Oscillations SHM

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Oscillations - SHM

• Chapter 13

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Oscillations

• In general an oscillation is simply aback and
  forth motion
• Since the motion repeats itself, it is called
  periodic
• We will look at a special type of oscillation -
  SHM

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Simple Harmonic Motion

• SHM is a result of the restoring force varying
  linearly with the displacement.
• In other words if you double the distance moved
  by the oscillator, there will be twice as much
  force trying to return it to its natural state

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SHM

• A mass on a spring is the prime example of SHM
  since the force that restores it to its
  equilibrium position is directly proportional to
  the amount by which the spring is stretched.

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Hooks Law

• The law that governs how spring stretches
• F -kx
• F force exerted by the spring (N)
• x the amount by which he string is stretched or
  compressed (m)
• k spring constant. A measure of how stiff the
  spring is (N/m)
• A small spring has a k 200 N/m
• the neg. sign indicate that displacement
  and force is in opposite direction

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Mass Oscillating on a Spring
(2)
(1)
(3)
(4)
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(5)
(6)
(7)
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New formula

• Since acceleration is not constant our Big 4
  Equations do not work
• F ma is still ok but a is always changing
• Calculating period
• T does not depend on amplitude does this make
  sense?
• Frequency

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SHM - Example

• Many skyscrapers use huge oscillating blocks of
  concrete to help reduce the oscillation of the
  building itself. In one such building the
  3.73x105 kg block completes one oscillation in
  6.80 s. What is the spring constant for this
  block?

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Additional Problems

• A block of mass 1.5 kg is attached to the end of
  a vertical spring of force constant k300 N/m.
  After the block comes to rest, it is pulled down
  a distance of 2.0 cm and released.(a) What is
  the frequency of the resulting oscillations?(b)
  What are the maximum and minimum amounts of
  stretch of the spring during the oscillations of
  the block?
• When a family of four people with a mass of 200
  kg step into their 1200 kg car, the car's springs
  compress 3 cm. (a) What is the spring constant
  of the car's springs, assuming they act as a
  single spring?(b) What are the period and
  frequency of the car after hitting a bump?
• A small insect of mass 0.30 g is caught in a
  spiderweb of negligible mass. The web vibrates
  with a frequency of 15 Hz.(a) Estimate the value
  of the spring constant for the web.(b) At what
  frequency would you expect the web to vibrate if
  an insect of mass 0.10 g were trapped?

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Energy Involved with SHM

• Lets look at a similar situation but concentrate
  on the energy of the object
• Energy stored in a spring (potential energy)
• U PE ½ kx2

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Energy of SHM
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3
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Energy Formula

• If total energy PE KE
• ½ kx2 ½ mv2
• Then Total energy PE (max) (KE 0)
• ½ kx2max
• Remember X(max) Amplitude

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Velocity

• If we combine all our formulas
• Which is useful when we solve for v

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Simple Pendulum

• Is this motion SHM?

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Pendulums

• Weight is broken into components
• Y component tension of rope
• X component restoring force ( brings the
  pendulum back resting position)
• F -mg(sin?) (neg. because it acts opp to
  displacement)
• If the angle is small and measured in radians the
  sin of angle basically equals the angle itself
  (only if ? lt 15)
• Therefore F-mg?

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• Now the arc length x is given by
• XL?
• Which means that we can replace the angle In the
  force formula with x/L
• F -mg (x/L)
• F (mg/L)x
• The formulas is just F -kx with the mg/L taking
  the place of k
• Therefore we do have SHM

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Period of a Pendulum

• Period of a Spring
• Replace the k with mg/L
• Period of a pendulum