1. Simple Harmonic Motion (S.H.M.)

2. S.H.M.
Definition
Properties
Forced Oscillation
Resonance

3. 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.

4. Definition
acceleration
a µ -x
displacement
Equil. position a a a a

5. Mathematical Expression
a µ -x
i.e. a = - w2 x
where w2 is a +ve const.

6. Example 1
Mass-Spring System a a a a
Equil. position

7. Example 2 a a a a
Simple Pendulum
Equil. position

8. Example 3 a
Floating Cylinder a a a
Equil. position

9. 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).

10. 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 .

11. 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]

12. Displacement-Time Graphx
x = A cos wt A t -A

13. Velocity-Time Graph v
v = - wA sin wt wA t
- wA

14. Acceleration-Time Graph
a = - w2A cos wt
w2A t
-w2A

15. Velocity-Displacement Graph
v = ± w (A2 - x2 )0.5 v wA t -A A
- wA

16. Acceleration-Displacement Graph
a = -w2x [the definition] a
w2A x -A A
-w2A

17. Phase Relationship x v a t

18. 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.

19. 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 :

20. 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.

21. Example (Mass-Spring System)
Periodic driving
force of freq. f
Oscillating with
natural freq. fo

22. 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.

23. Example 1
Breaking Glass
System : glass
Driving Force :
sound wave

24. Example 2 Collapse of the Tacoma Narrows
suspension bridge in America in 1940
System : bridge
Driving Force :
strong wind

25. Credits Projector Leader : Kok Tak Wing Members : Wan Chun Kong
Lam Mo Kit