Chapter 13 Oscillations about Equilibriumphysics.sfsu.edu/~wman/phy111hw/lecture notes/chapter13.pdfChapter 13 Oscillations about Equilibrium •Periodic Motion •Simple Harmonic

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

Copyright Dr. Weining Man

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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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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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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


An object in simple

harmonic motion has

the same motion as

one component of an

object in uniform

circular motion:


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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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The position as a function of time:

The angular frequency:


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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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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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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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• 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:


• Period of a mass on a spring:

• Total energy in simple harmonic motion:


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• Period of a simple pendulum:


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• A simple pendulum with small amplitude, ,

exhibits simple harmonic motion. degree


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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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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Chapter 13 Oscillations about Equilibriumphysics.sfsu.edu/~wman/phy111hw/lecture notes/chapter13.pdfChapter 13 Oscillations about Equilibrium •Periodic Motion •Simple Harmonic

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