Chapter 13 Oscillatory Motion. Periodic motion Periodic (harmonic) motion – self-repeating motion Oscillation – periodic motion in certain direction Period.
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Chapter 13
Oscillatory Motion
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)
Simple harmonic motion
• Simple harmonic motion – motion that repeats itself and the displacement is a sinusoidal function of time
)cos()( tAtx
Amplitude
• Amplitude – the magnitude of the maximum displacement (in either direction)
)cos()( tAtx
Phase
)cos()( tAtx
Phase constant
)cos()( tAtx
Angular frequency
)cos()( tAtx
)(coscos TtAtA 0
)2cos(cos )(cos)2cos( Ttt
T 2
T
2
f 2
Period
)cos()( tAtx
2
T
Velocity of simple harmonic motion
)cos()( tAtx
dt
tdxtv
)()(
)sin()( tAtv
dt
tAd )]cos([
Acceleration of simple harmonic motion
)cos()( tAtx
2
2 )()()(
dt
txd
dt
tdvta
)()( 2 txta
)cos(2 tA
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.
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
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
Energy in simple harmonic motion
)(sin)2/()(cos)2/( 2222 tkAtkA
)()( tKtU
)(sin)(cos)2/( 222 ttkA
)2/( 2kA )2/( 2kAKUE
Energy in simple harmonic motion
)2/( 2kAKUE
2/2/2/ 222 mvkxkA kmvxA /222
22 xAm
kv 22 xA
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?
Pendulums
• Simple pendulum:
• Restoring torque:
• From the Newton’s Second Law:
• For small angles
)sin( gFL
I
sin
I
mgL
)sin( gFL
Pendulums
• Simple pendulum:
• On the other hand
L
at
I
mgL
L
s s
I
mgLa
)()( 2 txta
I
mgL
Pendulums
• Simple pendulum:
I
mgL 2mLI
2mL
mgL
L
g
g
LT
22
Pendulums
• Physical pendulum:
I
mgh
mgh
IT
22
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?
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
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
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
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
Damped simple harmonic motion
bvFb Dampingconstant
Dampingforce
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
Questions?
Answers to the even-numbered problems
Chapter 13
Problem 200.15 Hz; 6.7 s
Answers to the even-numbered problems
Chapter 13
Problem 3865.8%; 76.4%
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