Simple Harmonic Motion S'H'M'

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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
So...?
Simple Harmonic Motion is a linear motion such
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.
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Definition
acceleration
a µ -x
displacement
0
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Mathematical Expression
a µ -x
i.e. a - w2 x where w2 is a ve const.
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Example 1
Mass-Spring System
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Example 2
Simple Pendulum
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Example 3
Floating Cylinder
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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
Definition
a -w2x where w2 is a constant
... integrating
... integrating
We obtain another four equations of motion
involving a , v , x and t .
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Equations of Motion (SHM)
x A cos wt
v - wA sin wt
a - w2A cos wt
v w (A2 - x2 )0.5
a -w2x the definition
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Displacement-Time Graph
x
x A cos wt
A
t
0
-A
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Velocity-Time Graph
v
v - wA sin wt
wA
t
0
- wA
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Acceleration-Time Graph
a
a - w2A cos wt
w2A
t
0
-w2A
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Velocity-Displacement Graph
v w (A2 - x2 )0.5
v
wA
t
0
A
-A
- wA
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Acceleration-Displacement Graph
a -w2x the definition
a
w2A
0
x
A
-A
-w2A
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Phase Relationship
t
0
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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 will oscillate
with a frequency which is called the natural
frequency ( fo ) of the system.
e.g. for a mass-spring system
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Forced Oscillation
When a system is disturbed by a periodic driving
force and then oscillate, this is called 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 driving force of freq. f
Oscillating with natural freq. fo
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Resonance
When a system is disturbed by a periodic driving
force which frequency is equal to the natural
frequency ( fo ) of the system, the system will
oscillate with LARGE amplitude. 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 Narrows suspension bridge
in America in 1940
System bridge
Driving Force strong wind
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Credits
Projector Leader Kok Tak Wing
Members Wan Chun Kong Lam Mo Kit