chapter 13 oscillatory motion. periodic motion periodic (harmonic) motion – self-repeating motion...
TRANSCRIPT
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Chapter 13
Oscillatory Motion
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Periodic motion
• Periodic (harmonic) motion – self-repeating motion
• Oscillation – periodic motion in certain direction
• Period (T) – a time duration of one oscillation
• Frequency (f) – the number of oscillations per unit time, SI unit of frequency 1/s = Hz (Hertz)
Tf
1
Heinrich Hertz(1857-1894)
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Simple harmonic motion
• Simple harmonic motion – motion that repeats itself and the displacement is a sinusoidal function of time
)cos()( tAtx
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Amplitude
• Amplitude – the magnitude of the maximum displacement (in either direction)
)cos()( tAtx
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Phase
)cos()( tAtx
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Phase constant
)cos()( tAtx
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Angular frequency
)cos()( tAtx
)(coscos TtAtA 0
)2cos(cos )(cos)2cos( Ttt
T 2
T
2
f 2
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Period
)cos()( tAtx
2
T
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Velocity of simple harmonic motion
)cos()( tAtx
dt
tdxtv
)()(
)sin()( tAtv
dt
tAd )]cos([
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Acceleration of simple harmonic motion
)cos()( tAtx
2
2 )()()(
dt
txd
dt
tdvta
)()( 2 txta
)cos(2 tA
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Chapter 13Problem 19
Write expressions for simple harmonic motion (a) with amplitude 10 cm, frequency 5.0 Hz, and maximum displacement at t = 0, and (b) with amplitude 2.5 cm, angular frequency 5.0 s-1, and maximum velocity at t = 0.
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The force law for simple harmonic motion
• From the Newton’s Second Law:
• For simple harmonic motion, the force is proportional to the displacement
• Hooke’s law:
maF
kxF
xm 2
m
k
k
mT 22mk
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Energy in simple harmonic motion
• Potential energy of a spring:
• Kinetic energy of a mass:
2/)( 2kxtU )(cos)2/( 22 tkA
2/)( 2mvtK )(sin)2/( 222 tAm
)(sin)2/( 22 tkA km 2
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Energy in simple harmonic motion
)(sin)2/()(cos)2/( 2222 tkAtkA
)()( tKtU
)(sin)(cos)2/( 222 ttkA
)2/( 2kA )2/( 2kAKUE
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Energy in simple harmonic motion
)2/( 2kAKUE
2/2/2/ 222 mvkxkA kmvxA /222
22 xAm
kv 22 xA
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Chapter 13Problem 34
A 450-g mass on a spring is oscillating at 1.2 Hz, with total energy 0.51 J. What’s the oscillation amplitude?
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Pendulums
• Simple pendulum:
• Restoring torque:
• From the Newton’s Second Law:
• For small angles
)sin( gFL
I
sin
I
mgL
)sin( gFL
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Pendulums
• Simple pendulum:
• On the other hand
L
at
I
mgL
L
s s
I
mgLa
)()( 2 txta
I
mgL
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Pendulums
• Simple pendulum:
I
mgL 2mLI
2mL
mgL
L
g
g
LT
22
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Pendulums
• Physical pendulum:
I
mgh
mgh
IT
22
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Chapter 13Problem 28
How long should you make a simple pendulum so its period is (a) 200 ms, (b) 5.0 s, and (c) 2.0 min?
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Simple harmonic motion and uniform circular motion
• Simple harmonic motion is the projection of uniform circular motion on the diameter of the circle in which the circular motion occurs
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Simple harmonic motion and uniform circular motion
• Simple harmonic motion is the projection of uniform circular motion on the diameter of the circle in which the circular motion occurs
)cos()( tAtx
dt
tdxtvx
)()(
)sin()( tAtvx
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Simple harmonic motion and uniform circular motion
• Simple harmonic motion is the projection of uniform circular motion on the diameter of the circle in which the circular motion occurs
dt
tdxtvx
)()(
)sin()( tAtvx
)cos()( tAtx
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Simple harmonic motion and uniform circular motion
• Simple harmonic motion is the projection of uniform circular motion on the diameter of the circle in which the circular motion occurs
2
2 )()(
dt
txdtax
)cos()( tAtx
)cos()( 2 tAtax
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Damped simple harmonic motion
bvFb Dampingconstant
Dampingforce
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Forced oscillations and resonance
• Swinging without outside help – free oscillations
• Swinging with outside help – forced oscillations
• If ωd is a frequency of a driving force, then forced
oscillations can be described by:
• Resonance:
)cos(),/()( tbAtx dd
d
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Questions?
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Answers to the even-numbered problems
Chapter 13
Problem 200.15 Hz; 6.7 s
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Answers to the even-numbered problems
Chapter 13
Problem 3865.8%; 76.4%