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Vibration & Waves: A Detailed Study in Physics Dynamics

Dive into the world of vibration and waves physics, from harmonic motion to wave interference. Learn about oscillators, conservation of energy, harmonic oscillations, and more. Explore the principles of transverse and longitudinal waves, frequency, amplitude, and wave interference.

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Vibration & Waves: A Detailed Study in Physics Dynamics

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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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