s imple h armonic m otion ( s.h.m.)
DESCRIPTION
S imple H armonic M otion ( S.H.M.). S.H.M. Definition Properties Forced Oscillation Resonance. Definition. So...?. Simple Harmonic Motion is a linear motion such that :. 1. its acceleration is directly proportional to its displacement from a fixed point - PowerPoint PPT PresentationTRANSCRIPT
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Simple Harmonic Motion (S.H.M.)
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S.H.M.
• Definition
• Properties
• Forced Oscillation
• Resonance
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Definition
Simple Harmonic Motion is a linear motionsuch that :
1. its acceleration is directly proportional to its displacement from a fixed point (the equilibrium position),
2. its acceleration always points towards the fixed point.
So...?
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Equil. position
Definition acceleration
displacement
0
a a a a
a x
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Mathematical Expression
a x
i.e. a x
where is a +ve const.
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Example 1
Mass-Spring System
aaaa
Equil. position
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Example 2
Simple Pendulum
aaa a
Equil. position
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aExample 3
Floating Cylindera
Equil. position
aa
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Notes
1. The acceleration is due to the resultant force acting.
2. The system will oscillate when disturbed. The maximum displacement is called the amplitude (A).
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Mathematical Derivations
a = x where is a constant
……... integrating………
……... integrating ………
Definition :
We obtain another four equations ofmotion involving a , v , x and t .
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Equations of Motion (SHM)
a = x [the definition]
x = Acos t
v = A sin t
a = A cos t
v = ± A x)0.5
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Displacement-Time Graph
x
t0
x = Acos tA
-A
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Velocity-Time Graph
v
t0
v = A sin tA
A
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Acceleration-Time Graph
a
t0
a = A cos tA
A
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Velocity-Displacement Graph
vv = ± A x)0.5
A
A
A-At0
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Acceleration-Displacement Graph
a
a = x [the definition]
A
A
A-Ax0
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Phase Relationship
0
x
v a
t
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Properties
1. S.H.M. is an oscillatory and periodic motion.
2. The time required for one complete oscillation is called the period.
3. The period is independent of the amplitude for a given system.
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Natural Frequency
When a system is disturbed, it willoscillate with a frequency which is calledthe natural frequency ( fo ) of the system.
e.g. for a mass-spring system :
m
kfo
2
1
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Forced Oscillation
When a system is disturbed by a periodicdriving force and then oscillate, this iscalled forced oscillation.
Note : The system will oscillate with its natural frequency ( fo ) which is independent of the frequency of the driving force.
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Example (Mass-Spring System)
Periodic drivingforce of freq. f
Oscillating withnatural freq. fo
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Resonance
When a system is disturbed by a periodicdriving force which frequency is equal tothe natural frequency ( fo ) of the system,the system will oscillate with LARGEamplitude.
Resonance is said to occur.
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Example 1
Breaking Glass
System : glass
Driving Force : sound wave
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Example 2
Collapse of the Tacoma Narrowssuspension bridge in America in 1940
System : bridge
Driving Force : strong wind