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Chapter 14 Vibrations and Wave

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**Springs Hooke’s Law F = -kx Potential Energy in a Spring PEsp=1/2 kx2**

The force exerted by a spring is equal to the spring constant times the distance the spring is compressed or stretched from its equilibrium Potential Energy in a Spring PEsp=1/2 kx2

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**Spring Practice Problem**

A spring stretches by 18 cm when a bag of potatoes weighing 56 N is suspended from its end. Determine the spring constant How much elastic potential energy is stored in the spring when it is stretched this far?

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Periodic Motion Equilibrium – When the forces on an object are balanced or equal zero and the acceleration is zero. Periodic Motion – Motion that repeats in a regular cycle Simple harmonic motion – when the force on an object is directly proportional to the displacement of the object

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Periodic Motion

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**Periodic Motion Period – (T) time for one complete cycle**

Amplitude – maximum distance that the object moves from equilibrium (measured in radians or meters) Frequency – (f) number of cycles or vibrations per unit of time (measured in hertz, Hz = s-1)

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**Pendulums Period of a Pendulum**

Example: A pendulum with a length of 36.9 cm has a period of 1.22 s. What is the acceleration due to gravity at the pendulum’s location

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**Springs Period of Spring**

Example: The body of a 1275 kg car is supported on a frame by four springs. Two people riding in the car have a combined mass of 153 kg. When driven over a pothole, the frame vibrates with a period of s. For the first few seconds, the vibration approximates SHM. Find the spring constant of a single spring. K=2.00 X 104 N/m

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**Resonance Swinging – How do you make yourself go higher?**

Occurs when small forces are applied at regular intervals to a vibrating or oscillating object and the amplitude of the vibration increases.

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Resonance Many objects have a natural frequency – vibrates in a regular pattern. Resonance occurs when whenever a sound wave has the same frequency as the natural frequency of an object. The sound will cause the object with the same natural frequency to vibrate.

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**Waves A disturbance that carries energy through matter or space.**

Types of Mechanical Waves Transverse Waves Longitudinal Waves Surface Waves Mechanical Waves – Waves that require a medium. Medium – A material that a disturbance travels

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**Wave Pulse – a single bump or disturbance that travels through a medium**

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**Periodic Wave – When a wave moves at the same rate**

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**Longitudinal Waves (think slinky)**

Longitudinal Waves – disturbance is in the same direction or parallel to, the direction of the wave’s motion. Transverse Waves (think rope) Transverse Wave – one that vibrates perpendicular to the direction of the wave’s motion

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Longitudinal wave On a longitudinal wave the area squeezed together is called the compression. The areas spread out are called the rarefaction. The wavelength is the distance from the center of one compression to the center of the next compression.

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**Surface Waves (think water)**

Surface Waves – Lake or ocean; Longitudinal at the surface, the particles move in a direction that is both parallel and perpendicular to the direction of wave motion.

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Measuring a Wave Wavelength (l) – shortest distance that the wave pattern repeats OR distance from peak to peak or trough to trough Phase – Same displacement and same velocity A crest and trough are exactly 180o out of phase. Period – time for one wavelength (T) Frequency - # of cycles per unit time (Hz)

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**Measuring a Wave Speed – displacement of wave peak over time.**

Amplitude – the distance of the wave peak/trough to equilibrium Crest – High Point of the wave Trough – Low Point of the wave

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Measuring a Wave Crest Crest Trough

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Wave Amplitude The amplitude of a wave is directly related to the energy of a wave. The amplitude of a longitudinal wave is determined by the closeness of the longitudinal waves. The closer the longitudinal waves and the farther the rarefaction lines.

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**Wave Amplitude - Longitudinal**

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Wave Problem The piano string tuned to middle C vibrates with a frequency of 264 Hz. Assuming the speed of sound in air is 343 m/s, find the wavelength of the sound waves produced by the string.

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**Wave Interactions Wave Interferences**

Superposition - when two or more waves come together, the result is the sum of the individual waves.

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Wave Interferences Constructive Interference – interference in which individual displacements on the same side of the equilibrium position are added together to form resultant wave

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Wave Interferences Destructive Interference – interference in which individual displacements on opposite sides of the equilibrium position are added together to form the resultant wave

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Wave Interferences

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Reflection Free End Fixed End

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Reflection Free End Fixed End

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**1. If you moved the slider to the far right, doubling the amplitude, the period would be…**

twice as big 1/2 as big Stays the same 1/4 times as big Not enough information to decide C they are independent From Pollock:

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**2. What will this wave look like after it reflects?**

B. c. D. Fixed end C

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**3. What will this wave look like after it reflects?**

B. c. D. Loose end B

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**Amp time t1 t2 t3 t4 4. What is the period of this wave? t1 t2 t2-t1**

None of the above Amp time t1 t2 t3 t4 D Adapted From Pollock at CU 1240 course His notes follow: Chris said that from here on out, participation was dwindling, I may have overdosed on CT’s today? But, this one got 85% correct, at least!

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**What is the wavelength (“”)?**

5.The picture shows “displacement as a function of location along a string” What is the wavelength (“”)? A B C D E none of these A Adapted from From Pollock at CU 1240 course Remember X axis is position not time Fundamentals of waves

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**6.The picture shows “displacement as a function of location along a string”**

What is the amplitude? A B C D E none of these C Adapted from From Pollock at CU 1240 course Remember X axis is position not time Fundamentals of waves

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**Not enough information**

7. Looking at the following waveform, what is the period? assume it repeats itself over and over 1 2 time (sec) 1 sec 2 sec 1 m/s 2 m/s Not enough information B Adapted From Pollock at CU 1240 course

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**Not enough information**

8. Looking at that same wave, what is its speed? 1 2 Time (sec) 1/2 m/s 2 m/s 5 m/s 20 m/s Not enough information E. Adapted From Pollock at CU 1240 course His notes follow: Obviously important that “not enough info” occasionally be the RIGHT answer! (80% correct) Now given that λ=10 m what is the speed of the wave?

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**E) None of the above/not enough info/not sure**

CT 9. The wavelength, λ, is 10 m. What is the speed of this wave? Time (sec) 1 1 m/s just under 7 m/s C) 10 m/s D) 15 m/s E) None of the above/not enough info/not sure D 10/(2/3) Adapted From Pollock at CU 1240 course His notes follow: Propaganda about talking to neighbors, kind of needed/helpful on a day like today with so many CT’s. Answer d only got 49% of vote! So, some “hints” might be appropriate here…

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**10. Which one of the following is most likely to be impossible?**

A. Transverse waves in a gas B. Longitudinal waves in a gas C. Transverse waves in a solid D. Longitudinal waves in a solid E. They all seem perfectly possible E From Pollock at CU 1240 course His notes follow: Last year Everyone got this wrong. It’s a good question, and serves to stimulate really good conversation /discussion. (E was 45%, but C and D both popular) A good one to talk about “why not”. This year we did a little better (30% said A, many of the rest said E) Ran out of time, but ended here.

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**Not enough information to decide**

11. If you moved the frequency slider to the left so that it changed from 500 to 250 the period would be twice as big 1/2 as big Stays the same 1/4 times as big Not enough information to decide A Adapted From Pollock at CU 1240 course

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Wave Reflection

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Wave Refractions

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What is diffraction? Diffraction occurs when an object causes a wave to change direction and bend around it.

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**Diffraction also occurs when passing through a small opening**

Diffraction also occurs when passing through a small opening. They diffract and spread out as they pass through the hole.

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Standing Waves A wave pattern that results when two waves of the same frequency, wavelength, and amplitude travel in opposite directions and interfere

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Standing Waves Node – a point in a standing wave that always undergoes complete destructive interference and therefore is stationary Antinode – a point in a standing wave, halfway between two nodes, at which the largest amplitude occurs

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