Transcript IB2_Day1a_SHMx

Simple Harmonic Motion
First things first…
 Permission slips for Weds? I need
them.
 We’ll meet out front at the start of the
day. Don’t go to your 1st period class.
 Choose one aspect of The Martian
that relates to physics and explain the
physics behind it. Typed – due 10/14.
At least one page.
Simple Harmonic Motion
Like all motion, it’s all about
displacement, velocity and
acceleration
Simple Harmonic Motion
Like all motion, it’s all about
displacement, velocity and
acceleration
I don’t care!
Get me out of
here!
Topic 4: Waves – Simple Harmonic Motion
4.1 – Oscillations
Oscillations
Oscillations are vibrations which repeat themselves.
v=0
v
=
0
EXAMPLE: Oscillations
v=0
v = vmax
v=0
EXAMPLE: Oscillations can
be driven internally, like a
mass on a spring.
FYI In all oscillations,
v = 0 at the extremes…
and v = vmax in the
middle of the motion.
v = vmax
can be driven externally,
like a pendulum in a
gravitational field.
x
Topic 4: Waves
4.1 – Oscillations
Oscillations
Oscillations are vibrations which repeat themselves.
EXAMPLE: Oscillations can be very rapid vibrations such as in
a plucked guitar string or a tuning fork.
Topic 4: Waves
4.1 – Oscillations
Time period, amplitude and displacement
Consider a mass on a
spring that is displaced
4 meters to the right
x0
and then released.
We call the maximum displacement x0 the amplitude.
In this example x0 = 4 m.
We call the point of zero displacement the equilibrium
position. Displacement x is measured from equilibrium.
The period T (measured in s) is the time it takes
for the mass to make one full oscillation or cycle.
For this particular oscillation, the period T is
about 24 seconds (per cycle).
x
Topic 4: Waves
4.1 – Oscillations
Time period and frequency
The frequency f (measured in Hz or cycles / s) is
defined as how many cycles (oscillations, repetitions)
occur each second.
Since period T is seconds per cycle, frequency must
be 1 / T.
f=1/T
T=1/f
relation between T and f
EXAMPLE: The cycle of the previous example repeated
each 24 s. What are the period and the frequency of the
oscillation?
SOLUTION:
The period is T = 24 s.
The frequency is f = 1 / T = 1 / 24 = 0.042 Hz
The period of a pendulum:
Always remember:
Do you see mass in
there?
What does it depend
on?
increase L = bigger T
decrease g = bigger T
Topic 4: Waves
4.1 – Oscillations
Phase difference
PRACTICE: Two identical mass-spring systems are
started in two different ways. What is their phase
difference?
Start stretched
and then release
x
Start unstretched
with a push left
x
SOLUTION:
The phase difference is one-quarter of a cycle.
Topic 4: Waves
4.1 – Oscillations
Phase difference
PRACTICE: Two identical mass-spring systems are
started in two different ways. What is their phase
difference?
Start stretched
and then release
x
Start unstretched
with a push right
x
SOLUTION:
The phase difference is three-quarters of a cycle.
Simple Harmonic Motion:
1. The object moves back and forth around
a reference point, or mean position.
2. A restoring force acts on the object.
3. At any given point, its acceleration is
directly proportional to its displacement in
magnitude, but is in the opposite direction
and always directed to the mean position.
Topic 4: Waves
4.1 – Oscillations
Conditions for simple harmonic motion
In simple harmonic motion (SHM), a and x are
related in a very precise way: Namely, a  -x.
a  -x
definition of SHM
PRACTICE: Show that a mass oscillating on a spring
executes simple harmonic motion.
x

As the object moves with –x, what is the acceleration’s sign?
Remember – the object is slowing and stopping at xmax.
F
Topic 4: Waves
4.1 – Oscillations
x x
0
Conditions for simple harmonic motion
a  -x
F
F and x oppose
each other.
definition of SHM
The minus sign in Hooke’s law, F = -kx, tells us that if
the displacement x is positive (right), the spring force F
is negative (left).
It also tells us that if the displacement x is negative
(left), the spring force F is positive (right).
Any force that is proportional to the opposite of a
displacement is called a restoring force.
For any restoring force F  -x.
Since F = ma we see that ma  -x, or a  -x.
All restoring forces can drive simple harmonic motion
(SHM).
x
Like this…
Try one.
• Change the length of the arm of the
pendulum.
• Measure the period
• What kind of relationship do we see?
Topic 4: Waves
4.1 – Oscillations
Conditions for simple harmonic motion
If we place a pen on the oscillating mass, and pull
a piece of paper at a constant speed past the pen,
we trace out the displacement vs. time graph of SHM.
x SHM traces out perfect sinusoidal waveforms.
t
Note that the period can be found from the graph:
Just look for repeating cycles.
Topic 4: Waves
4.1 – Oscillations
Qualitatively describing the energy changes taking
place during one cycle of an oscillation
Consider the pendulum to
the right which is placed in
position and held there.
Let the green rectangle
represent the potential energy
of the system.
Let the red rectangle represent
the kinetic energy of the system.
Because there is no motion yet, there is no kinetic
energy. But if we release it, the kinetic energy will grow
as the potential energy diminishes.
A continuous exchange between EK and EP occurs.
Topic 4: Waves
4.1 – Oscillations
Qualitatively describing the energy changes taking
place during one cycle of an oscillation
Consider the mass-spring
system shown here. The
mass is pulled to the right
and held in place.
Let the green rectangle represent the
potential energy of the system.
Let the red rectangle
FYI If friction and drag are
represent the kinetic
energy of the system. both zero ET = CONST.
A continuous exchange between EK and EP occurs.
Note that the sum of EK and EP is constant.
EK + EP = ET = CONST relation between EK and EP
x
Topic 4: Waves
4.1 – Oscillations
Qualitatively describing the energy changes taking
place during one cycle of an oscillation
EK + EP = ET = CONST relation between EK and EP
Energy
If we plot both kinetic
energy and potential
energy vs. time for either
system we would get the
following graph:
time
x
v=0
v = vMAX
v=0
Topic 4: Waves
4.1 – Oscillations
x
-2.0
0.0
2.0
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.
Or – in a pendulum
v=0
v = vMAX
v=0
Topic 4: Waves
4.1 – Oscillations
x
-2.0
0.0
2.0
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.
SHM and Circles…
SHM is a projection of circular
motion
Examples…
For more:
• The Physics Classroom (online) – Waves:
Lesson 0 – Vibrations
• Homer – pg 115 – 123
• For next time: Homer: 44 – 61 &
homework sheet.