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Chapter 25 VIBRATIONS AND WAVES

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Chapter 25VIBRATIONS AND WAVES

VIBRATIONS AND WAVES

Things that Wiggle and Jiggle

- A FEW IMPORTANT TERMS :
- Vibration = a wiggle in a period of time
- Wave = a wigglein an amount of space and. in a period of time.
- Pendulum = A weight swinging through a . . small arc at the end of a length . of string or wire.
- Period = The time it takes a pendulum to . . make one back-and-forth swing.

25.1 The Vibration of a Pendulum

- Pendulums swing back and forth with such regularity that they have been used for a long time to control the motions of clocks.

Galileo discovered that the time a pendulum takes to swing back and forth through small angles does not depend on the mass of the pendulum or the distance through which it swings,

but ONLY on the length (l)

of the pendulum and the acceleration due to

gravity (g).

L

g

- The exact relationship for the period of a simple pendulum is :
- T = 2 π
- Where L = length of the pendulum in meters
- T = time in seconds
- and g = acceleration due to gravity (9.8 m / sec2 )
- Thus, a long pendulum has a longer period than a short pendulum

- Try an example of calculating the period of a pendulum on the board .
- A pendulum has a length of 0.75 meters, what is its period ?
- T = 2π√(L/g)
- T = 2 (3.14159) √(0.75m / 9.8 m/sec2)
- T = 6.2832 √.0765 sec2
- T = 6.2832 x .277 sec
- T = 1.74 sec

- Simple Harmonic Motion= the back-and-forth swinging motion of a pendulum (or a spring! – also called oscillatory motion)
- Sine Wave = The shape of a curve made by simple harmonic motion graphed through time.
- Springs oscillate according to Hooke’s Law . F = k Δx . F = the force stretching or compressing a spring . k = the spring constant for that spring .Δx = the distance the spring is stretched or . compressed

WAVELENGTH

AMPLITUDE

- Terms in a sine wave
- Frequency = how often a vibration occurs . In cycles per second (hertz = Hz)

CREST

TROUGH

- If the frequency of a vibrating object is known, its period can be calculated.
- and vice versa.
- frequency =
- period =
- The transmissions of radio, TV, and cell antennas as well as sound travel in waves.

1

period

1

frequency

- What is the frequency in vibrations per second of a 100-Hz wave ?
- A 100-Hz wave vibrates 100 times per second.
- The Sears Tower in Chicago sways back and forth at a frequency of about 0.1 Hz. What is its period of vibration ?
- The period is : . 1 = 1 vib = 1 vib = 10 sec. . frequency 0.1 Hz 0.1 vib/sec
- Thus, each vibration takes 10 seconds.

- When energy is transferred by a wave from a vibrating source to a distant receiver, there is no transfer of matter between the two points.
- The energy is carried by a disturbance in a medium, not by matter moving from one place to another within the medium.
- The disturbance moves through (or across) the medium (water, a string, a slinky etc.), the parts of the medium only move up and down (or closer and farther apart).

- The frequency of a vibrating source is the same as the of the wave it produces !
- DUH !
- Seismologists (people who study earth quakes) use wave speed and motion to determine the location and power of an earth quake.
- How ?

- The speed of a wave depends on the medium through which it is moving.
- Sound moves at 331.4 m/sec in air, but about 4 times faster through water at 1500 m/sec.
- The general equation for calculating wave speed is :
- wave speed = wavelength x frequency
- Or v = λf
- Notice long wavelengths have lower frequencies and short wavelengths have high frequencies.

- Low notes high notes
- Same volume (amplitude) and velocity, different frequencies
- A wavelength of 1 meter passing a post at 1 wavelength per second has a wave speed of 1 m/sec.

- If a train of freight cars, each 10 m long, rolls by at a rate of 2 cars per second, find the speed of the train.
- From chapter 2 : v = d / t = (2 x 10m)/1 sec . = 20 m/sec
- From chapter 25: v = λf = (10 m) x (2 Hz) . = 20 m/sec
- If a water wave vibrates up and down two times each second and the distance between wave crests is 1.5 m, what is the frequency of the wave? What is its speed ?
- Frequency = 2 Hz; v = λf =(1.5 m) x ( 2 Hz) = 3 m/sec
- What is the wavelength of a 340 Hz sound wave when the speed of sound in air is 340 m/sec ?
- Λλ = v / f = (340 m/sec) / (340 Hz) = 1 m

- Calculate the frequency of your favorite radio station (NO! DON’T YELL THE CALL LETTERS OR DIAL POSITION OUT LOUD IN CLASS !)
- The wave speed will be 3 x 108 m/sec (the speed of light !).
- The dial position is in kHz (103 Hz) or MHz (106 Hz) x the dial numbers.
- EXAMPLE :
- 1000 kHz waves have a wavelength = λ = v / f = (3 x 108 m/sec) / (106 Hz) = 300 m long !

- Transverse Wave = When the motion of the medium (a rope or string, water or a spring) is at right angles to the direction that the wave is traveling.
- Waves in musical instruments (guitars, violins), on the surface of liquids, radio and light waves are transverse waves.

- Longitudinal Waves = A wave in which the particles of the medium move back and forth along the direction of motion of the wave rather than at right angles to it.
- Moving the end of a spring or slinky in and out or sound waves are examples of longitudinal waves.

- Waves from different sources can be in phase or out of phase.

Reinforcement

In phase

+

=

Cancellation

Out of phase

+

=

- Waves produced by 2 vibrating objects can overlap and form Interference Patterns
- Constructive Interference or reinforcement. When the crest of one wave overlaps the crest of another, the individual effects add together increasing the amplitude.
- Destructive Interference or cancellation. When the crest of one wave overlaps the trough of another, the individual effects are reduced

- Sheets of interference waves can make what is called a Moire` Pattern

- If the incident (original) wave and its reflected wave through a medium balance each other, a standing wave can be formed.
- In a standing wave, certain points of the medium don’t move, these are callednodes.
- The positions on a standing wave with the largest amplitude occur halfway between nodes and are called antinodes.
- Standing waves are the result of interference when 2 waves of equal amplitude and wavelength pass through each other.

- What makes a standing wave occur ?

INCIDENT WAVE

IN

PHASE

NODE

REFLECTED WAVE

OUT OF

PHASE

REFLECTED WAVE

NODE

INCIDENT WAVE

INCIDENT WAVE

IN

PHASE

NODE

REFLECTED WAVE

NODE

- Is it possible for one wave to cancel another wave so that the combined amplitude is zero?
- Yes, this is destructive interference. The nodes in a standing wave in a rope are places where this cancellation occurs.
- Suppose you set up a standing wave of 3 segments. If you then vibrate with twice the frequency, how many segments will occur in the new standing wave ? How many wavelengths will there be ?
- Twice the frequency produces twice the segments or 6 for 3 complete wavelengths.

- An observer of the wave crests behind or away from the direction of motion would identify a lower frequency due to a greater travel distance from the vibration source.
- This apparent change in frequency is called theDOPPLER EFFECTafter Christian Doppler (1803 – 1853)
- The greater the speed, the greater the Doppler effect
- The police and weather reporters use the Doppler effect of radar waves to measure the speeds of cars or water in clouds.

- Vibrating in a stationary position produces waves in concentric circles. (because the wave speed is the same in all directions)
- Vibrating while moving at a speed less than the wave speed produces nonconcentric circles. The wave crests in the direction of motion occur more often (have a higher frequency) due to a shorter travel distance.

- By bouncing radar waves off a moving object like a cloud or a car and measuring the return time and frequency of the reflected waves, the speed of the cloud or car can be determined.
- Astronomers use the Doppler effect to measure the distance and speed of stars and galaxies.
- Light from a star that is moving away from Earth is shifted toward the red end of the spectrum.
- Light from a star moving toward the Earth is shifted toward the blue end of the spectrum.

- When the vibration source is moving at the same speed as the wave front, a bow wave is produced.
- In aircraft this is called the “sound barrier”. It is not a real barrier, but is where the wave crests build up on each other at the speed of sound.
- Speedboats and supersonic aircraft travel in front of their bow waves.

BOOM !

- Shock waves produced by supersonic aircraft are the 3-dimensional version of the V-shaped bow wave produced by boats.
- On the ground this is noted as a sonic boom.