radoslaw bednarek, phd cytobiology and proteomics unit...

Radoslaw Bednarek, PhD

Cytobiology and Proteomics Unit

Biophysics seminars, 1 year 6MD, winter semester

Vibrations and Waves

1


• Simple Harmonic Motion (SHM)

• Energy in SHM

• The Period and Sinusoidal Nature of SHM

• The Simple Pendulum

• Waves Description

• Types of Waves

• Transverse Waves

• Longitudinal Waves

• Reflection and Transmission of Waves

• Wave Interference

• Standing Waves

• Refraction

• Diffraction

• Doppler Effect

• Energy Transported by Waves 2

Simple Harmonic Motion

© 2016 Pearson Education, Ltd.

If an object vibrates or

oscillates back and forth

over the same path, each

cycle taking the same

amount of time, the motion is

called periodic.

The mass and spring system

is a useful model for a

periodic system.

3

We assume that the surface is frictionless.

There is a point where the spring is neither

stretched nor compressed; this is the equilibrium

position.

We measure displacement from that point (x = 0 on

the previous figure).

The force exerted by the spring depends on the

displacement (often referred to as Hooke’s law):



4

Simple Harmonic Motion The minus sign on the force indicates that a restoring

force is in the direction opposite to

the displacement x.

External force on spring (a force exerted to stretch the

spring) has a plus sign: F = +kx

k is the spring constant, characterizes the stiffness of

the spring (the greater value – the greater force

needed to stretch the spring).

The force is not constant (varies with position), so the

acceleration is not constant either.


5

Simple Harmonic Motion Displacement (x) is measured from the

equilibrium point.

Amplitude (A) is the maximum

displacement (the greatest distance from

the equilibrium point).

A cycle is a full to-and-fro motion; this

figure shows half a cycle.

Period (T) is the time required to complete

one cycle.

Frequency (f) is the number of cycles

completed per second. © 2016 Pearson Education, Ltd.

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If the spring is hung vertically,

the only change is in the

equilibrium position, which is at

the point where the spring force

equals the gravitational force.

(a): Free spring, hung vertically.

(b): Mass m attached to spring

in new equilibrium position,

which occurs when:

ΣF = 0 = mg – kx0

so the spring stretches an extra

amount x0 = mg/k to be in

equilibrium.


7

Any vibrating system where the restoring

force is proportional to the negative of the

displacement (F = -kx) is in simple

harmonic motion (SHM), and is often called

a simple harmonic oscillator (SHO).



8

Energy in SHM

The potential energy of a spring (elastic potential

energy) is given by:

PE = ½ kx2

The total mechanical energy E of a mass-spring

system is the sum of the kinetic and potential

energies:

The total mechanical energy will be conserved, as

we are assuming the system is frictionless.


9

If the mass is at the limits of its

motion, the energy is all

potential.

If the mass is at the equilibrium

point, the energy is all kinetic.

We know what the potential

energy is at the turning points:


Energy in SHM

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The total energy is: ½ kA2

And we can write:

At the equilibrium point x = 0 and all the energy is kinetic. At intermediate

points energy is part kinetic and part potential because energy is conserved.

From this equation we can obtain the velocity as a function of position:

v2 = k/m (A2 – x2) = k/mA2( 1 – x2/A2).

From equations for kinetic and potential energy we have: ½ mv2max= ½ kA2

so: v2max = (k/m)A2.

Inserting this into equation above and taking the square root we have:

where:


Energy in SHM

11

If we look at the projection onto

the x axis of an object moving in

a circle of radius A at a constant

speed vmax, we find that the x

component of its velocity varies

as:

This is identical to SHM.

The Period and Sinusoidal

Nature of SHM


12

We can now determine the period of SHM because it is equal to that of

the revolving object making one complete revolution.

Therefore, we can use the period and frequency of a particle moving in

a circle to find the period and frequency.

The velocity vmax is equal to the circumference of the circle (distance)

divided by the period T:

vmax = 2πA/T = 2πAf

So period T:

T = 2πA/vmax

From energy conservation:

1/2mv2max=1/2kA2

so: A/Vmax = √m/k



Nature of SHM

13

14


Nature of SHM

Thus the period depends on the mass m and the spring

stiffness constant k, but not on the amplitude A:

We can write frequency f as the inversion of period:

We can similarly find the position as a function of time.

From the figure on slide number 12 we see that cos θ = x/A,

so the projection of the object’s position on the axis x is x = A cos θ.

The mass is rotating with angular velocity ω, we can write θ = ωt, so:

Angular velocity (specified in radians per second) can be written as ω = 2πf,

we can write :

or in terms of the period T = 1/f:



Nature of SHM

15


The top curve is a

graph of the previous

equation.

The bottom curve is

the same, but shifted

¼ period so that it is a

sine function rather

than a cosine.


Nature of SHM

16

The Simple Pendulum

A simple pendulum consists of a mass at the

end (the pendulum bob) of a lightweight cord.

We assume that the cord does not stretch,

and that its mass is negligible.

The motion of a simple pendulum resembles

SHM.

Is it really undergoing SHM? Is the restoring

force proportional to displacement?


17

The displacement of the pendulum along

the arc is given by x = lθ.

θ – the angle the cord makes with the

vertical, l – lenght of the cord.

In order to be in SHM, the restoring force

must be proportional to the negative of

the displacement.

If the restoring force is proportional to x

or to θ, the motion is simple harmonic.

Here we have that the restoring force is

the net force on the bob F = -mg sin θ

which is proportional to sin θ but not to θ

itself – the motion is not SHM.

The Simple Pendulum


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19

The Simple Pendulum

However, if the angle is small, sin θ ≈ θ.

Therefore, for small angles, the force is approximately

proportional to the angular displacement.

F = -mg sin θ ≈ -mg θ

substituting x = l × θ or θ = x/l we have:

F ≈ - mg/l x (this fits Hooke’s Law: F = -kx)

The effective force constant:

k = mg/l

The Simple Pendulum

The period and frequency are:


20

The Simple Pendulum

So, as long as the cord

can be considered

massless and the

amplitude is small, the

period does not depend

on the mass.


21

A 1-meter-long pendulum has a bob with a mass

of 1 kg. Suppose that the bob is now replaced with

a different bob of mass 2 kg, how will the period of

the pendulum change?

A.It will double.

B.It will halve.

C.It will remain the same.

D.There is not enough information.


The Simple Pendulum

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A 1-meter-long pendulum has a bob with a mass of 1 kg.

Suppose that the bob is now tied to a different string so that

the length of the pendulum is now 2 m. How will the period

of the pendulum change?

A.It will increase.

B.It will decrease.

C.It will remain the same.

D.There is not enough information.


The Simple Pendulum

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• Vibration – Wiggle in time

• Wave – Wiggle in space and

time


24

Wave Motion

A wave travels

along its medium,

but the individual

particles just move

up and down.


25

Wave Description

• Vibration and wave characteristics

– Crests

• high points of the wave

– Troughs

• low points of the wave


26

Wave characteristics:

• Amplitude, A: distance from the midpoint (equilibrium level) to the crest or to the trough

• Wavelength, λ: distance from the top of one crest to the top of the next crest, or distance between any successive identical parts of the wave

• Frequency f and period T

• Wave velocity


Wave Motion

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Wave Description

• Frequency

– Number of crests (or complete cycles)

passing any point per unit time

– Example:


2 vibrations occurring in 1 second is a

frequency of 2 vibrations per second.

28

Wave Description • Period

– Time to complete one vibration (or elapsed

between two successive crests passing by

the same point in space)

or, vice versa,

• Example: Pendulum makes 2 vibrations in 1

second. Frequency is 2 Hz. Period of

vibration is 1/2 second.

frequency

1 Period

period

1 Frequency


29

• Wave velocity

– Describes how fast a disturbance (eg. crest) moves

through a medium

– Related to frequency and wavelength of a wave

Wave velocity = wavelength x frequency

• Example:

– A wave with wavelength 1 meter and frequency of

1 Hz has a velocity of 1 m/s.


Wave Description

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Wave Description

A sound wave has a frequency of 500 Hz.

What is the period of vibration of the air

molecules due to the sound wave?

A.1 s

B.0.01 s

C.0.002 s

D.0.005 s


31

Wave Description

If the frequency of a particular wave is 20

Hz, its period is

A.1/20 second.

B.20 seconds.

C.more than 20 seconds.

D.None of the above.


32

A wave with wavelength 10 meters and time

between crests of 0.5 second is traveling in water.

What is the wave speed?

A.0.1 m/s

B.2 m/s

C.5 m/s

D.20 m/s


Wave Description

33

Types of Waves


The motion of particles in a wave can be:

• perpendicular to the wave direction (transverse);

• parallel to the wave direction (longitudinal). 34

Earthquakes produce both longitudinal and transverse waves.

Both types can travel through solid material, but only longitudinal waves can propagate through a fluid — in the transverse direction, a fluid has no restoring force.

Surface waves are waves that travel along the boundary between two media.


Types of Waves

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Transverse Waves

• Transverse wave

– Medium vibrates perpendicularly to direction of

energy transfer

– Side-to-side movement

– Examples:

• Vibrations in stretched strings of musical

instruments

• Radio waves

• Light waves

• S-waves that travel in the ground (providing

geologic information)


36

Transverse Waves

The distance between adjacent peaks in the direction of

travel for a transverse wave is its

A.frequency.

B.period.

C.wavelength.

D.amplitude.


37

Transverse Waves

The vibrations along a transverse wave move in a direction

A.along the wave.

B.perpendicular to the wave.

C.Both A and B.

D.Neither A nor B.


38

Longitudinal Waves

• Longitudinal wave

– Medium vibrates parallel to direction of

energy transfer

– Backward and forward movement consists of

• compressions (wave compressed)

• rarefactions (stretched region between

compressions)

• Examples: sound waves in solid, liquid, gas


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• Longitudinal wave

– Example:

• sound waves in solid, liquid, gas

• P-waves that travel in the ground (providing

geologic information)

Longitudinal Waves


40

Sound waves are longitudinal waves:


Types of Waves

41

Longitudinal Waves

The wavelength of a longitudinal wave is the distance

between

A.successive compressions.

B.successive rarefactions.

C.Both A and B.

D.None of the above.


42

Reflection and Transmission of Waves


A wave hitting an obstacle

will be reflected (a), and its

reflection will be inverted.

A wave reaching the end

of its medium, but where

the medium is still free to

move, will be reflected (b),

and its reflection will be

upright.

43


A wave encountering a

denser medium will be

partly reflected and

partly transmitted.

If the wave speed is less

in the denser medium,

the wavelength will be

shorter (frequency does

not change).


44


Two- or three-dimensional waves can be represented by

wave fronts, which are curves of surfaces where all the

waves have the same phase – wave crest.

Lines perpendicular to the

wave fronts are called rays;

they point in the direction

of propagation of the wave.


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The law of reflection: the angle of incidence

equals the angle of reflection.


46

Wave Interference

• Wave interference occurs when two or

more waves interact with each other

because they occur in the same place at

the same time.

• Superposition principle: the displacement

due the interference of waves is determined

by adding the disturbances produced by

each wave.


47

Wave Interference

Constructive interference:

When the crest of one wave

overlaps the crest of another,

their individual effects add

together to produce a wave of

increased amplitude.

Destructive interference: When

the crest of one wave overlaps

the trough of another, the high

part of one wave simply fills in the

low part of another. So, their

individual effects are reduced (or

even canceled out).


48

Wave Interference

• Example:

– We see the interference pattern made when two vibrating

objects touch the surface of water.

– The regions where a crest of one wave overlaps the trough of

another to produce regions of zero amplitude.

– At points along these regions, the waves arrive out of step, i.e.,

out of phase with each other.


49

Standing Waves

• If we tie a rope to a wall

and shake the free end up

and down, we produce a

train of waves in the rope.

• The wall is too rigid to

shake, so the waves are

reflected back along the

rope.

• By shaking the rope just

right, we can cause the

incident and reflected

waves to form a standing

wave.


50

Standing Waves

• Nodes are the regions of

minimal or zero

displacement, with

minimal or zero energy.

• Antinodes are the

regions of maximum

displacement and

maximum energy.

• Antinodes and nodes

occur equally apart from

each other.


51

Standing Waves • Tie a tube to a firm support.

Shake the tube from side to side

with your hand.

• If you shake the tube with the right

frequency, you will set up a

standing wave.

• If you shake the tube with twice

the frequency, a standing wave of

half the wavelength, having two

loops results.

• If you shake the tube with three

times the frequency, a standing

wave of one-third the

wavelength, having three loops

results.


52

Standing Waves; Resonance


The frequencies of the

standing waves on a

particular string are called

resonant (natural) frequencies.

They are also referred to as

the fundamental and harmonics. 53

Standing Waves

• Examples: – Waves in a guitar string

– Sound waves in a trumpet


54

Refraction


If the wave enters a medium where the wave speed is

different, it will be refracted — its wave fronts and rays will

change direction.

We can calculate the angle

of refraction, which depends

on both wave speeds

(the law of refraction):

55

Refraction


The law of refraction

works both ways — a

wave going from a slower

medium to a faster one

would follow the red line

in the other direction.

56

Diffraction


When waves encounter

an obstacle, they bend

around it, leaving a

“shadow region.”

This is called diffraction.

57

Diffraction


The amount of diffraction depends on the size of the

obstacle compared to the wavelength. If the obstacle is

much smaller than the wavelength, the wave is barely

affected (a). If the object is comparable to, or larger than,

the wavelength, diffraction is much more significant

(b, c, d).

58

• The change in frequency of a wave (or other periodic event) for an observer moving relative to its source.

• Named after the Austrian physicist Christian Doppler (proposed in 1842, Prague).

• Commonly heard when a vehicle sounding a siren approaches, passes, and recedes from an observer.

• Compared to the emitted frequency, the received frequency is higher during the approach, identical at the instant of passing by, and lower during the recession.

Doppler Effect

59

Doppler Effect

• The Doppler effect also applies to light.

– Increase in light frequency when light source

approaches you

– Decrease in light frequency when light source

moves away from you

– Star's spin speed can be determined by shift

measurement


60

Doppler Effect

• Doppler effect of light

– Blue shift

• increase in light frequency toward the blue end of

the spectrum

– Red shift

• decrease in light frequency toward the red end of

the spectrum

– Example: Rapidly spinning star shows a red

shift on the side facing away from us and a

blue shift on the side facing us.


61

Doppler Effect

The Doppler effect occurs for

A.sound.

B.light.

C.Both A and B.

D.Neither A nor B.


62

Energy Transported by Waves

Just as with the oscillation that starts it, the energy

transported by a wave is proportional to the square

of the amplitude.

Each particle (moves in SHM) has an energy:

E= ½ kA2

Definition of intensity (SI unit – W/m2):

The intensity is also proportional to the square of the

amplitude: © 2016 Pearson Education, Ltd.

63

If a wave is able to spread out three-dimensionally

from its source, and the medium is uniform, the

wave is spherical.

Intensity of a spherical wave: I=P/(4πr2)



Just from geometrical

considerations, as long as the

power output is constant, we

see:

64

By looking at the energy

of a particle of matter in

the medium of the wave,

we find:

Then, assuming the entire medium has the same density,

we find:

Therefore, the intensity is proportional to the square of the

frequency and to the square of the amplitude.



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radoslaw bednarek, phd cytobiology and proteomics unit...

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