Post on 22-May-2020
Radoslaw Bednarek, PhD
Cytobiology and Proteomics Unit
Biophysics seminars, 1 year 6MD, winter semester
Vibrations and Waves
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Vibrations and Waves
• 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.
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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):
Simple Harmonic Motion
© 2016 Pearson Education, Ltd.
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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.
© 2016 Pearson Education, Ltd.
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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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© 2016 Pearson Education, Ltd.
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.
Simple Harmonic Motion
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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).
© 2016 Pearson Education, Ltd.
Simple Harmonic Motion
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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.
© 2016 Pearson Education, Ltd.
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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:
© 2016 Pearson Education, Ltd.
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:
© 2016 Pearson Education, Ltd.
Energy in SHM
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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
© 2016 Pearson Education, Ltd.
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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
© 2016 Pearson Education, Ltd.
The Period and Sinusoidal
Nature of SHM
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The Period and Sinusoidal
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:
© 2016 Pearson Education, Ltd.
The Period and Sinusoidal
Nature of SHM
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© 2016 Pearson Education, Ltd.
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.
The Period and Sinusoidal
Nature of SHM
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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?
© 2016 Pearson Education, Ltd.
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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
© 2016 Pearson Education, Ltd.
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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
So, as long as the cord
can be considered
massless and the
amplitude is small, the
period does not depend
on the mass.
© 2016 Pearson Education, Ltd.
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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.
© 2015 Pearson Education, Ltd.
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.
© 2015 Pearson Education, Ltd.
The Simple Pendulum
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Vibrations and Waves
• Vibration – Wiggle in time
• Wave – Wiggle in space and
time
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Wave Motion
A wave travels
along its medium,
but the individual
particles just move
up and down.
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Wave Description
• Vibration and wave characteristics
– Crests
• high points of the wave
– Troughs
• low points of the wave
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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
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Wave Motion
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Wave Description
• Frequency
– Number of crests (or complete cycles)
passing any point per unit time
– Example:
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2 vibrations occurring in 1 second is a
frequency of 2 vibrations per second.
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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
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• 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.
© 2015 Pearson Education, Ltd.
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
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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.
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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
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Wave Description
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Types of Waves
© 2016 Pearson Education, Ltd.
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.
© 2016 Pearson Education, Ltd.
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)
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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.
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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.
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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
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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.
© 2015 Pearson Education, Ltd.
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Reflection and Transmission of Waves
© 2016 Pearson Education, Ltd.
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.
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© 2016 Pearson Education, Ltd.
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).
Reflection and Transmission of Waves
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© 2016 Pearson Education, Ltd.
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.
Reflection and Transmission of Waves
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© 2016 Pearson Education, Ltd.
The law of reflection: the angle of incidence
equals the angle of reflection.
Reflection and Transmission of Waves
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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.
© 2015 Pearson Education, Ltd.
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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).
© 2015 Pearson Education, Ltd.
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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.
© 2015 Pearson Education, Ltd.
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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.
© 2015 Pearson Education, Ltd.
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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.
© 2015 Pearson Education, Ltd.
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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.
© 2015 Pearson Education, Ltd.
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Standing Waves; Resonance
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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
© 2015 Pearson Education, Ltd.
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Refraction
© 2016 Pearson Education, Ltd.
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):
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Refraction
© 2016 Pearson Education, Ltd.
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.
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Diffraction
© 2016 Pearson Education, Ltd.
When waves encounter
an obstacle, they bend
around it, leaving a
“shadow region.”
This is called diffraction.
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Diffraction
© 2016 Pearson Education, Ltd.
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).
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• 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
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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
© 2015 Pearson Education, Ltd.
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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.
© 2015 Pearson Education, Ltd.
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Doppler Effect
The Doppler effect occurs for
A.sound.
B.light.
C.Both A and B.
D.Neither A nor B.
© 2015 Pearson Education, Ltd.
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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.
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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)
© 2016 Pearson Education, Ltd.
Energy Transported by Waves
Just from geometrical
considerations, as long as the
power output is constant, we
see:
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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.
© 2016 Pearson Education, Ltd.
Energy Transported by Waves
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