Chapter 14Vibrations and Wave

Springs Hooke’s Law F = -kx

› 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

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?

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

Periodic Motion

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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T =1

f

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f =1

T

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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T = 2πl

g

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 0.840 s. For the first few seconds, the vibration approximates SHM. Find the spring constant of a single spring.

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T = 2πm

k

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.

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.

Waves

A disturbance that carries energy through matter or space.

Types of Mechanical Waves1. Transverse Waves2. Longitudinal Waves3. Surface Waves

Mechanical Waves – Waves that require a medium.

Medium – A material that a disturbance travels

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

Periodic Wave – When a wave moves at the same rate

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

Longitudinal waveOn 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.

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.

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)

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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v =Δx

Δt= fλ

Measuring a Wave

CrestCrest

Trough

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.

Wave Amplitude - Longitudinal

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.

Wave Interactions Wave Interferences

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

Wave Interferences

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

Wave Interferences

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

Wave Interferences

Reflection

Free End Fixed End

Reflection

Free End Fixed End

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

A. twice as bigB. 1/2 as bigC. Stays the same D. 1/4 times as bigE. Not enough information to decide

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

A.

B.

c.

D.

Fixed end

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

Loose end

A.

B.

c.

D.

4. What is the period of this wave?

a) t1

b) t2

c) t2-t1

d) t3-t1

e) None of the above

t1 t2 t3

Amp

time

0

t4

Fundamentals of waves

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

What is the wavelength (“”)?

A

B

CD

E none of these

Remember X axis is position not time

Fundamentals of waves

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

What is the amplitude?

Remember X axis is position not time

A

B

CD

E none of these

7. Looking at the following waveform, what is the period? assume it repeats itself over and over

time (sec)1 2

A.1 secB. 2 secC. 1 m/sD. 2 m/sE.Not enough information

8. Looking at that same wave, what is its speed?

Time (sec)1 2

A.1/2 m/sB.2 m/sC.5 m/sD.20 m/sE.Not enough information

9. The wavelength, λ, is 10 m. What is the speed of this wave?

CT 2.1.10

1Time (sec)

A) 1 m/s B) just under 7 m/sC) 10 m/s D) 15 m/sE) None of the above/not enough info/not sure

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

A. Transverse waves in a gasB. Longitudinal waves in a gasC. Transverse waves in a solidD. Longitudinal waves in a solidE. They all seem perfectly possible

11. If you moved the frequency slider to the left so that it changed from 500 to 250 the period would be

A. twice as bigB. 1/2 as bigC. Stays the same D. 1/4 times as bigE. Not enough information to

decide

Wave Reflection

Wave Refractions

What is diffraction?Diffraction occurs when an object

causes a wave to change direction and bend around it.

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

Standing Waves

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

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