oscillations the force and motion of springs harmonically

OSCILLATIONS

Springs: supplying restoring force• When you pull on (stretch) a

spring, it pulls back (top picture)• When you push on (compress) a

spring, it pushes back (bottom)• Thus springs present a restoring

force:F = kx

• x is the displacement (in meters)

• k is the “spring constant” in Newtons per meter (N/m)

• the negative sign means opposite to the direction of displacement

Linear Restoring Forces and Simple Harmonic Motion

Equilibrium and Oscillation

The time to complete one full cycle of

oscillation is a Period.

T 1f

f 1T

The amount of oscillations per second is called frequency and is measured in Hertz.

Frequency and PeriodThe frequency of oscillation depends on physical properties of the oscillator; it does not depend on the amplitude of the oscillation.

Frequency of Oscillation

• Mass will execute some number of cycles per second (could be less than one)

• This is the frequency of oscillation (measured in Hertz, or cycles per second)

• The frequency is proportional to the square root of the spring constant divided by the mass.

• Larger mass means more sluggish (lower freq.)• Larger (stiffer) spring constant means faster (higher freq.)

It is easy to show that

is a more general solution of the equation of motion. The symbol is called the phase. It defines the

initial displacementx = A cos

Simple Harmonic MotionPhase

.

cos( )Ax t

Simple Harmonic MotionPosition, Velocity, Acceleration

cos( )Ax t

sin( )Av t

2 cos( )Aa t

Position

Velocity

Acceleration

cos( )Ax t

sin( )Av t

2 cos( )Aa t

Simple Harmonic MotionPosition, Velocity, Acceleration

Sinusoidal Relationships

Mathematical Description of Simple Harmonic Motion

SHM and Circular Motion

x(t) Acos

ddt

t

x(t) Acos t

t 0

x(t) Acos t 0

vx (t) Asin t 0

vx (t) vmax sin t 0

t 0

Energy in Simple Harmonic MotionAs a mass on a spring goes through its cycle of oscillation, energy is transformed from potential to kinetic and back to potential.

Energy Storage in Spring

• Applied force is kΔx (reaction from spring is kΔx)• starts at zero when Δx = 0• slowly ramps up as you push

• Work is force times distance• Let’s say we want to move spring a total distance of x

• would naively think W = kx2

• but force starts out small (not full kx right away) • works out that W = ½kx2

The Pendulum

Fnet t mgsin ma t

d2sdt 2 gsin

s L

The Pendulum

d2sdt 2

gsL

gL

(t) max cos t 0

x(t) Acos t 0

A Pendulum Clock

What length pendulum will have a period of exactly 1s?

gL

T 2 Lg

g T2

2

L

L 9.8m/s2 1s2

2

0.248m

A pendulum leaving a trail of ink

Damping

Resonance

Resonance• If you apply a periodic force to a system at

or near its natural frequency, it may resonate• depends on how closely the frequency

matches• damping limits resonance

• Driving below the frequency, it deflects with the force

• Driving above the frequency, it doesn’t do much at all

• Picture below shows amplitude of response oscillation when driving force changes frequency

Resonance Examples• Shattering wine glass

• if “pumped” at natural frequency, amplitude builds up until it shatters

• Swinging on swingset• you learn to “pump” at natural

frequency of swing• amplitude of swing builds up

• Tacoma Narrows Bridge• eddies of wind shedding of top

and bottom of bridge in alternating fashion “pumped” bridge at natural oscillation frequency

• totally shattered• big lesson for today’s bridge

builders: include damping

Example – ResonanceNovember 7, 1940 – Tacoma Narrows Bridge Disaster. At about 11:00 am the Tacoma Narrows Bridge, near

Tacoma, Washington collapsed after

hitting its resonant

frequency. The external driving force was the wind.

Resonance Resonance Applications:Applications:

Physics 201, Fall 2006, UW-Madison

Extended objects have more than one resonance frequency. When plucked, a guitar string transmits its energy to the body of the guitar. The body’s oscillations, coupled to those of the air mass it encloses, produce the resonance patterns shown.

Stop the SHM caused by winds on a high-rise

building

The weight is forced to oscillate at the same frequency as the buildingbut 180 degrees out of phase.

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