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**Simple Harmonic Motion**

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**Periodic Motion defined: motion that repeats at a constant rate**

equilibrium position: forces are balanced These ?’s are totally beyond the realm of science. Are they important? How do we answer them?

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Periodic Motion For the spring example, the mass is pulled down to y = -A and then released. Two forces are working on the mass: gravity (weight) and the spring. These ?’s are totally beyond the realm of science. Are they important? How do we answer them?

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**Periodic Motion for the spring: ΣF = Fw + Fs ΣFy = mgy + (-kΔy)**

These ?’s are totally beyond the realm of science. Are they important? How do we answer them?

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Periodic Motion Damping: the effect of friction opposing the restoring force in oscillating systems These ?’s are totally beyond the realm of science. Are they important? How do we answer them?

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Periodic Motion Restoring force (Fr): the net force on a mass that always tends to restore the mass to its equilibrium position These ?’s are totally beyond the realm of science. Are they important? How do we answer them?

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**Simple Harmonic Motion**

defined: periodic motion controlled by a restoring force proportional to the system displacement from its equilibrium position We must realize that we do this—we pre-suppose. There must be some starting point.

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**Simple Harmonic Motion**

The restoring force in SHM is described by: Fr x = -kΔx We must realize that we do this—we pre-suppose. There must be some starting point. Δx = displacement from equilibrium position

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**Simple Harmonic Motion**

Table 12-1 describes relationships throughout one oscillation We must realize that we do this—we pre-suppose. There must be some starting point.

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**Simple Harmonic Motion**

Amplitude: maximum displacement in SHM Cycle: one complete set of motions We must realize that we do this—we pre-suppose. There must be some starting point.

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**Simple Harmonic Motion**

Period: the time taken to complete one cycle Frequency: cycles per unit of time 1 Hz = 1 cycle/s = s-1 We must realize that we do this—we pre-suppose. There must be some starting point.

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**Simple Harmonic Motion**

Frequency (f) and period (T) are reciprocal quantities. We must realize that we do this—we pre-suppose. There must be some starting point. f = T 1 T = f 1

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**Reference Circle Circular motion has many similarities to SHM.**

Their motions can be synchronized and similarly described. These ?’s are totally beyond the realm of science. Are they important? How do we answer them?

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Reference Circle The period (T) for the spring-mass system can be derived using equations of circular motion: These ?’s are totally beyond the realm of science. Are they important? How do we answer them? T = 2π k m

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**Reference Circle This equation is used for Example 12-1.**

The reciprocal of T gives the frequency. These ?’s are totally beyond the realm of science. Are they important? How do we answer them? T = 2π k m

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**Periodic Motion and the Pendulum**

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Overview Galileo was among the first to scientifically study pendulums.

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Overview The periods of both pendulums and spring-mass systems in SHM are independent of the amplitudes of their initial displacements.

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Pendulum Motion An ideal pendulum has a mass suspended from an ideal spring or massless rod called the pendulum arm. The mass is said to reside at a single point.

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Pendulum Motion l = distance from the pendulum’s pivot point and its center of mass center of mass travels in a circular arc with radius l.

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**Pendulum Motion forces on a pendulum at rest: weight (mg)**

tension in pendulum arm (Tp) at equilibrium when at rest

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**Restoring Force Fr = Tp + mg**

When the pendulum is not at its equilibrium position, the sum of the weight and tension force vectors moves it back toward the equilibrium position. Fr = Tp + mg

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**Restoring Force Tp = Tw΄+ Fc , where: Tw΄ = Tw = |mg|cos θ Fc = mvt²/r**

Centripetal force adds to the tension (Tp): Tp = Tw΄+ Fc , where: Tw΄ = Tw = |mg|cos θ Fc = mvt²/r

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Restoring Force Total acceleration (atotal) is the sum of the tangential acceleration vector (at) and the centripetal acceleration. The restoring forces causes this atotal.

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Restoring Force A pendulum’s motion does not exactly conform to SHM, especially when the amplitude is large (larger than π/8 radians, or 22.5°).

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Small Amplitude defined as a displacement angle of less than π/8 radians from vertical SHM is approximated

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Small Amplitude For small initial displacement angles: T = 2π |g| l

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**Small Amplitude Longer pendulum arms produce longer periods of swing.**

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Small Amplitude The mass of the pendulum does not affect the period of the swing. T = 2π |g| l

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Small Amplitude This formula can even be used to approximate g (see Example 12-2). T = 2π |g| l

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Physical Pendulums mass is distributed to some extent along the length of the pendulum arm can be an object swinging from a pivot common in real-world motion

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Physical Pendulums The moment of inertia of an object quantifies the distribution of its mass around its rotational center. Abbreviation: I A table is found in Appendix F of your book.

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Physical Pendulums period of a physical pendulum: T = 2π |mg|l I

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**Oscillations in the Real World**

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Damped Oscillations Resistance within a spring and the drag of air on the mass will slow the motion of the oscillating mass.

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Damped Oscillations Damped harmonic oscillators experience forces that slow and eventually stop their oscillations.

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**β is a friction proportionality constant**

Damped Oscillations The magnitude of the force is approximately proportional to the velocity of the system: fx = -βvx β is a friction proportionality constant

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Damped Oscillations The amplitude of a damped oscillator gradually diminishes until motion stops.

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Damped Oscillations An overdamped oscillator moves back to the equilibrium position and no further.

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Damped Oscillations A critically damped oscillator barely overshoots the equilibrium position before it comes to a stop.

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Driven Oscillations To most efficiently continue, or drive, an oscillation, force should be added at the maximum displacement from the equilibrium position.

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Driven Oscillations The frequency at which the force is most effective in increasing the amplitude is called the natural oscillation frequency (f0).

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Driven Oscillations The natural oscillation frequency (f0) is the characteristic frequency at which an object vibrates. also called the resonant frequency

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**Driven Oscillations terminology: in phase pulses driven oscillations**

resonance

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**Driven Oscillations A driven oscillator has three forces acting on it:**

restoring force damping resistance pulsed force applied in same direction as Fr

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Driven Oscillations The Tacoma Narrows Bridge demonstrated the catastrophic potential of uncontrolled oscillation in 1940.

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Waves

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**Waves defined: oscillations of extended bodies**

medium: the material through which a wave travels

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**Waves disturbance: an oscillation in the medium**

It is the disturbance that travels; the medium does not move very far.

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Graphs of Waves Waveform graphs Vibration graphs

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Types of Waves longitudinal wave: disturbance that displaces the medium along its line of travel example: spring

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Types of Waves transverse wave: disturbance that displaces the medium perpendicular to its line of travel example: the wave along a snapped string

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**Longitudinal Waves Any physical medium can carry a longitudinal wave.**

Rarefaction zone: molecules are spread apart and have lower density and pressure Compression zone: molecules are pushed together and have higher density and pressure

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**Longitudinal Waves travel faster in solids than gases**

water waves have both longitudinal and transverse components—a “combination” wave

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Periodic Waves carry information and energy from one place to another

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Periodic Waves amplitude (A): the greatest distance a wave displaces a particle from its average position A = ½(ypeak - ytrough) A = ½(xmax - xmin)

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Periodic Waves wavelength (λ): the distance from one peak (or compression zone) to the next, or from one trough (or rarefaction zone) to the next

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Periodic Waves A wave completes one cycle as it moves through one wavelength. A wave’s frequency (f) is the number of cycles completed per unit of time

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**Periodic Waves wave speed (v): the speed of the disturbance**

for periodic waves: λf = v

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Sound Waves longitudinal pressure waves that come from a vibrating body and are detected by the ears cannot travel through a vacuum; must pass through a physical medium

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Sound Waves travel faster through solids than liquids, and faster through liquids than gases have three characteristics:

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Loudness the interpretation your hearing gives to the intensity of the wave intensity (Is): amount of power transported by the wave per unit area measured in W/m²

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Loudness a sound must be ten times as intense to be perceived as twice as loud sound is measured in decibels (dB)

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**Pitch related to the frequency**

high frequency is interpreted as a high pitch low frequency is interpreted as a low pitch 20 Hz to 20,000 Hz

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**Quality results from combinations of waves of several frequencies**

fundamental and harmonics why a trumpet sounds different than an oboe

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Sound Waves All three characteristics affect the way sound is perceived.

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Doppler Effect related to the relative velocity of the observer and the sound source actual sound emitted by the object does not change an approaching object has a higher pitch than if there were no relative velocity an object moving away has a lower pitch than if there were no relative velocity

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Mach Speed measurement is dependent on the composition and density of the atmosphere speed of sound changes with altitude

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