Download Simple Harmonic Motion

Simple Harmonic
Motion
Oscillations
•
Motion is repetitive (Periodic) and the oscillating
body moves back and forth around an equilibrium
position.
• Period: The time required for one full oscillation
•
•
We will focus on constant periods…
What are some examples of oscillating
bodies/systems?
Periodic Motion
• Objects that move back and forth periodically are described as oscillating.
• These objects move past an equilibrium position, O (where the body
would rest if a force were not applied) and their displacement from this
position changes with time.
• If the time period is independent of the maximum displacement, the
motion is isochronous.
O
E.g.
-Oscillating pendulums, watch springs or atoms can all be used to
measure time
Properties of oscillating bodies
A time trace is a graph showing the variation
of displacement against time for an oscillating
body.
Demo: Producing a time-trace of a mass on a
spring.
Distance sensor connected to computer
Amplitude (x0): The maximum displacement (in m) from the
equilibrium position (Note that this can reduce over time due to
damping).
Cycle: One complete oscillation of the body.
Period (T): The time (in s) for one complete cycle.
Frequency (f): The number of complete cycles made per second (in
Hertz or s-1). (Note: f = 1 / T)
Angular frequency (ω): Also called angular speed, in circular motion
this is a measure of the rate of rotation. In periodic motion it is a
constant (with units s-1 or rad s-1) given by the formula…
ω = 2π = 2πf
T
Q.
Calculate the angular speed of the hour hand of an analogue
watch (in radians per second).
Angle in one hour = 2π radians
Time for one revolution = 60 x 60 x 12 = 43200s
ω = 2π = 1.45 x 10-3 rad s-1
T
Simple Harmonic Motion (SHM)
Consider this example
Hooke’s Law
•
What is the relationship between the displacement of
the spring and the force applied to the spring?
• What characteristics of the spring will affect this?
• Hooke’s Law: up to its elastic limit, a spring will
experience a displacement that is proportional to the
force applied to the spring.
More on springs…
• Restoring Force:
• the tension in the spring that works to bring a
mass back to its rest position (equilibrium
position).
• The restoring force is the reaction force to
any applied force (i.e. the weight of
something hanging from the spring)
• This is the force that causes a mass to accelerate
around its rest position when it has experienced
a displacement away from equilibrium.
Simple Harmonic Motion
• A special case of periodic oscillations that
•
•
•
•
can be described by analyzing the forces
involved in the motion
Hooke’s law can be rewritten as… ma = -k x
We are going to define
angular frequency as: ω = √ (k / m)
Angular frequency has units of Hertz (s-1)
Defining relation for SHM:
a = -ω2 x
• “Simple harmonic motion takes place when a particle
that is disturbed away from its fixed equilibrium
position experiences an acceleration that is
proportional and opposite to its displacement” (from
the IB Physics text by Tsokos)
 Two requirements for SHM:
• Must have a fixed equilibrium point
• Acceleration, when displaced, must be
proportional to the amount of displacement
SHM--mathematics
• By using calculus, the defining relationship
•
•
•
•
becomes one that we can put in terms of the
angular frequency, the time that has passed,
the amplitude of the displacement, and the
“phase shift”
A = amplitude
x = A cos (ωt + Φ)
t = time
Φ = phase shift
ω = angular frequency
•
Amplitude: the maximum displacement away from
the equilibrium (rest) position.
• This occurs when the value of the cosine function
is equal to 1
• Φ = Phase Shift: recorded in radians; gives an
indication of the displacement at t = 0 s.
• See diagram on next slide…
• Phase difference:
ΔΦ = |Φ1 – Φ2|
Simple Pendulum…
• Is this simple harmonic motion? How do you
know?


NO! It is not SHM—the acceleration and the
displacement are not proportional to each other
Period of a pendulum can be found with:
T = 2π √(L/g)
Q1
Sketch a graph of acceleration against
displacement for the oscillating mass
shown (take upwards as positive.
displacement
o
a
x
Q2
Consider this duck, oscillating with SHM…
Where is…
i.
Displacement at a maximum?
ii.
Displacement zero?
C
iii. Velocity at a maximum?
iv. Velocity zero?
v.
C
A and E
Acceleration at a maximum?
vi. Acceleration zero?
A and E
C
A and E
Damped Oscillations
• Any oscillations that are taking place
with the presence of one or more
resistance forces, such as friction and air
resistance.
• As a result of resistance forces,
oscillations will eventually stop as the
energy of its motion is transformed into
other forms of energy—primarily thermal
energy of both the environment and the
system.
Types of damping effects:
•
Under-damping:
• Small resistance forces
• Oscillations continue, but with a slightly smaller
frequency than if force was not there
• Amplitude gradually reduces until it approaches 0 and
the oscillations stop
• The larger the damping force, the longer the period
and the faster it will stop oscillating
Types of damping effects:
•
Critical Damping:
• The system returns to its equilibrium state
as fast as possible without any oscillations
• Over-damping:
• The system returns to equilibrium without
oscillations, but much slower than in
critical damping
Graphical comparison of damping
effects
Oscillations and
Waves
Energy Changes During Simple
Harmonic Motion
Energy in SHM
velocity
Energy-time graphs
energy
KE
PE
Total
Note: For a spring-mass system:
KE = ½ mv2  KE is zero when v = 0
PE = ½ kx2  PE is zero when x = 0 (i.e. at vmax)
Energy–displacement graphs
energy
KE
PE
Total
-xo
+xo
displacement
Note: For a spring-mass system:
KE = ½ mv2  KE is zero when v = 0 (i.e. at xo)
PE = ½ kx2  PE is zero when x = 0
Kinetic energy in SHM
We know that the velocity at any time is
given by…
v = ω √ (xo2 – x2)
So if Ek = ½ mv2 then kinetic energy at an
instant is given by…
Ek = ½ mω2 (xo2 – x2)
Potential energy in SHM
If
a = - ω2 x
then the average force applied trying to pull
the object back to the equilibrium position as
it moves away from the equilibrium position
is…
F = - ½ mω2x
Work done by this force must equal the PE it
gains (e.g in the springs being stretched).
Thus..
Ep = ½ mω2x2
Total Energy in SHM
Clearly if we add the formulae for KE and PE in SHM we
arrive at a formula for total energy in SHM:
ET = ½ mω2xo2
Summary:
Ek = ½ mω2 (xo2 – x2)
Ep = ½ mω2x2
ET = ½ mω2xo2
teacherweb.com/CA/.../Berg/SimpleHarmonicMotionweb.ppt
sjhs-ibphysics.wikispaces.com/file/view/15+Kinematics+of+SHM.ppt
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