Date post: | 13-Jan-2017 |
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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.