1. Chapter (3)
Oscillations

2. Oscillations Mechanical Nonmechanical oscillation oscillation
Simple Harmonic Oscillation
Damped Harmonic Oscillation
Forced Harmonic Oscillation

3. Periodical Motion Amplitude A Period T Frequency F=1/T
Angular frequency ω = 2πF
Phase (ωt+φ)
Phase constant φ
X(t)=A sin ωt at t=0, x=0
X(t)=A sin (ωt+φ) at t=0, x≠0 A T φ

4. Simple Harmonic Oscillatorφ
Simple Harmonic Oscillator has the following characteristics:
X(t)=A sin (ωt+φ)
A is constant
f and T is independent of A

5. Displacement, Velocity, acceleration
X(t)=A sin(ωt+φ)
v(t)= ωA cos(ωt+φ)
a(t)= -ω2A sin(ωt+φ)
a(t)= -ω2 X(t)

6. Simple Harmonic Motion (SHM)
a(t)= -ω2 X(t) or
d2x/dt2 + ω2x(t)= 0
For SHM to occur, three conditions must be satisfied 1) there must be a position of equilibrium. 2) there must be no dissipation of energy. 3) the acceleration is proportional to X and opposite direction.

7. Hook’s law and Simple Harmonic Motion
F=-Kx, F= ma
 -kx= ma a=-(k/m) x
a= -ω2 X
ω2 =(k/m) or  

8. Energy conservation in SHM
In the absence of friction, the energy of the block-spring system is constant. Potential energy kinetic energy Since ω2 =(k/m) and sin2θ+cos2θ=1 total energy E=K+U=

9. The total energy of any SHM is constant and proportional to A2x U K
E=K+U -A A
energy t
E/2

10. Example of linear and angular SHM
Simple Pendulum
F=-mg sinθ,
for small θ, sinθ  θ x/L
F=-mgx/L = -(mg/L)x =-kx m L x mg
mg sinθ θ
mg cosθ T

11. Torsional Pendulum F  ζ (torque) x  θ (angular disp.)
m  I (moment of inertia)
k  k (torsional const.)
Thus, Hooke’s law takes the form ζ=-k θ M F