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

Springs as harmonic oscillators.

Fs = -k xSpring wants to push back to equilibrium

position.

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?

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”!

Pendula vs.

Springs

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==

Units of frequency!

mk

21

T1ƒ

π==

Units:1/s or Hz (Hertz!)

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

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

“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==

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

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°

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

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!

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=

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.