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## PowerPoint Slideshow about ' Simple Harmonic Motion' - dinah

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- defined: motion that repeats at a constant rate
- equilibrium position: forces are balanced

- 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.

- Damping: the effect of friction opposing the restoring force in oscillating systems

- Restoring force (Fr): the net force on a mass that always tends to restore the mass to its equilibrium position

- defined: periodic motion controlled by a restoring force proportional to the system displacement from its equilibrium position

- The restoring force in SHM is described by:

Fr x = -kΔx

- Δx = displacement from equilibrium position

- Table 12-1 describes relationships throughout one oscillation

- Amplitude: maximum displacement in SHM
- Cycle: one complete set of motions

- Period: the time taken to complete one cycle
- Frequency: cycles per unit of time
- 1 Hz = 1 cycle/s = s-1

- Circular motion has many similarities to SHM.
- Their motions can be synchronized and similarly described.

T = 2π

k

Reference Circle

- The period (T) for the spring-mass system can be derived using equations of circular motion:

T = 2π

k

Reference Circle

- This equation is used for Example 12-1.
- The reciprocal of T gives the frequency.

- Galileo was among the first to scientifically study pendulums.

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

- 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.

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

- forces on a pendulum at rest:
- weight (mg)
- tension in pendulum arm (Tp)
- at equilibrium when at rest

- 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

- Centripetal force adds to the tension (Tp):

Tp = Tw΄+ Fc , where:

Tw΄ = Tw = |mg|cosθ

Fc = mvt²/r

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

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

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

- 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

- 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.

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

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

- The magnitude of the force is approximately proportional to the velocity of the system:

fx = -βvx

β is a friction proportionality constant

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

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

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

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

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

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

- terminology:
- in phase
- pulses
- driven oscillations
- resonance

- A driven oscillator has three forces acting on it:
- restoring force
- damping resistance
- pulsed force applied in same direction as Fr

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

- defined: oscillations of extended bodies
- medium: the material through which a wave travels

- disturbance: an oscillation in the medium
- It is the disturbance that travels; the medium does not move very far.

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

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

- 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

- travel faster in solids than gases
- water waves have both longitudinal and transverse components—a “combination” wave

- carry information and energy from one place to another

- amplitude (A): the greatest distance a wave displaces a particle from its average position

A = ½(ypeak - ytrough)

A = ½(xmax - xmin)

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

- 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

- 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

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

- 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

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

- 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

- results from combinations of waves of several frequencies
- fundamental and harmonics
- why a trumpet sounds different than an oboe

- All three characteristics affect the way sound is perceived.

- related to the relative velocity of the observer and the sound source

- 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

- actual sound emitted by the object does not change

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

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