Transcript waves - DP Physics

WAVES
4-1 Oscillations
What is a wave?
A wave can be described as a disturbance that travels through a medium from one
location to another location.
The repeating and periodic disturbance that moves through a medium from one
location to another is referred to as a wave.
Types of Waves
A transverse wave is a wave in which particles of the medium move in a
direction perpendicular to the direction that the wave moves. Suppose that a slinky is
stretched out in a horizontal direction across the classroom and that a pulse is
introduced into the slinky on the left end by vibrating the first coil up and down. Energy
will begin to be transported through the slinky from left to right. As the energy is
transported from left to right, the individual coils of the medium will be displaced
upwards and downwards. In this case, the particles of the medium move
perpendicular to the direction that the pulse moves. This type of wave is a transverse
wave. Transverse waves are always characterized by particle motion
being perpendicular to wave motion.
Types of Waves
A longitudinal wave is a wave in which particles of the medium move in a
direction parallel to the direction that the wave moves. Suppose that a slinky is
stretched out in a horizontal direction across the classroom and that a pulse is
introduced into the slinky on the left end by vibrating the first coil left and right. Energy
will begin to be transported through the slinky from left to right. As the energy is
transported from left to right, the individual coils of the medium will be displaced
leftwards and rightwards. In this case, the particles of the medium move parallel to the
direction that the pulse moves. This type of wave is a longitudinal wave. Longitudinal
waves are always characterized by particle motion being parallel to wave motion.
Types of Waves
A surface wave is a wave in which particles of the medium undergo a circular motion.
Surface waves are neither longitudinal nor transverse. In longitudinal and transverse
waves, all the particles in the entire bulk of the medium move in a parallel and a
perpendicular direction (respectively) relative to the direction of energy transport. In a
surface wave, it is only the particles at the surface of the medium that undergo the
circular motion. The motion of particles tends to decrease as one proceeds further from
the surface.
Wave Properties
The main properties of waves are defined below.
Amplitude: the height of the wave, measured in meters.
Wavelength: the distance between adjacent crests, measured in meters.
Period: the time it takes for one complete wave to pass a given point, measured in
seconds.
Frequency: the number of complete waves that pass a point in one second, measured
in inverse seconds, or Hertz (Hz).
Speed: the horizontal speed of a point on a wave as it propagates, measured in
meters / second.
Amplitude
The amplitude of a wave refers to the maximum amount of displacement of a
particle on the medium from its rest position. In a sense, the amplitude is the distance
from rest to crest. Similarly, the amplitude can be measured from the rest position to
the trough position. In the diagram above, the amplitude could be measured as the
distance of a line segment that is perpendicular to the rest position and extends
vertically upward from the rest position to point A.
AMPLITUDE
Wavelength
The wavelength of a wave is simply the length of one complete wave cycle. If you
were to trace your finger across the wave in the diagram, you would notice that your
finger repeats its path. A wave is a repeating pattern. It repeats itself in a periodic and
regular fashion over both time and space. And the length of one such spatial
repetition (known as a wave cycle) is the wavelength. The wavelength can be
measured as the distance from crest to crest or from trough to trough. In fact, the
wavelength of a wave can be measured as the distance from a point on a wave to
the corresponding point on the next cycle of the wave.
WAVELENGTH
Period
Period refers to the time that it takes to do something. When an event occurs
repeatedly, then we say that the event is periodic and refer to the time for the event
to repeat itself as the period. The period of a wave is the time for a particle on a
medium to make one complete vibrational cycle. Period, being a time, is measured in
units of time such as seconds, hours, days or years.
Frequency
The frequency of a wave refers to how often the particles of the medium vibrate when
a wave passes through the medium. In mathematical terms, the frequency is the
number of complete vibrational cycles of a medium per a given amount of time.
Given this definition, it is reasonable that the quantity frequency would have units of
cycles/second, waves/second, vibrations/second, or something/second. Another unit
for frequency is the Hertz (abbreviated Hz) where 1 Hz is equivalent to 1 cycle/second.
Relationship between frequency and period
Recall that the relationship between frequency and period is that one is the inverse of
the other.
T
period
Period = 0.4 seconds
T = 1/f
f = 1/T = 1/0.4 = 2.5 cycles / second = 2.5 Hz
Speed/velocity
The speed of an object refers to how fast an object is moving and is usually expressed
as the distance traveled per time of travel. In the case of a wave, the speed is the
distance traveled by a given point on the wave (such as a crest) in a given interval of
time.
Speed = d/t
Speed = wavelength/period
Speed = l/T
Oscillation
Oscillation is going back and forth repeatedly between two positions or states. An
oscillation can be a periodic motion that repeats itself in a regular cycle, such as a sine
wave, the side to side swing of a pendulum, or the up and down motion of a spring
with a weight. An oscillating movement is around an equilibrium point or mean value.
Hooke’s Law
Hooke's Law is a principle of physics that states that the force needed to extend or
compress a spring by some distance is proportional to that distance.
The law is named after 17th century British physicist Robert Hooke, who sought to
demonstrate the relationship between the forces applied to a spring and its elasticity.
He first stated the law in 1660 as a Latin anagram, and then published the solution in
1678 as ut tensio, sic vis – which translated, means "as the extension, so the force" or
"the extension is proportional to the force").
This can be expressed mathematically as
F= -kX,
where F is the force applied to the spring (either in the form of strain or stress); X is the
displacement of the spring, with a negative value demonstrating that the
displacement of the spring once it is stretched; and k is the spring constant and details
just how stiff it is.
Simple Harmonic Motion
Simple harmonic motion (SHM) is any motion where a restoring force is applied that is
proportional to the displacement and in the opposite direction of that displacement.
Basic conditions to execute SHM are as under:
There must be an elastic restoring force acting on the system.
The system must have inertia
The acceleration of the system should be directly proportional to its displacement
and is always directed to mean position. (a  -x)
Combining Hooke’s Law with Newton’s Second Law of Motion
F = ma and
F = -kx
We can see that ma = -kx  acceleration is proportional to displacement
a  -x
Characteristics of SHM
The motion must be vibratory.
The motion should be a periodic motion.
The restoring force should be directly proportional to the displacement of the body
from its mean position.
Examples:
Motion of a body attached to the end of an elastic spring.
Motion of the bob of a simple pendulum if it is given a displacement.
Motion of an elastic strip.
Motion of the prongs of a tuning fork.
Motion of the wire of a guitar or violin.
Total Energy
The relationship between kinetic and
potential energy during oscillation.
EK + EP = ET = constant.
Example
EXAMPLE: The displacement x vs. time t for a 2.5-kg mass on a spring having spring
constant k = 4.0 Nm-1 is shown in the sinusoidal graph.
The time, T.
a. Find the period and frequency of the motion.
The period is the time it takes to complete one cycle.
60 ms = 6.0 x 10-2 seconds
Example
EXAMPLE: The displacement x vs. time t for a 2.5-kg mass on a spring having spring
constant k = 4.0 Nm-1 is shown in the sinusoidal graph.
Amplitude
b. Find the amplitude of the motion.
The amplitude is the maximum displacement.
As seen from the graph, the highest point is 2 mm above the equilibrium line.
Example
EXAMPLE: The displacement x vs. time t for a 2.5-kg mass on a spring having spring
constant k = 4.0 Nm-1 is shown in the sinusoidal graph.
c. Sketch the graph of x vs. t for the situation where the amplitude is cut in half.
The time behavior is still the same, and the only change is that the amplitude is less
by a factor of ½.
Example
EXAMPLE: The displacement x vs. time t for a 2.5-kg mass on a spring having spring
constant k = 4.0 Nm-1 is shown in the sinusoidal graph.
d. The blue graph shows an equivalent system is SHM. What is the phase difference
between the red and blue?
The time discrepancy between the two graphs.
Just think of it as the blue started later than the red but behaves the same way as
the red.
v=0
Topic 4: Waves
4.1 – Oscillations
v = vMAX
-2.0
v=0
0.0
2.0
x
Sketching and interpreting graphs of simple harmonic motion
examples
EXAMPLE: The displacement x vs. time t for a system
undergoing SHM is shown here.
x-black
(+)
( -)
(+)
( -)
(+)
v-red
(different
scale)
t
Sketch in red the velocity vs. time graph.
SOLUTION: At the extremes, v = 0.
At x = 0, v = vMAX. The slope determines sign of vMAX.
v=0
Topic 4: Waves
4.1 – Oscillations
-2.0
v = vMAX
0.0
v=0
2.0
x
Sketching and interpreting graphs of simple harmonic motion
examples
EXAMPLE: The displacement x vs. time t for a system
undergoing SHM is shown here.
x-black
v-red
(different
scale)
t
a-blue
(different
scale)
Sketch in blue the acceleration vs. time graph.
SOLUTION: Since a  -x, a is just a reflection of x.
Note: x is a sine, v is a cosine, and a is a – sine wave.
Topic 4: Waves
4.1 – Oscillations
-2.0
0.0
Sketching and interpreting graphs of simple harmonic motion
examples
EXAMPLE: The kinetic energy
vs. displacement for a system
undergoing SHM is shown in
the graph. The system consists
of a 0.125-kg mass on a spring.
(a) Determine the maximum
velocity of the mass.
SOLUTION:
When the kinetic energy is maximum, the velocity is
also maximum. Thus 4.0 = (1/ 2)mvMAX2 so that
4.0 = (1/ 2)(.125)vMAX2  vMAX = 8.0 ms-1.
2.0
x
Topic 4: Waves
4.1 – Oscillations
-2.0
0.0
2.0
Sketching and interpreting graphs of simple harmonic motion
examples
ET
EXAMPLE: The kinetic energy
vs. displacement for a system
undergoing SHM is shown in
EK
the graph. The system consists
of a 0.125-kg mass on a spring.
EP
(b) Sketch EP and determine the
total energy of the system.
SOLUTION:
Since EK + EP = ET = CONST, and since EP = 0 when
EK = EK,MAX, it must be that ET = EK,MAX = 4.0 J.
Thus the EP graph will be the “inverted” EK graph.
x
Topic 4: Waves
4.1 – Oscillations
-2.0
0.0
2.0
Sketching and interpreting graphs of simple harmonic motion
examples
EXAMPLE: The kinetic energy
vs. displacement for a system
undergoing SHM is shown in
the graph. The system consists
of a 0.125-kg mass on a spring.
(c) Determine the spring
constant k of the spring.
SOLUTION: Recall EP = (1/2)kx2.
Note that EK = 0 at x = xMAX = 2.0 cm. Thus
EK + EP = ET = CONST  ET = 0 + (1/ 2)kxMAX2 so that
4.0 = (1/ 2)k 0.0202  k = 20000 Nm-1.
x
Topic 4: Waves
4.1 – Oscillations
-2.0
0.0
2.0
Sketching and interpreting graphs of simple harmonic motion
examples
EXAMPLE: The kinetic energy
vs. displacement for a system
undergoing SHM is shown in
the graph. The system consists
of a 0.125-kg mass on a spring.
(d) Determine the acceleration
of the mass at x = 1.0 cm.
SOLUTION:
From Hooke’s law, F = -kx we get
F = -20000(0.01) = -200 N.
From F = ma we get -200 = 0.125a  a = -1600 ms-2.
x
Topic 4: Waves
4.1 – Oscillations
-2.0
0.0
Sketching and interpreting graphs of simple harmonic motion
examples
EXAMPLE: A 4.0-kg mass is
placed on a spring’s end and
displaced 2.0 m to the right.
The spring force F vs. its
displacement x from equilibrium
is shown in the graph.
(a) How do you know that the
mass is undergoing SHM?
SOLUTION:
In SHM, a  -x. Since F = ma, then F  -x also.
The graph shows that F  -x. Thus we have SHM.
2.0
x
Topic 4: Waves
4.1 – Oscillations
-2.0
0.0
2.0
Sketching and interpreting graphs of simple harmonic motion
examples
EXAMPLE: A 4.0-kg mass is
placed on a spring’s end and
displaced 2.0 m to the right.
The spring force F vs. its
displacement x from equilibrium
is shown in the graph.
(b) Find the spring constant of
the spring.
SOLUTION: Use Hooke’s law: F = -kx.
Pick any F and any x. Use k = -F / x.
Thus k = -(-5.0 N) / 1.0 m = 5.0 Nm-1.
F = -5.0 N
x = 1.0 m
x
Topic 4: Waves
4.1 – Oscillations
-2.0
0.0
Sketching and interpreting graphs of simple harmonic motion
examples
EXAMPLE: A 4.0-kg mass is
placed on a spring’s end and
displaced 2.0 m to the right.
The spring force F vs. its
displacement x from equilibrium
is shown in the graph.
(c) Find the total energy of the
system.
SOLUTION: Use ET = (1/2)kxMAX2. Then
ET = (1/2)kxMAX2 = (1/2)  5.0  2.02 = 10. J.
2.0
x
Topic 4: Waves
4.1 – Oscillations
-2.0
0.0
Sketching and interpreting graphs of simple harmonic motion
examples
EXAMPLE: A 4.0-kg mass is
placed on a spring’s end and
displaced 2.0 m to the right.
The spring force F vs. its
displacement x from equilibrium
is shown in the graph.
(d) Find the maximum speed of
the mass.
SOLUTION: Use ET = (1/2)mvMAX2.
10. = (1/2)  4.0  vMAX2
vMAX = 2.2 ms-1.
2.0
x
Topic 4: Waves
4.1 – Oscillations
-2.0
0.0
Sketching and interpreting graphs of simple harmonic motion
examples
EXAMPLE: A 4.0-kg mass is
placed on a spring’s end and
displaced 2.0 m to the right.
The spring force F vs. its
displacement x from equilibrium
is shown in the graph.
(e) Find the speed of the mass
when its displacement is 1.0 m.
SOLUTION: Use ET = (1/2)mv 2 + (1/2)kx 2. Then
10. = (1/2)(4)v 2 + (1/2)(5)12
v = 1.9 ms-1.
2.0
x
Example