Chapter 13 SHM?

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Chapter 13 SHM?

• WOD are underlined

2
Remember Hookes Law

• F - k ?x
• New Symbol k
• Spring constant.
• Stiffness of the spring.
• Depends on each springs dimensions and material.
• In N/m

3
Question

• If I let go, what will happen to the mass? Then
  what? Then what?

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Simple Harmonic Motion

• Repeating up and down motion, (like cos wave.)
  (Draw a picture.)
• Motion that occurs when the net force obeys
  Hookes Law
• The force is proportional to the displacement and
  always directed toward the equilibrium position
• Show Example with Spring
• The motion of a spring mass system is an example
  of Simple Harmonic Motion
• Are springs the only type of SHM?

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Simple Harmonic Motion

• The motion of a spring mass system is an example
  of Simple Harmonic Motion
• Are springs the only type of SHM
• No,
• Jump Rope, Sound Waves, Pendulum, Swing, up and
  down motion of an engine piston

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Motion of the Spring-Mass System

• Initially, ?x is negative and the spring pulls it
  up.
• The objects inertia causes it to overshoot the
  equilibrium position.
• ?x is positive now and the spring pushes it down.
  Again it will over shoot equilibrium.

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?x, v and a versus t graphs
What type of curve is this? For
Calculus, Derivative of sin is what? What
happens if you bump the spring?
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?x, v and a
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?x, v and a

• All three look like sinusoidal curves.
• V is shifted backwards from ?x
• a is shifted backwardwards from v.

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Acceleration of an Object in Simple Harmonic
Motion

• Remember F - k x F ma
• Set them equal to each other
• - k x ma
• Solve for a
• a -k?x / m
• The acceleration is a function of position
• Acceleration is not constant.
• So non-inertial frame of reference. So, the
  kinematic equations are not valid here.

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Amplitude New Symbol A

• Amplitude, A
• The amplitude is the maximum position of the
  object relative to the equilibrium position (Max
  Height)
• In the absence of friction, an object in simple
  harmonic motion will oscillate between the
  positions x A
• What friction is there?

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Amplitude New Symbol A

• Amplitude, A
• The amplitude is the maximum position of the
  object relative to the equilibrium position (Max
  Height)
• In the absence of friction, an object in simple
  harmonic motion will oscillate between the
  positions x A
• What friction is there?
• Air Resistance, Molecular Motion in Spring

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Period New Symbol T

• Period T
• uppercase T stands for period.
• Amount of time for the oscillator to go through 1
  complete cycle.
• (Time for 1 up and 1 down.)
• Often measured from Max to Max,
• But can be measured from start to start, etc.
• Measured in seconds.

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Frequency Another new symbol ƒ

• ƒ is for frequency.
• It is the number of cycles an oscillator goes
  through in one second.
• It is measured in 1/seconds
• 1/seconds gt New unit Hertz or Hz.
• What is the frequency of revolutions of a new M16
  bullet?

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Frequency Another new symbol ƒ

• ƒ is for frequency.
• It is the number of cycles an oscillator goes
  through in one second.
• It is measured in 1/seconds
• 1/seconds gt New unit Hertz
• What is the frequency of revolutions of a new M16
  bullet?
• Ans5100 Hz or Rev per Second.

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Period and Frequency

• The period, T, is the time per cycle.
• The frequency, ƒ, is cycles per time.
• Frequency is the reciprocal of the period
• ƒ 1 / T

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Quick Recap(Pic for WOD)

• A maximum distance from rest postion.
• T time for one complete cycle
• ƒ 1 / T

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• In the table, label each , -, or 0.

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Question

• When you compress (or stretch) a spring, you have
  to do work on it. You apply a force over some
  distance.
• Can you get that energy back?

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Elastic Potential Energy

• (Energy stored in a spring.
• Ability of a spring to do work.)
• Work done on a spring is stored as potential
  energy.
• The potential energy of the spring can be
  transformed into kinetic energy of the mass on
  the end.

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Energy Transformations

• Suppose a block is moving on a frictionless
  surface.
• Before it hits the spring, the total mechanical
  energy of the system is the kinetic energy of the
  block. What happens next?

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Energy Transformations, 2

• The spring is partially compressed.
• The mass has slowed down.
• S ME K.E. P.E.

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Energy Transformations, 3

• The spring is now fully compressed
• The block momentarily stops
• The total mechanical energy is stored as elastic
  potential energy of the spring

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At all times, total Mechanical Energy is constant
KE PE
(Put into notes) Equations for SHM Energy KE ½
mv2 PE ½ kx2
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Keep in mind.

• It takes the same energy to stretch a spring as
  compress it.
• PE ½ kx2
• Is the same as
• ½ k(-x)2
• So PE is same at Max or Min A.

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Back to Period and Frequency

• Period
• Frequency
• What variable is not in these equations?

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Back to Period and Frequency

• Period
• Frequency
• What variable is not in these equations? A. T
  and f do not depend on Amplitude.

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Problem

• A 1 kg block is dropped from a height of 1 m onto
  a spring with k 55 N/m. How far will the
  spring compress?

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Problem (revisited)

• A 1 kg block is dropped from a height of 1 m onto
  a spring with k 55 N/m. What will its
  frequency and period of oscillation be?

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Problem

• A 1 kg block is dropped from a height of 1 m onto
  a spring with k 55 N/m.
• Q1. How far will the spring compress?
• Q2. What will its frequency and period of
  oscillation be?