Physics 218 Chapter 15

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Physics 218Chapter 15

• Prof. Rupak Mahapatra

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Checklist for Today

• Midterm 3 average 61
• Collect your exams from your TAs.
• Last lecture next Monday
• Will cover up to Chpater 18
• Ch 14 and 15 Home work due Wed this week
• Ch 18 Home work due Mon next week

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Angular Quantities

• Position ? Angle q
• Velocity ? Angular Velocity w
• Acceleration ? Angular Acceleration a
• Force ? Torque t
• Mass ? Moment of Inertia I
• Today well finish
• Momentum ? Angular Momentum L
• Energy

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Rotational Kinetic Energy

• KEtrans ½mv2
• ? KErotate ½Iw2
• Conservation of Energy must take rotational
  kinetic energy into account

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Rotation and Translation

• Objects can both Rotate and Translate
• Need to add the two
• KEtotal ½ mv2 ½Iw2
• Rolling without slipping is a special case where
  you can relate the two
• V wr

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Rolling Down an Incline

• You take a solid ball of mass m and radius R and
  hold it at rest on a plane with height Z. You
  then let go and the ball rolls without slipping.
• What will be the speed of the ball at the bottom?
• What would be the speed if the ball didnt roll
  and there were no friction?

Note Isphere 2/5MR2
Z
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A bullet strikes a cylinder

• A bullet of speed V and mass m strikes a solid
  cylinder of mass M and inertia I½MR2, at radius
  R and sticks. The cylinder is anchored at point 0
  and is initially at rest.
• What is w of the system after the collision?
• Is energy Conserved?

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Rotating Rod

• A rod of mass uniform density, mass m and length
  l pivots at a hinge. It has moment of inertia
  Iml/3 and starts at rest at a right angle. You
  let it go
• What is w when it reaches the bottom?
• What is the velocity of the tip at the bottom?

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Person on a Disk

• A person with mass m stands on the edge of a disk
  with radius R and moment ½MR2. Neither is moving.
  
• The person then starts moving on the disk with
  speed V.
• Find the angular velocity of the disk

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Same Problem Forces

• Same problem but with Forces

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Chapter 18 Periodic Motion

• This time
• Oscillations and vibrations
• Why do we care?
• Equations of motion
• Simplest example Springs
• Simple Harmonic Motion
• Next time
• Energy

Concepts

?The math
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What is an Oscillation?

• The good news is that this is just a fancy term
  for stuff you already know. Its an extension of
  rotational motion
• Stuff that just goes back and forth over and over
  again
• Stuff that goes around and around
• Anything which is Periodic
• Same as vibration
• No new physics

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Examples

• Lots of stuff Vibrates or Oscillates
• Radio Waves
• Guitar Strings
• Atoms
• Clocks, etc
• In some sense, the Moon oscillates around the
  Earth

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Why do we care?

• Lots of engineering problems are oscillation
  problems
• Buildings vibrating in the wind
• Motors vibrating when running
• Solids vibrating when struck
• Earthquakes

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Whats Next

• First well model oscillations with a mass on a
  spring
• Youll see why we do this later
• Then well talk about what happens as a function
  of time
• Then well calculate the equation of motion using
  the math

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Simplest Example Springs

• What happens if we attach a mass to a spring
  sitting on a table at its equilibrium point
• (I.e., x 0) and let go?
• What happens if we attach a mass, then stretch
  the spring, and then let go?

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Questions

• What are the forces?
• Hookes Law F -kx
• Does this equation describe our motion?
• x x0 v0t ½at2

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The forces
No force
Force in x direction
Force in x direction
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More Detail
Time
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Some Terms
Amplitude Max distance Period Time it takes
to get back to here
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Overview of the Motion

• It will move back and forth on the table as the
  spring stretches and contracts
• At the end points its velocity is zero
• At the center its speed is a maximum

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Simple Harmonic Motion

• Call this type of motion
• Simple Harmonic Motion
• (Kinda looks like a sine wave)
• Next The equations of motion
• Use SF ma -kx
• (Here comes the math. Its important that you
  know how to reproduce what Im going to do next)

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Equation of Motion

• A block of mass m is attached to a spring of
  constant k on a flat, frictionless surface
• What is the equation of motion?

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Summary Equation of Motion

• Mass m on a spring with spring constant k
• x A sin(wt f)
• Where
• w2 k/m
• A is the Amplitude
• is the phase
• (phase just allows us to set t0 when we want)

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Simple Harmonic Motion

• At some level sinusoidal motion is the definition
  of Simple Harmonic Motion
• A system that undergoes simple harmonic motion
• is called a
• simple harmonic oscillator

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Understanding Phase Initial Conditions

• A block with mass m is attached to the end of a
  spring, with spring constant k. The spring is
  stretched a distance D and let go at t0
• What is the position of the mass at all times?
• Where does the maximum speed occur?
• What is the maximum speed?

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Check

• This looks like a cosine. Makes sense

Spring and Mass Paper which tells us what happens
as a function of time
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Example Spring with a Push

• We have a spring system
• Spring constant K
• Mass M
• Initial position X0
• Initial Velocity V0
• Find the position at all times

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What is MOST IMPORTANT?

• Simple Harmonic Motion
• X A sin(wt f)
• What is the amplitude?
• What is the phase?
• What is the angular frequency?
• What is the velocity at the end points?
• What is the velocity at the middle?