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Section 1 Simple Harmonic Motion

Chapter 11

- Periodic (vibrating) motion is repeated motion.
- swing
- pendulum
- mass on a spring
- metronome
- pendulum clocks
- oscillations
- waves

Chapter 11

Vibrations and Waves

Table of ContentsSection 1 Simple Harmonic Motion

Section 2 Measuring Simple Harmonic Motion

Section 3 Properties of Waves

Section 4 Wave Interactions

Section 1 Simple Harmonic Motion

Chapter 11

The Vibrating Spring- One example ofperiodic motionis the motion of a mass attached to a spring.
- The direction of the force acting on the mass (Felastic) is always opposite the direction of the mass’s displacement from equilibrium (x = 0).
- Felis a restoring force, because it acts to restore the system to equilibrium

Section 1 Simple Harmonic Motion

Chapter 11

Hooke’s Law- Measurements show that thespringforce, or restoring force,isdirectly proportionalto thedisplacementof the mass.
- This relationship is known asHooke’s Law:

Felastic = –kx

spring force = –(spring constant displacement)

- The quantitykis a positive constant called thespring constant with SI unit N/m.

PE elastic = 1/2kx2

PE max = 1/2kx2max

- The stiffer the spring the greater the spring constant.

Section 1 Simple Harmonic Motion

Chapter 11

Sample ProblemHooke’s Law

If a mass of 0.55 kg attached to a vertical spring stretches the spring 2.0 cm from its original equilibrium position, what is the spring constant?

Section 1 Simple Harmonic Motion

Chapter 11

- Determine the spring constant.

- Unknown:
- k = ?

- Diagram:

- Given:
- m = 0.55 kg
- x = – 2.0 cm = – 0.02 m
- g = 9.81 m/s2
- Fg = mg = 0.55 kg x 9.81 m/s2 = 5.4 N
- F = -kx

Section 1 Simple Harmonic Motion

Chapter 11

When the mass is attached to the spring, the equilibrium position changes. At the new equilibrium position, the net force acting on the mass is zero. So the spring force (given by Hooke’s law) must be equal and opposite to the weight of the mass.

Fnet = 0 = Felastic + Fg

(Fel is up and Fg is down)

Felastic + Fg= 0

Felastic= –kx

Fg = – 5.4 N

–kx– 5.4 N = 0

–kx= 5.4 N

k = Felastic /–x

k = 5.4 N /–(-.02 m)

k = 270 N/m

Section 1 Simple Harmonic Motion

Chapter 11

Evaluate your answer.

The value of k implies that 270 N of force is required to displace the spring 1 m. A 270 N force is the weight of a 27.5 kg mass.

Practice A p. 371

- a. 15 N/m b. less stiff
- 3.2 x 102 N/m
- 2.7 x 103 N/m
- 81 N

Section 1 Simple Harmonic Motion

- All vibrating systems that obey Hooke’s law demonstrate simple harmonic motion.
- Simple harmonic motion describes any periodic motion that is the result of a restoring force that is proportional to displacement. (Hooke’s Law)
- Because simple harmonic motion involves a restoring force, every simple harmonic motion is a back-and-forth motion over the same path.

Harmonic Oscillators

A harmonic oscillator is an oscillator with a restoring force proportional to its displacement from equilibrium. Its period of oscillation depends only on the stiffness of that restoring force and on its mass, not on its amplitude of oscillation.

Period (T) is the time for 1 complete oscillation and in measured in seconds.

Any harmonic oscillator can be thought of as having a restoring force component that drives the motion and an inertial component that resists the motion.

http://upload.wikimedia.org/wikipedia/commons/2/21/Pendulum_animation.gif

http://upload.wikimedia.org/wikipedia/commons/e/ea/Simple_Harmonic_Motion_Orbit.gif

Clocks are based on Simple Harmonic Oscillators

Pendulum and Balance Clocks

The accuracy of pendulum and balance clocks is limited by friction, air resistance, and thermal expansion to about 10 s/year.

Electronic Clocks

Quartz crystals are piezoelectric materials that respond mechanically to electrical stress and electrically to mechanical stress.

Loses or gains less than a 0.1 s/year

Atomic Clocks (Gain / lose 1 s/ 10 million years)

The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.

The meter is the length of the path travelled by light emitted by krypton-86 in vacuum during a time interval of 1/299 792 458 of a second.

Section 2 Measuring Simple Harmonic Motion

Chapter 11

Amplitude, Period, and Frequency in SHM- In SHM, the maximum displacement from equilibrium is defined as theamplitude of the vibration.
- Apendulum’s amplitude is measured by the angle between the pendulum’s equilibrium position and its maximum displacement.
- For amass-spring system,the amplitude is the maximum amount the spring is stretched or compressed from its equilibrium position.
- TheSI unitsof amplitude are theradian (rad)and themeter (m).

Section 2 Measuring Simple Harmonic Motion

Chapter 11

Amplitude, Period, and Frequency in SHM- Theperiod(T)is the time that it takes a complete cycle to occur.
- The SI unit of period is seconds (s).
- The frequency(f)is the number of cycles or vibrations per unit of time. (cycles/s)
- The SI unit of frequency is hertz (Hz).
- Hz = s–1

Section 2 Measuring Simple Harmonic Motion

Chapter 11

Amplitude, Period, and Frequency in SHM- Period and frequency areinversely related:

- Any time you know the period or the frequency, you can calculate the other value.

Simple Harmonic Motion

http://webphysics.davidson.edu/physlet_resources/bu_semester1/c18_SHM_graphs.html

y = A sinwt

y = A sin 2pf

Section 1 Simple Harmonic Motion

Chapter 11

Simple Harmonic Motion- Why must the angle of displacement be small for a pendulum to exhibit simple harmonic motion?
- For SHM condition the restoring force must be proportional to the displacement.
- F must be proportional to q and not sin q.

360o = 2P radians

15o x 2P radians/ 360o

Chapter 11

Objectives- Distinguishlocal particle vibrations from overall wave motion.
- Differentiatebetween pulse waves and periodic waves.
- Interpret waveforms of transverse and longitudinal waves.
- Applythe relationship among wave speed, frequency, and wavelength to solve problems.
- Relateenergy and amplitude.

Chapter 11

Wave Motion- Awaveis the motion of a disturbance and is produced by a vibration. A wave transfers energy.
- Amediumis a physical environment through which a disturbance can travel. For example, water is the medium for ripple waves in a pond.
- Waves that require a medium through which to travel are calledmechanical waves.Water waves and sound waves are mechanical waves.
- Electromagnetic wavessuch as visible light do not require a medium.

Chapter 11

Wave Types- A wave that consists of a single traveling pulse is called apulse wave.
- Whenever a wave’s source is periodic motion, such as a vibration or the motion of your hand moving up and down repeatedly, aperiodic waveis produced.
- A wave generated by a simple harmonic oscillator is called asine wave.

Chapter 11

Relationship Between SHM and Wave MotionAs the sine wave created by this vibrating blade travels to the right, a single point on the string vibrates up and down with simple harmonic motion.

Chapter 11

Wave Types- Atransverse waveis a wave whose particles vibrateperpendicularlyto the direction of the wave motion.
- Thecrestis the highestpoint above the equilibrium position, and thetroughis thelowestpoint below the equilibrium position.
- Thewavelength (l)is the distance between two adjacent similar points of a wave.
- Amplitude is maximum displacement from rest or equilibrium.

Chapter 11

Wave Types- Alongitudinal wave is a wave whose particles vibrate parallel to the direction the wave is traveling.
- A longitudinal wave on a spring at some instant in time, t, can be represented by a graph. Thecrestscorrespond to compressed regions, and thetroughscorrespond to stretched regions.
- The crests are regions of high density and pressure (relative to the equilibrium density or pressure of the medium), and the troughsare regions of low density and pressure.

Chapter 11

Period, Frequency, and Wave Speed- The frequency of a wavedescribes the number of waves that pass a given point in a unit of time.
- Theperiod of a wavedescribes the time it takes for a complete wavelength to pass a given point.
- The relationship between period and frequency in SHM holds true for waves as well; the period of a wave isinversely relatedto its frequency.

Chapter 11

Amplitude, Period, and Frequency- Theperiod(T)is the time that it takes a complete cycle of a wave to occur.
- The SI unit of period is seconds (s).
- The frequency(f)is the number of waves or vibrations per unit of time. (waves/s)
- The SI unit of frequency is hertz (Hz).
- Hz = s–1

Chapter 11

Amplitude, Period, and Frequency- Period and frequency areinversely related:

- Any time you know the period or the frequency, you can calculate the other value.

Chapter 11

Period, Frequency, and Wave Speed, continued- The speed of a mechanicalwave is constant for any given medium.
- Thespeed of a waveis given by the following equation:

v = fl

wave speed = frequency wavelength

- This equation applies to both mechanical and electromagnetic waves.

Chapter 11

Waves and Energy Transfer- Waves transfer energy by the vibration of matter.
- Waves are often able to transport energy efficiently.
- The rate at which a wave transfers energy depends on theamplitude.
- The greater the amplitude, the more energy a wave carries in a given time interval.
- For a mechanical wave, the energy transferred is proportional to the square of the wave’s amplitude.
- E a amp2
- The amplitude of a wave gradually diminishes over time as its energy is dissipated.

Practice D p. 387

- 0.081 m <l< 12 m
- a. 3.41 m b. 5.0 x 10-7 m c. 1.0 x 10-10 m
- 4.74 x 1014 Hz
- a. 346 m/s b. 5.86 m

Chapter 11

Objectives- Apply the superposition principle.
- Differentiatebetween constructive and destructive interference.
- Predictwhen a reflected wave will be inverted.
- Predictwhether specific traveling waves will produce a standing wave.
- Identifynodes and antinodes of a standing wave.

Chapter 11

Wave Interference- Two different material objects can never occupy the same space at the same time.
- Because mechanical waves are not matter but rather are displacements of matter, two waves can occupy the same space at the same time.
- The combination of two overlapping waves is calledsuperposition.

Chapter 11

Wave Interference, continuedIn constructive interference,individual displacements on the same side of the equilibrium position are added together to form the resultant wave.

Chapter 11

Wave InterferenceIn destructive interference,individual displacements on opposite sides of the equilibrium position are added together to form the resultant wave.

Chapter 11

Reflection- What happens to the motion of a wave when it reaches a boundary?
- At afree boundary, waves arereflected.
- At afixedboundary, waves arereflectedand inverted.

Free boundary Fixed boundary

PhET Wave on a String - Interference, Harmonic Motion, Frequency , Amplitude

Chapter 11

Standing Waves- A standing waveis a wave pattern that results when two waves of the samefrequency, wavelength, and amplitude travel in opposite directions and interfere.
- Standing waves have nodes and antinodes.
- A nodeis a point in a standing wave that maintainszero displacement.
- Anantinodeis a point in a standing wave, halfway between two nodes, at which thelargest displacementoccurs.

Chapter 11

Standing Waves, continued- Only certain wavelengthsproduce standing wave patterns.
- Theendsof the string must benodesbecause these points cannot vibrate.
- A standing wave can be produced for any wavelength that allows both ends to be nodes.
- In the diagram, possible wavelengths include2L(b),L(c), and2/3L(d).

Chapter 11

Standing WavesThis photograph shows four possible standing waves that can exist on a given string. The diagram shows the progression

of the second standing wave for one-half of a cycle.

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