Simple Harmonic Motion (S.H.M.) S.H.M. Definition Properties Forced Oscillation Resonance.
-
Upload
christian-mcbride -
Category
Documents
-
view
227 -
download
0
Transcript of Simple Harmonic Motion (S.H.M.) S.H.M. Definition Properties Forced Oscillation Resonance.
Simple Harmonic Motion (S.H.M.)
S.H.M.
• Definition
• Properties
• Forced Oscillation
• Resonance
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...?
Equil. position
Definition acceleration
displacement
0
a a a a
a x
Mathematical Expression
a x
i.e. a x
where is a +ve const.
Example 1
Mass-Spring System
aaaa
Equil. position
Example 2
Simple Pendulum
aaa a
Equil. position
aExample 3
Floating Cylindera
Equil. position
aa
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).
Mathematical Derivations
a = x where is a constant
……... integrating………
……... integrating ………
Definition :
We obtain another four equations ofmotion involving a , v , x and t .
Equations of Motion (SHM)
a = x [the definition]
x = Acos t
v = A sin t
a = A cos t
v = ± A x)0.5
Displacement-Time Graph
x
t0
x = Acos tA
-A
Velocity-Time Graph
v
t0
v = A sin tA
A
Acceleration-Time Graph
a
t0
a = A cos tA
A
Velocity-Displacement Graph
vv = ± A x)0.5
A
A
A-At0
Acceleration-Displacement Graph
a
a = x [the definition]
A
A
A-Ax0
Phase Relationship
0
x
v a
t
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.
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
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.
Example (Mass-Spring System)
Periodic drivingforce of freq. f
Oscillating withnatural freq. fo
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.
Example 1
Breaking Glass
System : glass
Driving Force : sound wave
Example 2
Collapse of the Tacoma Narrowssuspension bridge in America in 1940
System : bridge
Driving Force : strong wind
Credits
Projector Leader : Kok Tak Wing
Members : Wan Chun Kong Lam Mo Kit