Download pkt 6 oscillations and waves

Oscillations and Waves
Oscillation:
Wave:
Examples of oscillations:
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2.
3.
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mass on spring (eg. bungee jumping)
pendulum (eg. swing)
object bobbing in water (eg. buoy, boat)
vibrating cantilever (eg. diving board)
earthquake
bouncing ball
musical instruments (eg. strings, percussion, brass, woodwinds, vocal chords)
heartbeat
Mean Position (Equilibrium Position) – position of object at rest
Displacement (x, meters) – distance in a particular direction of a particle from its mean position
Amplitude (A or x0, meters) – maximum displacement from the mean position
Period (T, seconds) – time taken for one complete oscillation
Frequency (f, Hertz) – number of oscillations that take place per unit time
Phase Difference – difference in phase between the particles of two oscillating systems
Relationship between period
and frequency:
Angular Frequency -
Formula:
Symbol:
Units:
1. A pendulum completes 10 swings in 8.0 seconds.
a) Calculate its period.
b) Calculate its frequency.
c) Calculate its angular frequency.
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Example of an Oscillating System
A mass oscillates on a horizontal spring without friction. At each position, analyze its displacement, velocity and acceleration.
Restoring Force:
Force from the Spring:
1. When is the velocity of the mass at its maximum value?
2. When is the acceleration of the mass at its maximum value?
Simple Harmonic Motion (SHM) – motion
that takes place when the acceleration of an
object is proportional to its displacement
from its equilibrium position and is always
directed toward its equilibrium position
Defining Equation for SHM:
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The Displacement Function
A mass on a spring is allowed to oscillate up and down about its mean position without friction.
Two traces of the displacement (x) of the mass versus time (t) are shown.
Initial condition:
Function:
Initial condition:
Function:
Analyzing the Displacement Function
1. Analyze the displacement function shown at right.
a) What is the amplitude?
b) What is the period?
2. What is the displacement of the mass when:
c) What is the frequency?
d) What is the angular frequency?
a) t = 1.0 s?
b) t = 2.0 s?
c) t = 2.5 s?
e) Write the displacement function.
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Equations of Motion for Simple Harmonic Motion
a) Displacement Function
b) Velocity Function
c) Acceleration Function
Defining Equation for SHM:
Alternate Velocity Function
Alternative Equations of Motion
When would these equations be used?
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1. The graph shown at right shows the displacement of an object in SHM.
a) Use the graph to find the:
i) amplitude of oscillation
ii) period of oscillation
iii) angular frequency
iv) displacement function
v) maximum velocity
vi) maximum acceleration
b) At what time(s) does the maximum velocity occur?
c) At what time(s) does the maximum acceleration occur?
d) Determine the velocity of the object at 1.3 seconds. (Use both formulas.)
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Example of SHM – Mass on a Horizontal Spring
A mass m oscillates horizontally on a spring
without friction, as shown. Is this SHM?
Angular frequency, period, and frequency for a mass on a spring
1. A 2.00 kg mass oscillates back and forth 0.500m from its rest position on a horizontal spring
whose constant is 40.0 N/m.
a) Calculate the angular frequency, period and frequency of this system.
b) Write the displacement, velocity and acceleration functions for this system.
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Example of SHM – Simple Pendulum
A mass is allowed to swing freely from the end of a
light-weight string. This motion is approximately
simple harmonic motion if the angle of vibration is not
too large. The angular frequency is given by
!=
g
L
1. Determine the period and frequency for the pendulum.
2. A 20.0 g pendulum on an 80.0 cm string is pulled back 5.0 cm and then swings. Determine its:
a) angular frequency
b) maximum velocity
d) displacement function
c) maximum acceleration
e) velocity function
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Energy and Simple Harmonic Motion
A mass oscillates back and forth on a spring. Analyze the energy in the system at each location.
When the mass is at its
mean position . . .
When the mass is at its
extreme positions . . .
When the mass is at any
position . . .
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1. A 2.00 kg mass is oscillating on a spring and its
displacement function is shown.
a) At what time(s) does the mass have the most kinetic energy?
b) Determine the maximum kinetic energy of the mass.
c) At what time(s) does the mass have maximum potential energy? Determine this value.
d) Determine the kinetic and potential energy of the system at 1.5 seconds.
2. The graph at right shows the potential energy of a 2.5 kg
object in SHM as a function of its displacement.
a) Sketch on the graph how its kinetic energy varies.
b) What is the total energy of the object?
c) What is the angular frequency of the object?
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Damping in Oscillations
Damping:
Effect of damping:
Sketch the displacement function for a system without and with damping.
Without Damping
With Damping
Degrees of Damping
Light damping (under-damping):
small resistive force so only a small
percentage of energy is removed each
cycle – period is not affected – can take
many cycles for oscillations to die out
eg. – car shock absorbers
Heavy damping (over-damping): large resistive force –
can completely prevent any oscillations from taking place
– takes a long time for object to return to mean position
eg.- oscillations in viscous fluid
Critical damping: intermediate resistive force so time
taken for object to return to mean position is minimum –
minimal or no “overshoot”
eg. – electric meters with pointers, automatic door closers
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Resonance
Natural Frequency of Vibration: when a system is displaced from equilibrium and allowed to oscillate
freely, it will do so at its natural frequency of vibration
Forced Oscillations – a system may be forced to oscillate at any given frequency by an outside driving
force that is applied to it
Resonance –
Amplitude vs. frequency graph
for forced oscillations
Factors that affect the frequency response and
sharpness of curve:
1)
2)
3)
4)
1. Sketch the frequency response for a lightly
damped system whose natural frequency is
20 Hz that experiences forced oscillations.
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Waves
Both pulses and traveling waves:
Pulse – single oscillation or disturbance
Continuous traveling wave – succession of
oscillations (series of periodic pulses)
Mechanical Waves: require a medium to transfer energy
eg. – sound waves, water waves, waves on strings, earthquake waves
Electromagnetic Waves: do not require a medium to transfer energy
eg. – light waves, all EM waves
A transverse wave is
one in which the
direction of the
oscillation of the
particles of the medium
is perpendicular to the
direction of travel of the
wave (the energy).
Examples: light, violin
and guitar strings, ropes,
earthquake S waves
transfer energy though there is
no net motion of the medium
through which the wave passes.
A longitudinal wave is
one in which the direction
of the oscillation of the
particles of the medium is
parallel to the direction of
travel of the wave (the
energy).
Example: sound,
earthquake P waves
Compression: region where particles of medium are close together
Rarefaction: region where particles of medium are far apart
Note that transverse mechanical waves cannot propagate (travel) through a gas – only longitudinal waves can.
Displacement (x, meters) – distance in a particular direction of a particle
from its mean position
Amplitude (A or x0, meters) – maximum displacement from the mean
position
Period (T, seconds) – time taken for one complete oscillation
- time for one complete wave (cycle) to pass a given point
Frequency (f, Hertz) – number of oscillations that take place per unit time
Wavelength (λ, meters) – shortest distance along the wave between two points that are in phase
-the distance a complete wave (cycle) travels in one period.
Compare the motion of a single particle to
the motion of the wave as a whole (the
motion of the energy transfer).
Particle Speed:
Wave Speed:
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Motion of the Wave
1.
2.
Motion of a Particle
Control variable: in one medium - wave speed
Control variable: across a boundary - frequency
Wave speed depends on the properties of the medium, not
how fast the medium vibrates. To change wave speed, you
must change the medium or its properties.
As a wave crosses a boundary between two different media, the
frequency of a wave remains constant not the speed or
wavelength.
Light:
Sound:
Waves in Two Dimensions
Wavefront – line (or arc) joining
neighboring points that have the same phase
or displacement
At great distances,
the wavefronts are
approximately
parallel and are
known as plane
waves.
Ray – line indicating direction of wave
motion (direction of energy transfer).
Rays are perpendicular to wavefronts.
Intensity -
NOTE:
Symbol:
Formula:
Units:
12 x 10-5 W of sound power pass through each surface as shown. Surface 1 has area 4.0 m2 and
surface 2 is twice as far away from the source. Calculate the sound intensity at each location.
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Reflection and Refraction
Sketch the incident and reflected rays as well as the reflected wavefront.
Law of Reflection
The angle of incidence is
equal to the angle of
reflection when both
angles are measured with
respect to the normal line
(and the incident ray,
reflected ray and normal
all lie in the same plane).
Mirror
Refraction: the change in direction of a wave (due to a change in speed) when it crosses a boundary between
two different media at an angle
Air to water:
Water to air:
Refractive Index (Index of refraction)(n):
ratio of sine of angle of incidence to sine of angle
of refraction, for a wave incident from air
Snell’s Law: the ratio of the sine of the angle of
incidence to the sine of the angle of refraction is a
constant, for a given frequency
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