Physics 2111, 2008 Fundamentals of Physics 1 Chapter 13 Oscillations 1.Oscillations 2.Simple...

Physics 2111, 2008 Fundamentals of Physics 1

Fundamentals of Physics

Chapter 13 Oscillations

1. Oscillations

2. Simple Harmonic Motion

Velocity of SHM

Acceleration of SHM

3. The Force Law for SHM

4. Energy in SHM

5. An Angular Simple Harmonic Oscillator

6. Pendulums

The Simple Pendulum

The Physical Pendulum

Measuring “g”

7. SHM & Uniform Circular Motion

8. Damped SHM

9. Forced Oscillations & Resonance


Oscillations

Oscillations - motions that repeat themselves.

Oscillation occurs when a system is disturbed from a position of stable equilibrium.

– Clock pendulums swing

– Boats bob up and down

– Guitar strings vibrate

– Diaphragms in speakers

– Quartz crystals in watches

– Air molecules

– Electrons

– Etc.


Oscillations

Oscillations - motions that repeat themselves.


Simple Harmonic Motion

Harmonic Motion - repeats itself at regular intervals (periodic).

Frequency - # of oscillations per second

1 oscillation / s = 1 hertz (Hz)

Period - time for one complete oscillation (one cycle)

T

T

Tf

1



Position T

ime



Angles are in radians.

2T

f 2

s

cyclescycle

radianss

radians

fT

1


Amplitude, Frequency & Phase

The frequency of SHM is independent of the amplitude.

txtx m cos

txtx m cos

txtx m 2cos

4cosmx t x t

m mx x


Velocity & Acceleration of SHM

In SHM, a(t) is proportional to x(t) but opposite in sign.

txtx m cos

txtv m sin

dt

dxtv

txta m cos2

txta 2

The phase of v(t) is shifted ¼ period relative to x(t),

mxv max

mxa 2max

dt

dvta


The Force Law for SHM

Simple Harmonic Motion is the motion executed by a particle of mass m subject to a force proportional to the displacement of the particle but opposite in sign.

“Linear Oscillator”: F ~ - x

F t k x t

SimpleHarmonicMotion/HorizSpring.html

Hooke’s Law:


The Differential Equation that Describes SHM

Simple Harmonic Motion is the motion executed by a particle of mass m subject to a force proportional to the displacement of the particle but opposite in sign. Hooke’s Law!

txktF

amF Newton’s 2nd Law:

The general solution of this differential equation is:

txtx m cos

022

2

2

2

xdt

xd

xkdt

xdm

m

k2where


What is the frequency?

k = 7580 N/m

m = 0.245 kg

f = ?


xm without m falling off?

m = 1.0 kg

M = 10 kg

k = 200 N/m

s = 0.40

Maximum xm without slipping



SimpleHarmonicMotion/HorizSpring.html

http://positron.ps.uci.edu/~dkirkby/music/html/demos/SimpleHarmonicMotion/SpringWithGraph.html

http://positron.ps.uci.edu/~dkirkby/music/html/demos/SimpleHarmonicMotion/SpringWithGraph.html


Vertical Spring Oscillations


Energy in Simple Harmonic Motion

UKE

xkUvmK

2

212

21

txktU

txktU

m

m

222

1

22

1

cos

cosk m 2

22

1 sin txmtK m

txktK m22

21 sin

212 mE k x constant


Energy in Simple Harmonic Motion

turningpoint

range of

motion

22

1 xkU

constant22

1 mxkE

turningpoint


Gravitational Pendulum

SimpleHarmonicMotion/pendulum

Simple Pendulum: a bob of mass m hung on an unstretchable massless string of length L.


Simple Pendulum

Simple Pendulum: a bob of mass m hung on an unstretchable massless string of length L.

a t x t

T

2

2

TLg

2SHM for small

2LmI

L F L F

I

mg L

I

g gsin

acceleration ~ - displacement

SHM


A pendulum leaving a trail of ink:


Physical Pendulum

A rigid body pivoted about a point other than its center of mass (com). SHM for small

Pivot Point

Center of Mass

h F h F

I

mg h

I

g gsin

acceleration ~ - displacement

SHM

a t x t

T

2

2

TI

mg h2

quick method to measure g


Angular Simple Harmonic Oscillator

Torsion Pendulum:

2I

T

m I

k

2

2

dI I

dt

Hooke’s Law

Spring:



Any Oscillating System:

“inertia” versus “springiness”

2m

Tk

T 2 inertia

springiness

TL

g2 2

IT

T

I

mg h2


SHM & Uniform Circular Motion

The projection of a point moving in uniform circular motion on a diameter of the circle in which the motion occurs executes SHM.

The execution of uniform circular motion describes SHM.

http://positron.ps.uci.edu/~dkirkby/music/html/demos/SimpleHarmonicMotion/Circular.html



radius = xm

x(t)

The reference point P’ moves on a circle of radius xm.

The projection of xm on a diameter of the circle executes SHM.

txtx m cos

tangle

UC Irvine Physics of Music Simple Harmonic Motion Applet Demonstrations



The reference point P’ moves on a circle of radius xm.

The projection of xm on a diameter of the circle executes SHM.

x(t) v(t) a(t)

radius = xm mxv

mxa 2

txtx m cos txtv m sin txta m cos2



The projection of a point moving in uniform circular motion on a diameter of the circle in which the motion occurs executes SHM.

Measurements of the angle between Callisto and Jupiter:

Galileo (1610)

earth

planet


Damped SHM

SHM in which each oscillation is reduced by an external force.

F k x

F bvD Damping Force

In opposite direction to velocity

Does negative work

Reduces the mechanical energy

Restoring Force

SHM


Damped SHM

netF m a

k x bv m a

differential equation

2

2

dx d xk x b m

dt dt


Damped Oscillations

md x

dtk x b

dx

dt

2

20

2nd Order Homogeneous Linear Differential Equation:

Eq. 15-41

Solution of Differential Equation: x t x e tm

b

mt

( ) cos

2

where: k

m

b

m

2

24

b = 0 SHM


Damped Oscillations

x t x e tm

b

mt

( ) cos

2

km

2

12

b

m

1 small damping2

b

m

b

mcritically damped

21 0

" "

21 0 " "2

boverdamped

m

“the natural frequency”

Exponential solution to the DE


Auto Shock Absorbers

Typical automobile shock absorbers are designed to produce slightly under-damped motion


Forced Oscillations

Each oscillation is driven by an external force to maintain motion in the presence of damping:

F td0 cos

d = driving frequency


Forced Oscillations

Each oscillation is driven by an external force to maintain motion in the presence of damping.

2nd Order Inhomogeneous Linear Differential Equation:

md x

dtk x m

dx

dtF td

2

2

2

0 cos

0 cos

net

d

F m a

k x bv F t m a

k

m

“the natural frequency”


Forced Oscillations & Resonance

= natural frequency


2nd Order Homogeneous Linear Differential Equation:

Steady-State Solution of Differential Equation:

x t x tm( ) cos

where:

xF

m b

b

m

m

d d

d

d

0

2 2 2 22 2

2 2

tan

md xdt

k x mdxdt

F td

2

2

2

0 cos

km



= natural frequency


When = d resonance occurs!

km

The natural frequency, , is the frequency of oscillation when there is no external driving force or damping.

less damping

more damping

x

F

m bm

d d

0

2 2 2 22 2


Oscillations


Resonance


Stop the SHM caused by winds on a high-rise building

The weight is forced to oscillate at the same frequency as the building

but 180 degrees out of phase.

400 ton weight mounted on a spring on a high floor of the Citicorp building in New York.



md xdt

k x mdxdt

F td

2

2

2

0 cos

tqCdt

dqR

dt

qdL dsin

1m2

2

E

Mechanical Systems

Electrical Systems

e.g. the forced motion of a mass on a spring

e.g. the charge on a capacitor in an LRC circuit

Physics 2111, 2008 Fundamentals of Physics 1 Chapter 13 Oscillations 1.Oscillations 2.Simple...

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