Physics 11: Vibrations and Waves

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Physics 11 Vibrations and Waves

• Christopher Chui

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Simple Harmonic Motion (SHM)

• Any spring has a natural length at which it
  exerts no force on the mass is called equilibrium
• If stretched, the restoring force F -kx,
  called SHM
• The stretched distance, x, is displacement
• The max displacement is called amplitude, A
• One cycle is one complete to-and-fro (-A to A)
  motion
• Period, T, is the time for one complete cycle
• Frequency, f, is the number of complete cycles in
  one second. T 1/f and f 1/T

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Energy in SHO

• PE ½ kx2 k is called the spring constant
• Total mechanical energy, E ½ mv2 ½ kx2
• At the extreme points, E ½ kA2
• At the equilibrium point, E ½ mvo2 vo is max
• Using conservation of energy, we find at any
  time, the velocity v - vo sqrt(1 x2/A2)

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The Period and Sinusoid of SHM

• The period does not depend on the amplitude
• For a revolving object making one revolution, vo
  circumference / time 2pA / T 2pAf
• Since ½ kA2 ½ mvo2, T 2p sqrt(m/k)
• Since f1/T, f 1/(2p) sqrt(k/m)
• x Acos q Acos wt Acos 2pft Acos 2pt/T
• v -vo sin 2pft -vo sin 2pt/T
• A F/m -kx/m -kA/m cos 2pft -aocos2pft

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The Simple Pendulum of length L

• The restoring force, F - mg sin q
• For small angles, sin q is approx to q
• F -mg q -mg x/L -kx, where k mg/L
• The period, T 2 p sqrt (L/g)
• The frequency, f 1/T 1/(2 p) sqrt (g/L)

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Damped Harmonic Motion

• Automobile spring and shock absorbers provide
  damping so that the car wont bounce up and down
• Overdamped takes a long time to reach equilibrium
• Underdamped takes several bounces before coming
  to rest
• Critical damping reaches equilibrium the fastest

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Forced Vibrations and Resonance

• A system with a natural frequency may have a
  force applied to it. This is a forced vibration
• If the applied force its natural frequency,
  then we have resonance. This freq is resonance
  freq. This will lead to resonant collapse

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Wave Motion

• Waves are moving oscillations, not carrying
  matter along
• A simple wave bump is a wave pulse
• A continuous or periodic wave has at its source a
  continuous and oscillating disturbance
• The amplitude is the max height of a crest
• The distance between two consecutive crests is
  called the wavelength, l
• The frequency, f, is the number of complete
  cycles
• The wave velocity, v lf, is the velocity at
  which wave crests move, not the velocity of the
  particle
• For small amplitude, v sqrt FT/(m/L) , m/L
  mass/length

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Transverse and Longitudinal Waves

• Particles vibrate up and down transverse wave
• Particles vibrate in the same direction
  longitudinal wave, resulting in compression and
  expansion
• The velocity of longitudinal wave sqrt (elastic
  force factor / inertia force factor)sqrt (E/ r)
• For liquid or gas, v sqrt (B/ r), r is the
  density

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Energy of Waves

• Wave energy is proportional to the square of
  amplitude
• Intensity, I energy/time/area power/area
• For a spherical wave, I P/4pr2
• For 2 points at r1 and r2, I2/I1 r12 / r22
• For wave twice as far, the amplitude is ½ as
  large, such that A2/A1 r1 /r2

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Reflection and Interference

• The law of reflection the angle of incidence
  the angle of reflection
• Interference happens when two waves pass through
  the same region at the same time
• The resultant displacement is the algebraic sum
  of their separate displacements
• A crest is positive and a trough is negative
• Superposition results in either constructive or
  destructive
• 2 constructive waves are in phase destructive
  waves are out of phase

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Standing Wave and Resonance

• 2 traveling waves may interfere to give a large
  amplitude standing wave
• The points of destructive interference are nodes
• Points of constructive interference are antinodes
• Frequencies at which standing waves are produced
  are natural freq or resonance freq
• Only standing waves with resonant frequencies
  persist for long such as guitar, violin, or piano
• The lowest frequency is the fundamental freq 1
  antinode, L 1st harmonic ½ l1
• The other natural freq are overtones, multiples
  of fundamental frequencies, L nln/2 n 1, 2,
  3, ...