Simple Harmonic Motion (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 (the equilibrium position), 2. its acceleration always points towards the fixed point.

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

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

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

Example 2 a a a a Simple Pendulum Equil. position

Example 3 a Floating Cylinder a a a Equil. position

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 Definition : a = -w2x where w2 is a constant ……... integrating……… ……... integrating ……… We obtain another four equations of motion involving a , v , x and t .

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]

Displacement-Time Graph x x = A cos wt A t -A

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

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

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

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

Phase Relationship 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 will oscillate with a frequency which is called the natural frequency ( fo ) of the system. e.g. for a mass-spring system :

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.

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

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.

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

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

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