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