Simple Harmonic Motion

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Simple Harmonic Motion Elasticity

• Chapter 10

2
Elastic Potential Energy

• What is it?
• Energy that is in materials as a result of
  their .
• Where is it found?

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Law

• A spring can be or with a .
• The by which a spring is compressed or
  stretched is to the magnitude of the
  ( ).
• Hookes Law
• Felastic
• Where
• spring constant of spring ( )
• displacement

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Hookes Law

• What is the graphical relationship between the
  elastic spring force and displacement?
• Felastic -kx

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Hookes Law

• A force acting on a spring, whether stretching or
  compressing, is always .
• Since the spring would prefer to be in a
  relaxed position, a negative force will
  exist whenever it is deformed.
• The force will always attempt to bring
  the spring and any object attached to it back to
  the position.
• Hence, the restoring force is always .

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Example 1

• A 0.55 kg mass is attached to a vertical spring.
  If the spring is stretched 2.0 cm from its
  original position, what is the spring constant?
• Known
• m
• x
• g
• Equations
• Fnet (1)
• (2)
• (3)
• Substituting 2 and 3 into 1 yields
• k
• k
• k

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Elastic in a Spring

• The exerted to put a spring in tension or
  compression can be used to do . Hence the
  spring will have Elastic .
• Analogous to kinetic energy

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Example 2

• What is the maximum value of elastic potential
  energy of the system when the spring is allowed
  to oscillate from its relaxed position with no
  weight on it?

• A 0.55 kg mass is attached to a vertical spring
  with a spring constant of 270 N/m. If the spring
  is stretched 4.0 cm from its original position,
  what is the Elastic Potential Energy?
• Known
• m 0.55 kg
• x -4.0 cm
• k 270 N/m
• g 9.81 m/s2
• Equations
• PEelastic
• PEelastic
• PEelastic

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Elastic Potential Energy

• What is area under the curve?

A A A A Which you should see equals the

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Simple Harmonic Motion Springs

• Simple Harmonic Motion
• An around an will occur when an
  object is from its equilibrium position and
  .
• For a spring, the restoring force F -kx.
• The spring is at equilibrium
• when it is at its relaxed length.
• ( )
• Otherwise, when in tension or
• compression, a restoring
• force exist.

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Simple Harmonic Motion Springs

• At displacement ( )
• The Elastic Potential Energy will be at a
• The force will be at a .
• The acceleration will be at a .
• At (x )
• The Elastic Potential Energy will be
• Velocity will be at a .
• Kinetic Energy will be at a
• The acceleration will be , as will the
  force.

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10.3 Energy and Simple Harmonic Motion
Example 3 Changing the Mass of a Simple Harmonic
Oscilator
A 0.20-kg ball is attached to a vertical spring.
The spring constant is 28 N/m. When released
from rest, how far does the ball fall before
being brought to a momentary stop by the spring?
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10.3 Energy and Simple Harmonic Motion
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Simple Harmonic Motion of Springs

• Oscillating systems such as that of a spring
  follow a pattern.
• Harmonic Motion of Springs 1
• Harmonic Motion of Springs (Concept Simulator)

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Frequency of Oscillation

• For a spring oscillating system, the frequency
  and period of oscillation can be represented by
  the following equations
• Therefore, if the of the spring and the
  are known, we can find the and
  at which the spring will oscillate.
• k and mass equals frequency of
  oscillation (A spring).

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Harmonic Motion The Simple Pendulum

• Simple Pendulum Consists of a massive object
  called a suspended by a string.
• Like a spring, pendulums go through
• as follows.
• Where
• Note
• This formula is true for only of .
• The period of a pendulum is of its mass.

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Conservation of ME The Pendulum

• In a pendulum, is converted into
  and vise-versa in a continuous repeating pattern.
• PE mgh
• KE ½ mv2
• MET PE KE
• MET
• Note
• kinetic energy is achieved at the point of
  the pendulum swing.
• The potential energy is achieved at the
  of the swing.
• When is , , and when
  is , .

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Key Ideas

• Elastic Potential Energy is the in a spring
  or other elastic material.
• Hookes Law The of a spring from its
  is the applied.
• The of a vs. is equal to the
  .
• The under a vs. is equal to the
  done to compress or stretch a spring.

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Key Ideas

• Springs and pendulums will go through oscillatory
  motion when from an position.
• The of of a simple pendulum is of
  its of displacement (small angles) and .
• Conservation of energy Energy can be converted
  from one form to another, but it is
  .