Oscillations

1. Different types of motion:•Uniform motion•1D motion with constant acceleration•Projectile motion•Circular motion•Oscillations

2. Different types of oscillations:

•Periodic oscillations

•Chaotic oscillations

A motion is called periodic when the system comes back to the same physical conditions every time interval T.

•Harmonic oscillations (The simplest, the most important type of periodic oscillations)

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3. Harmonic oscillations

Relation between circular motion and simple harmonic oscillations

x

y

fT

22

xa 2

tAa

tAv

tAx

cos

sin

cos

2

r=A

Period of SHO is independent from amplitude!

linear equation

t T2

Hzssf 1/11 1 Units:

T – periodf – frequencyω – angular frequencyA – amplitude

cosAx

2

A=xmax

T

max

max

0

aa

v

xx

0

0

max

a

vv

x

0

0

max

a

vv

x

x

t

t

v

a

t

max

max

0

aa

v

xx

3

4. Hook’s law and simple harmonic motion

kxma

k

m

fT

221

Hook’s law:

Newton’s second law:

Simple harmonic motion:

maF

kxF

xa 2 xm

ka

m

k2

Question: Two identical masses hang from two identical springs. In case 1, the mass is pulled down 2 cm and released. In case 2, the mass is pulled down 4 cm and released. How do the periods of their motions compare?

A. T1 < T2 B. T1 = T2 C. T1 > T2

Period is independent of amplitude!This is in fact general to all SHM (not only for springs)! 4

Question: Mass m attached to a spring with a spring constant k. If the mass m increases by a factor of 4, the frequency of oscillation of the mass

Question: A 2.0 kg mass attached to a spring with a spring constant of 200 N/m. The angular frequency of oscillation of the mass is __ rad/s.A. 2 B. 10 C. 60 D. 100

1. is doubled

2. is multiplied by a factor of 4

3. is halved

4. is multiplied by a factor of 1/4

sradsradkg

mN

m

k/10/100

2

/200

k

m

fT

221

m

k2

5

5. The simple pendulum

sinmgFma

If 1 than sin xxL

ga 2

0cos)( tAtx

Hzf 70.02

mg

L

L

x

Example:

mL 5.01

2

4.45.0

/8.9 sm

sm

L

g

Lxga sin

g

L

fTf

L

g 2

1 ;

2 ;

sg

L

fT 4.12

1

Period and frequency are independent fromamplitude!

x

t

6

Example: A person swings on a swing. When the person sits still, the swing moves back and forth at its natural frequency. If, instead, the person stands on the swing, the new natural frequency of the swing is:

A. Greater B. The same C. Smaller

L

g

If the person stands, L becomes smaller

Example: Grandpa decides to move to the Moon, and he naturally takes his old pendulum clock with him. But gravity on the Moon is approximately g/6... As a result, his clock is:

A. Too fast B. Too slowC. Too fast until noon, too slow after noon

The period of the pendulum is longer on the Moon. So each “second according to this clock” is then longer than a real second.

To tune the clock you can move the clock’s disk up.

g

LT

22

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6. Damped Harmonic Motion

x(t)

t

7. Resonance

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1. There are many examples of waves:• Seismic waves• Ripples on a pond• Sound• Electromagnetic wave including light

Waves

• Periodic waves

2. Types of Waves

Longitudinal wave: the oscillations are along the direction the wave travels.

Transverse wave: the oscillations are transverse (perpendicular) to the direction the wave travels.

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• Wave pulses (waves that have short duration – just a quick pulse)

• Special type of periodic waves – sinusoidal waves

3. Sinusoidal waves

t

y T

)sin(),( kxtAtxy

fT 2/2

x

y λA

Amplitude: APeriod: T Wave length:

Tf /1Frequency:

Wave number:

/2k

Angular frequency:

Sinusoidal waves are periodic

a) A periodic wave is periodic in time at fixed position as the wave passes that position. The period of the wave is T. Each particle in the medium oscillates with the same period and in the same way as the wave passes the particle.

b) A periodic wave is also periodic in space at fixed time. The period of the wave in space is called the wavelength .

Wave at fixed x: Wave at fixed t:

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4. Wavelength, period and speed of waves

• The “speed” of a wave is the rate of movement of the disturbance. • It is not the speed of the individual particles!• The speed is determined by the properties of the medium.

λ - wave lengthT - period

kfTv //

Question 1: If you double the wavelength of a periodic wave and the wave speed does not change, what happens to the wave frequency f ?1) f is unchanged 2) f is halved 3) f is doubled

Question 3: A sinusoidal wave has a wavelength of 4.00 m and an angular frequency of 3.14 rad/s. What is the speed of the wave?

smsmfv 00.2214.300.42 1

Question 2: A sinusoidal wave has a wavelength of 4.00 m and a period of 2.00s. What is the speed of the wave?

smsmTv 00.200.200.4/

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5. Interference – combination of waves (an interaction of two or more waves arriving at the same place)

Important: principle of superposition

a) Constructive interference: the interfering waves add up so that they reinforce each other (the total wave is larger)

b) Destructive interference: the interfering waves add up so that they cancel each other (the total wave is smaller or even zero)

Valley

Valley Peak

(a)(b)

Waves source

(a)No shift or shift by

(b)

Shift by

,...2,1,012

m

mrr 21

12 mrr

),(),(),( 21 trtrtr

12

Example: Two speakers S1 and S2 are driven by the same signal generator and are different distances from a microphone P as shown.

The minimum frequency for constructive interference to occur at point P is __ Hz. (The speed of sound is v = 340 m/s.)

,...2,1,012

n

nrr Hzmm

sm

rr

vfvf 200

70.140.3

/340/

12min

A. 100 B. 200 C. 400 D. 800

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The waves spread out from the opening!

6. Diffraction

What it is? The bending of waves behind obstacles or apertures into the ”shadow region”, that can be considered as interference of many waves.

Haw to observe?Diffraction is most pronounced when the wavelength of the wave is similar to the size of the obstacle or aperture. For example, the diffraction of sound waves is commonly observed because the wavelength of sound is similar to the size of doors.

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Sound

Acoustic waves in the range of frequencies: 20Hz - 20,000Hz

1. Sound waves:• can travel in any solid, liquid or gas• in liquids and gases sound waves are longitudinal ONLY!• longitudinal and transversal sound waves could propagate in in solids

Sound in air is a longitudinal wave that contains regions of low and high pressure

Vibrating tuning fork

These pressure variations are usually small – a “loud” sound changes the pressure by 2.0x10-5 atm

Pressure sensor

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Speed of Sound in Some Common Substances

1. Air (20 oC) 344

Substance Speed (m/s)

3. Water (0 oC) 1,402

5. Human tissue 1,540

6. Aluminum 6,420

7. Iron and steel 5,941

4. Lead 1,200

2. Helium (20 oC) 999

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2. Perception of Sound

3) Tone quality - how we distinguish sounds of the same pitch and loudness (how we perceive the qualities of the waveform)

4) Duration

1) Pitch - how “high” or “low” we perceive sound

(it is directly related to how we perceive frequency)Combinations of notes that are “pleasing” to the ear have frequencies that are related by a simple whole-number ratio (Pythagoras)

2) Loudness - how we perceive the amplitude of the sound wave

Loudness manly depends on amplitude and frequency

We use four subjective characteristics to describe how we perceive sound:pitch, loudness, tone quality and duration

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3. Doppler Effect

What do you hear when a sound-emitting object (train, car) passes you?The sound changes in pitch (frequency) as the object goes.Why does this happen?

Because the speed of a wave in a medium is constant, change in will affect change in f.

S

LSL vv

vvff

vL or vS is “+” if in the same direction as from listener to source and is “-” otherwise

•As the object travels towards you, the distance between wavefronts is compressed; this makes it seem like is smaller and frequency is higher.

•As the object travels away from you, the distance between wavefronts is extended; this makes it seem like is larger and frequency is lower.

fv

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Velocity of listener (L): vL

Velocity of source (S): vS

Velocity of sound: v

4. Slightly mismatched frequencies cause audible “beats”

12 fffbeats

A) Increase f1; B) Increase f2; C) Decrease f1;D) Decrease f2; E) There is not enough information to choose

Question: The beat frequency between tones with frequencies f1 and f2 is 2.0 Hz. In order to increase the beat frequency, one must __.

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