Download Chapter 13 Oscillations about Equilibrium

Chapter 13 Oscillations about
Equilibrium
• Periodic Motion
• Simple Harmonic Motion
• Connections between Uniform Circular
Motion and Simple Harmonic Motion
• The Period of a Mass on a Spring
• Energy Conservation in Oscillatory
Motion
• The Pendulum
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Units of Chapter 13
Optional, not required.
• Damped Oscillations
• Driven Oscillations and Resonance
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13-1 Periodic Motion
Period, T: time required for one cycle of
periodic motion.
Frequency, f: number of oscillations per unit
time (per second)
This unit is
called the Hertz:
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Periodic motion around “Equilibrium” point
ΔX>0 ,
resorting force to
the “left” ,
ΔX<0 ,
restoring force to
the “right”
Amplitude A:
the maximum displacement from equilibrium.
(Within each period T, total distance traveled d= 4A )
Restoring force: brings object back to equilibrium
position. It can be spring force, mg component.
It always points to Equilibrium position and is
always opposite to displacement .
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13-2 Simple Harmonic Motion
A spring exerts a restoring force that is
proportional to the displacement from
equilibrium:
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13-2 Simple Harmonic Motion
Here, A is called
the amplitude of
the motion.
A mass on a
spring has a
displacement as
a function of time
that is a sine or
cosine curve:
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13-2 Simple Harmonic Motion
If we call the period of the motion T – this is the
time to complete one full cycle – we can write
the position as a function of time:
It is then straightforward to show that the
position at time t + T is the same as the
position at time t, as we would expect.
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Simple harmonic motion:
SHM is a special oscillation, whose restoring force
is always proportional to the displacement.
F = constant*ΔX
SHM’s displacement is a function of time of a sine
or cosine curve. x vary as cos(ωt) or sin(ωt).
Example: Mass on spring
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1. Object bouncing between 2 walls is not SHM
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13-3 Connections between Uniform Circular
Motion and Simple Harmonic Motion
An object in simple
harmonic motion has
the same motion as
one component of an
object in uniform
circular motion:
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13-3 Connections between Uniform Circular
Motion and Simple Harmonic Motion
Here, the object in circular motion has an
angular speed of
where T is the period of motion of the
object in simple harmonic motion.
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13-3 Connections between Uniform Circular
Motion and Simple Harmonic Motion
The position as a function of time:
The angular frequency:
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13-3 Connections between Uniform Circular
Motion and Simple Harmonic Motion
The velocity as a function of time:
And the acceleration:
Both of these are found by taking
components of the circular motion quantities.
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Simple Harmonic Motion
Set equilibrium point at x=0, F = constant*x
Acceleration: a=F/m is proportional to displacement x
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13-4 The Period of a Mass on a Spring
Since the force on a mass on a spring is
proportional to the displacement, and also to
the acceleration, we find that
.
Substituting the time dependencies of a and x
gives
1, Stiffer spring. Higher k, shorter T; higher ω, faster
2, More mass, (more inertia, harder to oscillate) 15
longer
T; lower
ω, slower
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Man
Greater m, more inertia, slower oscillation, longer T. Smaller ω
Greater k, stronger resorting force, quicker oscillation, greater
ω, shorter T
When k becomes
4 times,
becomes
twice big,
T becomes half,
Frequency is
doubled.
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When m becomes
4 times,
becomes
twice big,
T becomes twice
longer, Frequency
becomes half. 16
If spring or mg component is the restoring force,
If no friction or other loss, E=K+U stay constant at all time.
Energy converts only between KE and PE in SHM.
Moving away from Equilibrium:
Potential Energy , U ￪;
Kinetic Energy , K ￬; Speed ￬;
Resorting force and acceleration
have opposite direction to v.
Moving toward Equilibrium:
Potential Energy , U ￬;
Kinetic Energy , K ￪; Speed ￪;
Resorting force and acceleration
have SAME direction as v.
At equilibrium point, Δx=0. Potential Energy =0;
Kinetic Energy KE reach maximum; speed = vmax
At maximum displacement , x=A or x=-A
Potential Energy U reach Umax; KE=0, Speed =0
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13-5 Energy Conservation in Oscillatory
Motion
In an ideal system with no nonconservative
forces, the total mechanical energy is
conserved. For a mass on a spring:
Since we know the position and velocity as
functions of time, we can find the maximum
kinetic and potential energies:
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13-5 Energy Conservation in Oscillatory
Motion
As a function of time,
So the total energy is constant; as the
kinetic energy increases, the potential
energy decreases, and vice versa.
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13-5 Energy Conservation in Oscillatory
Motion
This diagram shows how the energy
transforms from potential to kinetic and
back, while the total energy remains the
same.
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T is independent to amplitude A :
(greater A, means longer distance to cover,
but it has more total energy, more kinetic energy,
higher speed everywhere.
The beauty of SMH is that Force, acceleration and x match
exactly, so that the period of SHM is only determined by the
ratio between restoring force and mass.
SHM’s period is independent to amplitude A and initial
condition. )
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13-6 The Pendulum
A simple pendulum consists of a mass m (of
negligible size) suspended by a string or rod of
length L (and negligible mass).
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13-6 The Pendulum
Looking at the forces on the
pendulum bob, we see that
the restoring force is
proportional to sin θ,
However, for small angles, sin
θ and θ in radius unit are
approximately equal.
Hence the restoring force
is proportional to the
displacement θ.
(which is θ or s in this
case).
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13-6 The Pendulum
When angle is less than 22 degree,
Restoring force mg sin θ = mgθ , proportional to
angular displacement θ and linear displacement s=Lθ.
Restoring force has opposite direction respect to
Displacement. F=mgθ=mgs/L
As if there is a spring with spring constant
k= F/s=mg/L to drag it back to the center position
The period of a pendulum depends only on the
length of the string:
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Demo: simple pendulum
Two pendulum comparison:
(small maximum angle)
1) more mass, same T, same ω
2) larger amplitude A, same T, same ω
3) longer L, longer T, lower ω
4) If larger g, (more restoring force),
shorter T, higher ω.
T & ω are only determined by L & g.
Why are T & ω independent of m and Amplitude
θmax or hmax as you saw in lab and class?
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T is independent to amplitude:
With bigger θmax, it has longer distance to cover
but it also has more total energy mghmax,
hence higher velocity every where )
(still same time for one cycle).
Each pendulum has its own period, independent to
Initial released position.
T is also independent to mass for simple
pendulum;
greater mass => greater inertia (slower) =>
but also greater resorting force(quicker)
Final total effect of m cancels . Still same time
for one cycle, same T.
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T is inherent system properties, only
determined by L and g
(this is how clock measures time)
Greater L, slower oscillation, longer T!
Greater g, stronger resorting force, quicker
oscillation, shorter T!
On earth, to double T period of pendulum,
you need to make L FOUR times longer.
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13-7 Damped Oscillations
This exponential decrease is shown in the
figure:
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13-8 Driven Oscillations and Resonance
An oscillation can be driven by an oscillating
driving force; the frequency of the driving force
may or may not be the same as the natural
frequency of the system.
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13-8 Driven Oscillations and Resonance
If the driving frequency is close to the
natural frequency, the amplitude can
become quite large, especially if the
damping is small. This is called resonance.
Example: instruments (same key. Play this
instrument you can get resonant at another
one for the same key…)
Example: Bridges…. Go search Tacoma
Narrows bridge on youtube.com!
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Summary of Chapter 13
• Period: time required for a motion to go
through a complete cycle
• Frequency: number of oscillations per unit time
• Angular frequency:
• Simple harmonic motion occurs when the
restoring force is proportional to the
displacement from equilibrium.
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Summary of Chapter 13
• The amplitude A is the maximum displacement
from equilibrium.
• Position as a function of time:
• Velocity as a function of time:
• Acceleration as a function of time:
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Summary of Chapter 13
• Period of a mass on a spring:
• Total energy in simple harmonic motion:
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Summary of Chapter 13
• A simple pendulum with small amplitude,
exhibits simple harmonic motion.
,
degree
• Period of a simple pendulum:
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Summary of Chapter 13
Not required.
• Oscillations where there is a nonconservative
force are called damped.
• Underdamped: the amplitude decreases
exponentially with time:
• Critically damped: no oscillations; system
relaxes back to equilibrium in minimum time
• Overdamped: also no oscillations, but
slower than critical damping
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Summary of Chapter 13
Not required.
• An oscillating system may be driven by an
external force
• This force may replace energy lost to friction,
or may cause the amplitude to increase greatly
at resonance
• Resonance occurs when the driving frequency
is equal to the natural frequency of the system
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