Vibrations and WavesHonors Physics

Biblical ReferenceThen we will no longer be infants, tossed back and forth by the waves.

Ephesians 4:14

Simple Harmonic Motion Back and forth motion

that is caused by a force that is directly proportional to the displacement.

The displacement centers around an equilibrium position.

xFs

Springs – Hooke’s Law

One of the simplest type of simple harmonic motion is called Hooke's Law.

This is primarily in reference to springs.

kxorkxF

k

k

xF

s

s

N/m):nitConstant(U Spring

alityProportion ofConstant

Springs – Hooke’s Law

The negative sign tells us that “F” is a restoring force; it works in the opposite direction of the displacement.

Common formulas which are set equal to Hooke's law are Newton’s Second Law and weight.

Hooke’s Law

Example

kxFs

kxmg

mgFg sg FF

m

n

msm

kg

x

mgk 15

36.0

)81.9)(55.0( 2

A 0.55 kg mass is attached to a vertical spring, which stretches 36 cm from it’s original equilibrium position. What is the spring constant?

Example

kxF

m

N

m

N

x

Fk 1000

05.0

)50(

Nmm

NkxF 110)11.0)(1000(

A load of 50 N attached to a spring hanging vertically stretches the spring 5.0 cm. The spring is now placed horizontally on a table and stretched 11.0 cm. What force is required to stretch the spring this amount?

Conservation of Energy in Springs

Springs are like Waves and CirclesThe amplitude, A, of a wave is the same as the displacement ,x, of a spring. Both are in meters.

Crest

Trough

Equilibrium Line

PeriodCREST

Trough

Equilibrium Line

• Period (T): the time for one revolution or one complete oscillation (one crest and trough).

• Oscillations could also be called vibrations and cycles.

• Ts = sec/cycle

• In the wave above we have 1.75 cycles or waves (vibrations or oscillations).

• Assume that the wave crosses the equilibrium line in one second intervals.

• T = 3.5 seconds/1.75 cycles. T = 2 sec.

Frequency The Frequency of a wave is the inverse of Period. That means that the frequency is cycles/sec. The commonly used unit is Hertz (HZ).

Tf

fT

Hzsccyc

fFrequency

scyc

sTPeriod

11

5.05.0sec5.3

75.1

seconds

cycles

275.1

5.3

cycles

seconds

The period of a Spring-Mass System is:◦ Proportional to 2

◦ Inversely proportional to the square root of the spring constant

◦Proportional to the square root of the mass on the spring

Measuring SHM for a Spring

k

mT 2

• The greater the mass, the larger the period

• The greater the spring constant (more stiff), the smaller the period

A 125 N object vibrates with a period of 3.56 seconds when hanging from a spring. Find the spring constant.

Example

k

mT 2 2

24

T

mk

m

N

ssmN

k 7.39)56.3(

)81.9

125(4

2

2

2

Remember the Pendulum…

The period of a pendulum is:◦ Proportional to 2 (it’s sweeping out an arc of a circle)

◦ Inversely proportional to the square root of gravity

◦Proportional to the square root of the length of the pendulum

Measuring SHM for a Pendulum

g

LT 2

The height of a tower is unknown, but a pendulum, extending from the ceiling almost touches the floor. If the period of the pendulum is 12 s, what is the approximate height of the tower?

Example

g

LT 2

2

2

4gT

L

msm

sL 36

4

)81.9()12(

2

22

What is a wave

A Wave is a vibration or disturbance in space.

A Medium is the substance that all sound waves travel through and need to have in order to move.

Longitudinal Waves

Longitudinal Wave - A fixed point will move parallel with the wave motion

2 areas:

Compression - an area of high molecular density and pressure

Rarefaction - an area of low molecular density and pressure

Transverse Waves

Transverse Wave - A fixed point will move perpendicular with the wave motion.

Wave parts: Crest, Trough, Wavelength, Amplitude, Frequency, Period

All waves have 4 basic properties:

Amplitude

Wavelength λ lambda

Frequency f

Speed c

Amplitude – the maximum distance the wave moves up and down.

The more energy a wave has the greater the amplitude.

Wavelength – the distance between two corresponding parts of a wave

Short Waves can complete more cycles than Long Waves in the same amount of time.

Frequency – the number of complete waves that pass a given point

Frequency is measured in the unit called Hertz (Hz).

A wave that occurs every second has a frequency of 1 Hz.

Speed – the distance a wave travels in a given amount of time.

The speed of sound through air is 331 m/s.

Wave Speed

You can find the speed of a wave by multiplying the wave’s wavelength in meters by the frequency (cycles per second).

Since a “cycle” is not a standard unit this gives you m/s.

ExampleA harmonic wave is traveling along a rope. It is observed that the oscillator that generates the wave completes 40.0 vibrations in 30.0 s. Also, a given maximum travels 425 cm along a rope in 10.0 s . What is the wavelength?

fv

mHz

sm

f

v319.0

34

425.0

s

m

s

mv 425.0

0.10

25.4Hz

s

cyclesf

3

4

0.30

0.40

Superposition - The combination of two overlapping waves

Wave Behavior

Interference - The result of superposition

Standing Waves

Two main parts of standing waves:

Antinodes – Areas of maximum amplitude

Nodes – Areas of zero amplitude

A standing wave is produced when a wave that is traveling is reflected back upon itself.

Interference is the interaction between waves that meet

There are two types of interference: Constructive and Destructive

When an object hits a surface it can not pass, it bounces back.

This is called reflection.

The bending of waves due to a change in speed is called refraction.

When a wave moves around a barrier or through an opening in a barrier, it bends and spreads out. This is

known as diffraction.