Simple Harmonic Motion Notes

1

• Simple Harmonic Motion Notes
• 1.Simple Harmonic motion is a constant vibratory
  (oscillatory-back and forth motion) that is
  continuous (no losses in __________) that has the
  requirement of a restoring force (force that
  pulls the vibrating material back to an
  equilibrium position (see handout).
• 2. The restoring force is ____________
  proportional to the displacement of the vibrating
  material from the equilibrium position.
• 3. The time required for 1 complete back and
  forth vibration (wave cycle) is called the
• ________ (T) and is measured in units of seconds
  (s).
• 4. The ___________ of the period is the
  frequency which is measured in units of hertz
• (hz 1/s).
• 5. The displacement of the vibrating material is
  referred to as the ___________ (A).
• 6. There is never a true harmonic motion for
  there will always be losses in ___________. The
  degree to which the oscillator loses energy is
  referred to as damping. See figure below for
  differing degrees of damped oscillation.
• 7. One method of sustaining a damped oscillation
  is by periodically pumping __________
• into the vibrating system called forced
  oscillation (plucking a guitar string, striking
• a bell or having a tuning fork force a marker
  board to vibrate). Figure 26.8 pg. 394.
• 8. Another method of sustaining damped
  oscillation is by _____________. Resonance
  occurs when a forced vibrations frequency
  matches the natural frequency of a glass causing
  the glass to __________ or soldiers breaking step
  when crossing the bridge so the frequency of the
  march wont match the ________ vibrating
  frequency of the bridge.
• 9. Resonance can also occur from forced
  vibrational frequencies that are ___________ of
  the natural frequency of the resonating material
  (1/2, ¼, 1/8th the natural frequency). Resonance
  wont occur at multiples of the natural
  frequency.

energy
directly
period
inverse
amplitude
energy
energy
resonance
break
natural
factors
2

• Simple Harmonic Motion
• A pendulum could be considered a simple harmonic
  oscillator for it oscillates with the presence of
  a ___________ force (gravity) if there were no
  losses in energy during its oscillation. Using
  what we know about centripetal acceleration (ac
  v2/r and v 2p r/T), we can derive an equation
  to determine the period of the oscillation.
• Frequency and period of oscillation are
  ___________________ proportional to one another.
• A mass hanging on a spring could also be
  considered to be another form of a simple
  harmonic oscillator for it oscillates with the
  presence of a restoring force (the force constant
  of the spring pulling the mass to the equilibrium
  position). The period of oscillation can be
  determined as follows
• The restoring force of a spring is directly
  proportional to the displacement of the object
  from the equilibrium position and can be
  determined as follows (this is called Hooks Law).

restoring
ac (2pr / T)2 / r ac g for a
pendulum
g 4p2 r / T2
___________
T v 4p2 r /g
______
T 2p v r /g
inversely
f 1/T
_________
T 2p v m / k
F kx
3

• Displacement of 200. g mass __________ cm
• 2. Show calculation of force constant __________
  N/m
• F kx
• k F/x F mg
• so k mg / x
• k (200. g (kg/1000g)(9.80 m/s2) / (6.57 cm (m /
  100cm))
• k 29.833 N/m
• 3. Determine theoretical period __________ s
• T 2p (m / k)1/2
• T 2 p (200. g (kg/1000g) / (29.833 N/m))1/2
• T 0.51445 s Time for 30.0 swings __________
  s
• 5. Period for one swing __________ s
• 17.0 s / 30.0 swings 0.56667 s or 0.567 s

6.57
29.8
0.514
17.0
0.567
4
10.

• 6. error _________
• error TV EV . 100
• TV
• error 0.51445 s 0.56667 s . 100
• 0.51445 s
• error (0.05222 s / 0.51445 s) .100
• error 10.15

5

• Vibrations and waves pg. 362-368 498 (chapter
  25 of old book)
• 1. Waves carry __________ (which can contain
  information).
• 2. A wave is a vibration projected through space
  and _________.
• Wave Characteristics
• 1. Amplitude (A) - displacement of vibrating
  material from the ________________ position.
• 2. Wavelength (?) distance between successive
  _______________ points on a wave (such as the
  distance between the crests or troughs).
• 3. Frequency (f) the number of wave cycles per
  __________ (hz). The frequency is inversely
  related to the __________ (T).
• Wave Types
• 1. Electromagnetic waves (fig. 26.3 pg. 498)
  require no ___________ in which to propagate
  (transfer energy).
• 2. Mechanical waves require a medium in which to
  propagate (___________ is a mechanical wave).
  When waves travel between 2 points, only the wave
  _________ is transferred (there is no transfer of
  the medium between the 2 points).
• Wave Forms (pg. 367-368)
• 1. Longitudinal waves medium is displaced ___
  to the direction of wave propagation.
• 2. Transverse waves medium is displaced ___ to
  the direction of wave propagation.
• Wave Speed pg. 366
• 1. The speed of a mechanical wave depends upon
  the type of _________ through which the wave
  pulse propagates. Sound travels faster in solids
  than in liquids and faster in liquids than in
  gases (sound will not travel at all in a
  ___________).
• 2. Wave speed is the __________ of wavelength (?)
  and frequency (f).
• v ? f

energy
time
equilibrium
identical
seconds
period
medium
sound
pulse

-
medium
vacuum
product
6

• Properties of all waves (all waves exhibit these
  properties)
• 1. Reflection the bouncing of all or a portion
  of a wave when it reaches a ________________
  between 2 media (pg. 384 531-532).
• Law of Reflection angle of incidence is
  ___________ to the angle of reflection when
  measured relative to the normal.
• 2. Refraction the change in direction of wave
  when it reaches a boundary between 2 media due to
  traveling at different ___________ in each
  medium. (pg. 385-386 535-540)
• 3. Diffraction-_________ of a wave around a
  barrier or through a narrow slit causing it to
  spread.
• (pg. 560-562)
• 4. Interference-the formation of a _________
  waveform from the superposition of 2 or more
  waves (superposition principle-the amplitude of
  superimposed waves is the sum of the amplitudes
  at each point).
• Constructive interference superimposed waves
  result in an ______________ in amplitude
  (destructive interference results when there is a
  decrease in amplitude). pg. 369 562-563
• Standing Waves pg. 370-371
• 1. A standing wave is an _________________
  pattern produced from an incident wave
  interfering with a reflected wave of same
  amplitude and wavelength. The waveform produced
  has regions of constant maximum amplitude
  (antinodes) and regions where the amplitude is
  zero (nodes).
• 2. Standing waves can be produced by vibrating
  the medium at different frequencies.
• 3a. A standing wave with 1 antinode is referred
  to as the fundamental and occurs at the
  _______________ frequency (fo).
• b. A standing wave with 2 antinodes is referred
  to as the ____ harmonic (____ overtone of the
  fundamental) and occurs at a frequency _________
  the natural frequency.
• c. A standing wave with 3 antinodes is referred
  to as the _____ harmonic (____ overtone of the
  fundamental) and occurs at a frequency _________
  the natural frequency.
• Equations for standing waves
• fn n fo ?n 2 L / n v ?nfn fn nv /
  2L

boundary
equal
speeds
bending
new
increase
interference
fundamental
2nd
1st
twice
3rd
2nd
triple
7

• v ?nfn ?n 2L/n fn nfo nv/(2L)
• 1a. Example 1 A standing wave with 5 segments
  has a frequency of 3.50.103 Hz.What is the
  fundamental (natural) frequency of the string?
• fo f5 / 5
• fo 3.50.103 hz / 5
• fo 700. hz
• b. What would be the frequency of the 2nd
  overtone?
• f3 3 fo
• f3 3 (700.00 hz)
• f3 2.10.103 hz
• c. If the string is 2.50 m long, what is the
  speed of the wave?
• f5 5v / (2L)
• v 2Lf5 / 5
• v 2(2.50 m)(3.50.103 hz) / 5
• v 3.50.103 m/s
• d. What would be the wavelength of the 4th
  harmonic?
• ?4 2L / 4
• ?4 2 (2.50 m) / 4
• ?4 1.25 m

8

• Example 2 The frequency of the 2nd harmonic of
  a wave is 440. Hz. What would be the frequency
  of the 4th overtone?
• fo fn/n
• f5 / 5 f2 / 2
• f5 5/2 f2
• f5 5/2 (440. hz)
• f5 1.10.103 hz
• Example 3 A standing wave, with 3 antinodes,
  is set up in a 3.5 m long string that has a
  fundamental frequency of 345 Hz. a. What is the
  wavelength of the 3rd harmonic?
• ?3 2L / 3
• ?3 2(3.5 m) / 3
• ?3 2.3 m
• b. What is the wave speed?
• fo 1v / (2L) or v ?3f3
• v fo2L v ?33fo
• v (345 hz (2 (3.5 m))) v (2.333 m)3(345
  hz)
• v 2400 m/s v 2400 m/s