Vibration and waves

A restoring force always pushes or pulls the object toward the equilibrium position Simple harmonic motion -occurs when the net force is proportional to the displacement from the equilibrium point and is always directed toward the equilibrium point

The amplitude A – is the maximum distance of the object from its equilibrium position The period T- is the time it takes the object to move through one compete cycle of motion ( from x=A to x=-A and back to x=A) The frequency f is the number of complete cycles or vibrations per unit of time (f=1/T) The harmonic oscillator equation: a=(-k/m)x A ranges over the values –kA/m and +kA/m

Elastic potential energy: PEs=1/2kx2 Conservation of energy: (KE +PEg+PEs)i= (KE +PEg+PEs)f

Velocity as a Function of position ½ kA2+ 1/2mv2+1/2kx2 v=±√k/m(A2-x2)

Comparing harmonic motion with circular motion: v= C√ (A2-x2) sinθ=v/vo sin θ= (√A2-x2)/A v/vo=(√A2-x2)/A v= vo(√A2-x2)/A =C √ (A2-x2)

Period and frequency vo=2πA/T T=2πA/vo Conservation of energy1/2kA2=1/2mv2 A/vo =√m/k T=2π√m/k f= (1/2π)√k/m The angular frequency ω=2πf =√k/m

Position, velocity and acceleration as a fct. of time x=Acosθ θ =ωt ω=Δθ/Δt =2π/T= 2πf x= Acos(2πft) v=-Aωsin(2πft) ω=√k/m a=-Aω2cos(2πft)

Motion of a pendulum Ft=-mg sinθ =-mg θ Ft= mg sin(s/L) Ft=-(mg/L)s Ft=-kx k= mg/L ω=2πf= √k/m ω=√mgL/m=√g/L T= 2π√g/L

Simple harmonic motion for an object-spring system, and its analogy, the motion of a simple pendulum

Wave-the motion of a disturbance Transverse waves- each segment of the rope that is disturbed moves in a direction perpendicular to the wave motion Longitudinal waves- the elements of the medium undergo displacements parallel to the direction of wave motion (sound waves)

Frequency, Amplitude and wavelength: v=Δx/Δt Δx=λ; Δt=T v= λ/T v=f λ The wavelength λ- is the distance between 2 successive points that behaves identically The speed of waves on strings: v=√F/μ (F-tension, μ- linear density-mas of the string/unit length

Interference of waves Two traveling waves can meet and pass through each other without being destroyed or even altered The superposition principle: when 2 or more traveling waves encounter other while moving through a medium, the resultant wave is found by adding togherther the displacements of the individual waves point by point (constructive interference or destructive interference)