PRE Chap 13 SHM and Springs

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PRE Chap 13SHM and Springs

• Simple Harmonic Motion SHM
• SHM is a vibratory motionthat is a repeated
  motion as a pendulum or a mass on a spring.
• SHM obeys Hookes Law
• To be SHM there must be some restoring force
  acting to restore equilibrium
• To describe SHM we need to define F and T

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T

• T the period
• The period is the TIME iut takes for one complete
  vibration or one oscillation or one wave or one
  cycle
• So as in a pendulum we begin at point A when
  the bob returns to point A that is one cycle
• How long it takes to make one cycle is the
  period, T, measured in seconds.

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F Frequency

• Frequency is the NUMBER of waves or vibrations or
  swings or cycles that take place in a unit of
  time, usually in a second
• Frequency then is the number of cycles per
  second.
• 1 cycle/sec is 1 Hertz 1 Hz. The unit of F is
  the Hz.
• The relationship between period and frequency is
  F 1/T

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Graphing Vibratory Motion

• Such a graph say of a mass bobbing up and down at
  the end of a spring, would produce a sinusoidal
  motiona sine curve.
• Transverse waves are a good depiction of such
  sine curves.
• SHM is a disturbance from the equilibrium
  position to some amplitudeie max distance from
  the rest or equilibrium position.

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Restoring Force

• What defines SHM is the requirement that there be
  some restoring force.
• There must be some force which opposes the
  displacement from equilibrium
• There must be some force that is trying to
  restore the system to equilibrium.
• The restoring force always pushes or pulls back
  toward the equilibrium position.

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Hookes Law

• SHM is the vibratory motion that obeys HOOKES
  LAW.
• IN SHM a system will vibrate at a single,
  constant, frequency. Thats what makes it
  simple.
• Hookes Law demands that the system is one
  that, when distorted, tries to return to its
  original configuration once it is released.

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F - KX

• A Hookean System is one that can be distorted and
  can return to its original configuration because
  there is some restoring force trying to return it
  to equilibrium.
• The amount of restoring force is -kx.

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-k

• F -kx
• k is called the spring constant. It is a number
  which is a relative measure of its stiffness.
  High number very stiffcar springs. Low number
  less stiffa slinky.
• K is negative because it always acts opposite the
  direction of displacement. Recall that signs
  simply indicate direction.
• We can look up k values on a chart.
• The unit of k is N/m. How many N does it take to
  displace a spring so many meters.

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x

• F -kx
• Let us take stretching the spring some distance X
  as the positive direction.
• Let us take the compression of a spring some
  distance a negative value.

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Elastic PE

• PE is energy stored up by virtue of work done to
  change its position relative to some equilibrium
  frame of reference
• PE is energy stored up due to work done to
  destable-ize a system.
• I stretch a spring. I did work. I have changed
  its position form equilibrium.
• Now the work is stored up waiting to do work or
  return to equilibrium so that PE 1/2kxx
• MAX PE is when distortion is at MAX amplitude.

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Consider a pendulum

• There is an energy interchange in a swinging
  pendulum.
• KE PE constant
• ½ m vv ½ kxx constant.

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Acceleration in SHM

• Hookes Law is F-kx
• Newton is F ma
• Once displaced and then let go a restoring force
  drives the system so we combine the two eq.
• A - k/m (X)
• Minus indicates that both a and F are always
  in the direction opposite displacement x

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Period in SHM

• The period in a spring system or pendulum or
  other oscillating systems is
• T 2 pi times the square root of m/k
• Acceleration in terms of the period becomes
• A - 4 pi squared / T squared times the
  displacement distance (x)

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A simple Pendulum

• The equation to solve for T of a pendulum takes
  into account that Gravity acts as the restoring
  force.
• Because the external force gravity acts as the
  restoring force, a pendulum is not a true Hookean
  System. It is not perfect SHM.
• However, we will consider it such since it is
  very close so long as the angle of swing is not
  too large.

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Last slide in this series!!!

• The period of a pendulum is
• T 2 pi times the square root of the length of
  the pendulum string / gravity.
• Notice that mass is NOT a factor!!
• Since gravity it pulling down on the mass,
  remember that any two objects dropped under the
  influence of G from the same height at same time
  will hit ground at same timemass is NOT a
  factor!
• Its basically the same thing!