Mass-Spring System

  • The Torsional Oscillator

  • The Pendulum

  • The Physical Pendulum

The Vertical Mass-Spring System

Spring stretched by x1 when loaded.

mass m oscillates about the new equil. pos.

with freq

The Torsional Oscillator

= torsional constant

Used in timepieces

The Pendulum

Small angles oscillation:

Simple pendulum (point mass m):

Example 13.3. Rescuing Tarzan

Tarzan stands on a branch as a leopard threatens.

Jane is on a nearby branch of the same height, holding a 25-m-long vine attached to a point midway between her & Tarzan.

She grasps the vine & steps off with negligible velocity.

How soon can she reach Tarzan?

Time needed:

GOT IT? 13.2.

  • What happens to the period of a pendulum if

  • its mass is doubled,

  • it’s moved to a planet whose g is ¼ that of Earth,

  • its length is quadrupled?

no change



Conceptual Example 13.1. Nonlinear Pendulum

  • A pendulum becomes nonlinear if its amplitude becomes too large.

  • As the amplitude increases, how will its period changes?

  • If you start the pendulum by striking it when it’s hanging vertically,

    • will it undergo oscillatory motion no matter how hard it’s hit?

  • If it’s hit hard enough,

  • motion becomes rotational.

(a) sin increases slower than 

 smaller  

 longer period

The Physical Pendulum

Physical Pendulum = any object that’s free to swing

Small angular displacement  SHM

Example 13.4. Walking

When walking, the leg not in contact of the ground swings forward,

acting like a physical pendulum.

Approximating the leg as a uniform rod, find the period for a leg 90 cm long.

Table 10.2

Forward stride = T/2 = 0.8 s

13.4. Circular & Harmonic Motion

Circular motion:

2  SHO with same A &  but  = 90

x =  R

x = R

x = 0

Lissajous Curves

GOT IT? 13.3.

The figure shows paths traced out by two pendulums swinging with different frequencies in the x- & y- directions.

What are the ratios x : y ?

1 : 2

3: 2

Lissajous Curves

13.5. Energy in Simple Harmonic Motion


= constant

Energy in SHM

Potential Energy Curves & SHM

Linear force:

 parabolic potential energy:

Taylor expansion near local minimum:

 Small disturbances near equilibrium points  SHM

GOT IT? 13.4.

  • Two different mass-springs oscillate with the same amplitude & frequency.

  • If one has twice as much energy as the other, how do

  • their masses & (b) their spring constants compare?

  • (c) What about their maximum speeds?

  • The more energetic oscillator has

  • twice the mass

  • twice the spring constant

  • (c) Their maximum speeds are equal.

13.6. Damped Harmonic Motion

sinusoidal oscillation

Damping (frictional) force:

Damped mass-spring:

Amplitude exponential decay


 At t = 2m / b, amplitude drops to 1/e of max value.

(a) For

 is real, motion is oscillatory ( underdamped )

(c) For

 is imaginary, motion is exponential ( overdamped )

(b) For

 = 0, motion is exponential ( critically damped )

Damped & Driven Harmonic Motion

Example 13.6. Bad Shocks

A car’s suspension has m = 1200 kg & k = 58 kN / m.

Its worn-out shock absorbers provide a damping constant b = 230 kg / s.

After the car hit a pothole, how many oscillations will it make before the amplitude drops to half its initial value?

Time  required is

# of oscillations:

bad shock !

13.7. Driven Oscillations & Resonance

External force  Driven oscillator


d= driving frequency

( long time )

Prob 75:

= natural frequency

Damped & Driven Harmonic Motion


Buildings, bridges, etc have natural freq.

If Earth quake, wind, etc sets up resonance, disasters result.

Collapse of Tacoma bridge is due to self-excitation described by the van der Pol equation.

Tacoma Bridge

Resonance in microscopic system:

  • electrons in magnetron  microwave oven

  • Tokamak (toroidal magnetic field)  fusion

  • CO2 vibration: resonance at IR freq  Green house effect

  • Nuclear magnetic resonance (NMR)  NMI for medical use.

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