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Vibration and waves

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.

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Vibration and waves

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  1. Vibration and waves

  2. 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

  3. 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

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

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

  6. 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)

  7. 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

  8. 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)

  9. 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

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

  11. 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)

  12. 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

  13. 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)

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