Post on 20-Apr-2018
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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 PendulumCopyright Dr. Weining Man
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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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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 .
Periodic motion around “Equilibrium” point
ΔX>0 ,
resorting force to
the “left” ,
ΔX<0 ,
restoring force to
the “right”
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
A mass on a
spring has a
displacement as
a function of time
that is a sine or
cosine curve:
Here, A is called
the amplitude of
the motion.
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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
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
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
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1, Stiffer spring. Higher k, shorter T; higher ω, faster
2, More mass, (more inertia, harder to oscillate)
longer T; lower ω, slower
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Greater m, more inertia, slower oscillation, longer T. Smaller ω
Greater k, stronger resorting force, quicker oscillation, greater
ω, shorter T
When m becomes
4 times,
becomes
twice big,
T becomes twice
longer, Frequency
becomes half.
When k becomes
4 times,
becomes
twice big,
T becomes half,
Frequency is
doubled.
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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.
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.
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
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. )
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,
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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).
13-6 The Pendulum
The period of a pendulum depends only on the
length of the string:Copyright Dr. Weining Man
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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
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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.
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.
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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!
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:
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• Acceleration as a function of time:
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
• Period of a simple pendulum:
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• A simple pendulum with small amplitude, ,
exhibits simple harmonic motion. degree
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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