More Oscillations! (Today: Harmonic

Oscillators)

Movie assignment reminder!• Final due THURSDAY April 20

• Submit through eCampus

• Different rubric; remember to check iteven if you got 100% on your draft:

http://sarahspolaor.faculty.wvu.edu/home/physics-101

Needs to be typed! Can still write equations in by hand if you want (should be essay format, but calculations can be done like they are formatted, for instance, in the text book for derivations). Then scan and submit through eCampus

Springs as harmonic oscillators.

Fs = -k xSpring wants to push back to equilibrium

position.

Remember we spoke about hooke’s law for springs, and that all ideal springs, when pulled away from their resting position, will want to go back to that x=0 point, or that equilibrium point. But, if you pull a block attached to a spring outwards to an amplitude A, if there’s no friction that restoring force will ensure it oscillates indefinitely between A and -A. We spoke about this as harmonic oscillation. That’s what we’ll talk about today.

Graphing the Motion of Springs

A. B.

C. D.

Q113The paper moves at a constant speed underneath the pencil. If we were to

graph what we observe, what would the position versus time graph look like?

ANSWER: A

The periodic motion of a spring is called sinusoidal motion, since it follows a sine or cosine relation.

This periodic motion is Simple Harmonic Motion.

I wanted to remind you of this to say that we’re in part excited about springs because they are a nice way to introduce waves.

Today’s Main Ideas

• Simple harmonic oscillators.

• Position, velocity, and acceleration with time.

• Relation to pendulums.

Simple Harmonic Motion• Any vibrating system with F proportional to -x like

Hooke’s law (F=-kx) undergoes SHM

• Called a simple harmonic oscillator (SHO)

Examples: Spring; pendulum (for small amplitudes), a person on a swing, vibrating strings, sound (next few lectures), a car stuck in a ditch being ``rocked out”!

SHO: Just like a spring, the greater force you pull it away from equilibrium the larger restoring force it has. Think of examples; swinging, twanging a guitar string, etc.

Pendula vs.

Springs

We’re also going to talk about how simple harmonic motion relates to something rotating at constant angular velocity; you can see by these graphics if you project downward the shadow of the spinning ball on the circle, the pendulum, and the mass on the spring, you’d see their shadows all moving at the same displacement, velocity, and acceleration over time in the x dimension. And just like in the moving paper example before, they’d all be tracing out a sinusoidal wave pattern. Hence, all of these systems can be described by the topic we’re talking about today, simple harmonic motion.

Springs as harmonic oscillators.

Time it takes for one cycle

(“period”):

km2T π=

Note: not dependent on amplitude A!

Period and Frequency of a Spring

• Period

• Frequency

– The frequency, ƒ, is the number of complete cycles or vibrations per second

km2T π=

mk

21

T1ƒ

π==

Going to start by throwing some equations at you. Your book derives these. Think of frequency in units of REVS or CYCLES per second.

Units of frequency!

mk

21

T1ƒ

π==

Units:1/s or Hz (Hertz!)

Talk about things doing some number cycles per second. Dentist drill spins 700 times per second (700 Hz). I can spin at 1 time per second (1 Hz). You’ll note I’m talking about spinning here, and that certainly relates to this lecture on harmonic motion, because of the effect I noted before.

Side View of Circular Motion

mk

21

T1ƒ

π==

Motion around a circle as viewed from the side has a the same position dependence as a spring

ω

Bug

Ignore the mathy doodles on this figure and just think about what’s happening. Say you’re looking at the side of a record player with a bug sitting on the disk as it goes around. If you look from the side, that bug’s x position will appear to undergo simple harmonic motion. You will see it oscillate at a frequency equal to the number of revolutions per second of the disk.What is the bug’s angular velocity in radians per second?

mk

21

T1ƒ

π==

ω

Bug

For a spin rate of f = 100 Hz, (100/s or 100rev/s) what is the bug’s angular velocity, ω, in radians per second?

A. 100 B. 100/(2π) C. 2π(100) D. 2(100)

Q114

I wanted to give you an easy throw-back to units conversion that will come into play here, and ask… What is the bug’s angular velocity in radians per second?[See light board notes]

“Angular Frequency”

• Explicit definitions for SHM:• The frequency gives the number of cycles per second

• The angular velocity/speed (or angular frequency) gives the number of radians per second

mkƒ2 =π=ω

mk

21

T1ƒ

π==

We actually use angular velocity when discussing simple harmonic oscillators, and that’s because of that explicit link to circular motion.

Use of a reference circle allows a description of SHM over time!

• x is the position at time t• x varies between +A and -A

Motion as a Function of Time

[See light board notes]

Graphical Representation of Motion

• When x is a maximum or minimum, velocity is zero

• When x is zero, the speed is a maximum (slope of x)

• Acceleration vs. time is the slope the of velocity graph. When x is max in the positive direction, a is max in the negative direction

a

x

v

Summary of Formulas

tAx ωcos=

tAv ωω sin−=

tAa ωω cos2−=

mkAa =max

mkf == πω 2

Calculator Warning!

•What are the units of ω t ?Thus, your calculator will either need to be in radians to give the correct answer, or you need to convert ω t to degrees.

tAx ωcos=tAv ωω sin−=tAa ωω cos2−=

2π radians = 360°Make sure you’re putting the right units into the cos and sin functions!!! Your calculator should match your inputs! Make sure if you’re putting in degrees, your calculator is set to use degrees. Same goes for radians.

Calculator CheckYou’ve connected a 10kg mass to a horizontal spring on a frictionless surface, and pull it out to a distance 0.1m from its equilibrium point.

You then let go and see that it does one whole oscillation once every 5 seconds. What is its x position after 23s?

tAx ωcos=

tAv ωω sin−=

tAa ωω cos2−=mkf == πω 2

DANGER!!

[See light board notes]

Simple Pendulum Compared to a Spring

The Simple PendulumGravity causes restoring force for oscillations. If θ is small (small

amplitude oscillations):

xLmg

Fpendulum −=

What causes it to swing back and forth?

Fs = -k xShould look familiar!

Pendulum = Simple Harmonic Motion

xLmg

Fpendulum −=

Restoring force is proportional to negative of displacement (Fspring= -kx)

Effective “spring constant” is keff = mg/L

effspring k

mT π2=gLTpendulum π2=

The period of simple pendulum is independent of mass or amplitude; instead depends ONLY

on the length of cord!Show period lack of dependence on mass with a few different masses

For a pendulum clock, the timing mechanism is designed by adjusting L

A simple pendulum has mass 2 kg and length 1 m. What is the period of the pendulum?

A) 2.0 sB) 2.8 sC) 4.4 sD) 8.9 sE) 19.7 s

Q115

effspring k

mT π2=gLTpendulum π2=

A

Note: Damped OscillationsWhy does a child stop swinging if

not continuously pushed?When work is done by a dissipative force (friction or air resistance), not all of the mechanical energy is conserved.

This means not all of her potential energy at the top of each swing is converted into kinetic energy so her next swing is not as high.

The period of oscillations stays the same. The amplitude decreases with time.