Microsoft PowerPoint - Lecture Notes 1 - 213 (Oscillations)

7/29/2019 Microsoft PowerPoint - Lecture Notes 1 - 213 (Oscillations)

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Oscillations about equilibrium(Synonym:Vibration)

What means Oscillation?Oscillation is the periodic variation, typically in time, of some measure

as seen, for example, in a swinging pendulum.

Many things oscillate/vibrate: Periodic motion

(a motion that repeats itself over and over)Pulse Oscillations are the origin of thesensation of musical tone

.. in Aerospace: OrbitsElectrical/Computer: LRC resonance in circuits

Physics:Atomic Vibrations, String Theory, Electromagnetic Waves

Why does something vibrate/oscillate?

Whenever the system is displaced from equilibrium, a restoring force pulls it back,but it overshoots the equilibrium position.


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more examples

.heart beat, breathing, sleeping, taking shower, eating, chewing, blinking, drinking

.motion of planets, stars, motion of electrons, atoms

.wind (Tacoma Bridge)

.vocal cords, ear drums


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Parameter used to describe vibrations

Period T Time taken to complete one

cycle of the vibration. Units: s

Frequency f = 1/T

Number of vibration cycles per

second. Units: 1/s (Hz, Hertz)

Amplitude A Maximumdisplacement from equilibrium

position

One cycle is time take for a pendulum:

center right center left

center

Pendulum

T


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Special cases of periodic motion

Simple harmonic motion (SHM)

occurs when the restoring force

(the force directed toward a stable

equilibrium point) is proportional tothe displacement from equilibrium.

For instance when the restoringforce is F = - k x.(Hooks Law)


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A

Period T

Simple harmonic motionThe displacement from equilibrium can be describes as a

cosinusoidal function

0

0

2

2

2

2

2

2

2

=+

=+

=

xtx

xm

k

t

x

kxt

x

m

Equation of motion

for SHM


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Simple harmonic motion is the projection of circular

motion on the x-axis

Angular velocity

is NOT necessarily

the same as

Angular frequency


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Displacement, Velocity and Acceleration of SHM

( )

( )

( ) ( )txtAdt

tdvta

tAdttdxtv

tAtx

22 )cos()(

)sin()(

)cos(

=+==

+==

+=

A is the amplitude of the

motion, the maximum

displacement from

equilibrium, A=vmax, andA2 =amax.

Mass-Spring Java applet


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Mass-Spring-SystemA Spring always pushes or pulls mass back towards equilibrium

position. The time period can be calculated from Hookes Law:

k

mTor

m

k

tAktAm

kxmamaFkxF

2

)]cos([)]cos([

2

2

==

=

===

Independent from amplitude!

(Application: measure the mass of astronauts in

space) Heavier mass slower oscil lationsStiffer spring (greater k) rapid oscillations

The period of oscillation is

21==

fT


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Mass Spring-System in vertical setup

When a mass-spring system is oriented

vertically, it will exhibit SHM with the sameperiod and frequency as a horizontally

placed system.

Same formulae as for the horizontal setup

but the system oscillates around a newequilibrium position y0.

y0 = mg/k


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Energy: E = U + K

U: Potential Energy

K: Kinetic Energy

SpringEpot = U = k x

2

Ekin = K = m v2

Turning points:

E = Umax + 0

(Displacement and U at maximum)Minimum:

E = Kmax + 0 (Velocity and K at maximum)

Total energy of system

E = U + K = k (A cos(t))2

+ m (A sin(t))2

= k A2 cos2(t) + m A2 2 sin2(t)

E = k A2 cos2(t) + m A2 k/m sin2(t) = k A2 (cos2(t)+

sin2(t))

And therefore:

E = k A2

Potential Energy of simple harmonicmotion

Energy Conservation in Oscillatory Motion


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Exercise 1: The period of oscillation of an object in an ideal

mass-spring system is 0.50 sec and the amplitude is 5.0 cm.

What is the speed at the equilibrium point?

At equilibrium x=0:2222

2

1

2

1

2

1

2

1mvkAkxmvUKE ==+=+=

Since E=constant, at equilibrium (x = 0) the KE must be

a maximum. Here v = vmax = A.

( )( )cm/sec8.62rads/sec6.12cm5.0and

rads/sec6.12s50.0

22

===

===

Av

T

The amplitude A is given, but is not.


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Exercise 2: The diaphragm of a speaker has a mass of 50.0 g

and responds to a signal of 2.0 kHz by moving back and forthwith an amplitude of 1.810-4 m at that frequency.

(a) What is the maximum force acting on the diaphragm?

( ) ( ) 2222maxmax 42 mAffmAAmmaFF =====The value is Fmax=1400 N.

(b) What is the mechanical energy of the diaphragm?

Since mechanical energy is conserved, E = KEmax = Umax.

2

maxmax

2

max

21

2

1

mvKE

kAU

=

=The value of k is unknown so use KEmax.

( ) ( )

2222

maxmax2

2

1

2

1

2

1fmAAmmvKE ===

The value is KEmax= 0.13 J.


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Exercise 3: The displacement of an object in SHM is given

by:( ) ( ) ( )[ ]tty rads/sec57.1sincm00.8=

What is the frequency of the oscillations?

Comparing to y(t)= A sint givesA = 8.00 cm and = 1.57 rads/sec.The frequency is:

Hz250.02

rads/sec57.1

2===

f

( )( )

( )( )

222

max

max

max

cm/sec7.19rads/sec57.1cm00.8

cm/sec6.12rads/sec57.1cm00.8

cm00.8

===

===

==

Aa

Av

Ax

Other quantities can also be determined:

The period of the motion is sec00.4rads/sec57.1

22===

T


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A torsional pendulum is an oscillator for which

the restoring force is torsion. For example,suspending a barfrom a thin wire and winding it

by an angle , a torsional torque

is produced, where is a characteristic property of the

wire, known as the torsional constant. Therefore, the

equation of motion is

where I is the moment of inertia. But this is just a

simple harmonic oscillatorwith equation of

motion

where is the initial angle,

is the angular frequency, and is the phase constant.


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PendulumA mass, called a bob, suspended from a fixed point so that it can

swing in an arc determined by its momentum and the force of gravity.The length of a pendulum is the distance from the point of

suspension to the center of gravity of the bob. Chance observation of

a swinging church lamp led Galileo to find that a pendulum made

every swing in the same time, independent of the size of the arc. He

used this discovery in measuring time in his astronomical studies. His

experiments showed that the longer the pendulum, the longer is thetime of its swing.

If we assume the angle is small, for then we can approximate sin with (expressed in radian measure). (As an example, if = 5.00 =

0.0873 rad, then sin = 0.0872, a difference of only about 0.1%.)With that approximation and some rearranging, we then have

02

2

=+

I

mgL

t

Physical Pendulum,

Small amplitude


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UPendulum = mgh = mgL (1-cos)

For smaller displacements, the movement of

an ideal pendulum can be described

mathematically as simple harmonic motion

(like the mass-spring), as the change in

potential energy at the bottom of a circular arc

is nearly proportional to the square of the

displacement. Real pendulums do not have

infinitesimal displacements, so their behaviour

is actually of a non-linear kind.


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The Physical Pendulum

A "physical" pendulum hasextended size and is a

generalization of the bob pendulum.

An example would be a bar rotating

around a fixed axle. A simple

pendulum can be treated as a

special case of a physical pendulum

with moment of inertia I. ( I = miri2)

Period of a physical pendulum

(Note: l is now the length from thesuspension point to the center of

mass CM instead of L)

Simple Pendulum

All the mass of a simplependulum is concentrated in the

mass m of the particle-like bob,

which is at radius L from the

pivot point. Thus, we cansubstitute I = mL2 for the

rotational inertia of the

pendulum.

gLT 2=

for small amplitudes!!

Example:

Simple Pendulum: I = mL2Leg: I = 1/3 mL2


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Exercise 4:A clock has a pendulum that performs one full

swing every 1.0 sec. The object at the end of the stringweights 10.0 N. What is the length of the pendulum?

( )( ) m25.04s0.1m/s8.9

4L

2

2

22

2

2

===

=

gT

gLT

Solving for L:


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CM

Pivot

mg

Length: L

Mass: M

ICM= 1/12 ML2

Parallel-Axis TheoremIPivot= 1/12 ML

2 + M ( L)2 = 1/3 ML2

g

L

LgM

ML

gMl

IT

3

22

)

2

1(

3

1

22

2

===

The period is


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Ring

CM

r

IPivot= Mr2 + Mr2 = 2Mr2

g

rT

gMr

Mr

gMl

IT

22

222

2

=

==

Disc

CM

r

IPivot= Mr2 + Mr2 = 3/2 Mr2

g

rT

gMr

Mr

gMl

IT

2

32

2

3

22

2

=

==


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http://lectureonline.cl.msu.edu/%7Emmp/applist/damped/d.htm

http://physics.usask.ca/~pywell/p121/Images/tacoma.avi

http://www.walter-fendt.de/ph14e/resonance.htm

Microsoft PowerPoint - Lecture Notes 1 - 213 (Oscillations)

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