16.105 Physics I : Mechanics

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16.105 Physics I Mechanics Motion of
particles described by vectors r(x,y,z,t)
x(t) î y(t) ? z(t) k Motion is the sum of
motion along three perpendicular
directions Point-like particle moves according
to Newtons Laws
F mdp/dt m a
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16.107 Physics II Waves and Modern
Physics Oscillations building blocks are
periodic functions ?(x,t) ?n an
cos( knx) cos(?nt) Motion is the sum of periodic
functions. ?(x,t) is a solution of Schrödinger
equation -(?2/2m)?2?(x,t)/?2 x V(x) ?(x,t)
i ? ? ?(x,t)/ ?t Solutions are wave-like
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Chapter 17 Oscillations

• Oscillations are everywhere
• vibrations in your eardrum, cell phone, CD
  player, quartz watches, heart beat
• some of these vibrations are obvious to us and
  some are not even detectable
• however, the mathematical description is
  basically the same

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Oscillations

• Some oscillations occur in a medium such as sound
  waves or water waves
• light, radio waves, x-rays are also oscillatory
  phenomena but do not involve motion of a medium
  but rather electric and magnetic fields
• sound waves need a medium
• light waves do not need a medium

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Oscillations

• Vibrations do not continue forever unless we
  continually pump the system
• Pendulumclock?springQuartz watch ? battery
• we need to balance losses due to damping or
  friction
• energy is converted into heat and lost eg.
  slinky

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• Snapshots of oscillating system
• frequency f of oscillations each second
• 1 hertz 1 Hz 1 s-1
• period T 1/f is the time for one complete
  oscillation

Mass
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Simple Harmonic Motion
x(t) xm cos(?t ?)

• xm is the amplitude (maximum value)
• ? is the phase angle (determines x(0))phi
• ? is ? omega
• x(t T) x(t) ------gt ?(tT) ? ?t ?
• ?(tT) ?t 2? ----gt ?T 2?
• ? 2?/T 2? f angular frequency rads/sec
• recall circular motion ???/?t

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Periodic motionorHarmonic motion

• Motion is repeated at regular intervals
• x(t) xm cos(?t ?)
  simple harmonic motion
• xm (positive constant) is the amplitude
• ?t ? is the phase of the motion
• ? is the phase constant(or phase angle)
• x(0) xm cos(? )

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Both curves have the same period and phase
constant
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Both curves have the same amplitude and phase
constant
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xm occurs when phase ?t?0
Both curves have the same amplitude and period
?t ? is the phase x(t) xm cos( ?t
?) points with the same phase have the same value
of x a decrease in ? shifts the red curve to the
right
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SHM
x(t) xm cos(?t ?)

• xm is a constant
• ? is a constant phi
• ? is a constant omega
• ?t ? varies with time t
• x(t) varies with time t
• lets try some examples using maple

shm.mws
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Velocity
x(t) xm cos(?t ?)

• v(t) - ? xm sin(?t ? )
• But cos(AB) cos(A)cos(B)-sin(A)sin(B)
• Hence cos(?t ? ?/2) - sin(?t ?)
• v(t) ? xm cos(?t ? ?/2)
• phase constant increased --gt shift to left

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Acceleration
x(t) xm cos(?t ?) v(t) - ? xm sin(?t ?
)

• v(t) ? xm cos(?t ? ?/2)
• Calculate acceleration
• a(t) - ?2 xm cos(?t ?)
• a(t) ?2 xm cos(?t ? ?/2 ?/2)
• phase constant increased again --gt shift left