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Physics 218 Chapter 15

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Title: Physics 218 Chapter 15


1
Physics 218Chapter 15
  • Prof. Rupak Mahapatra

2
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

3
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

4
Rotational Kinetic Energy
  • KEtrans ½mv2
  • ? KErotate ½Iw2
  • Conservation of Energy must take rotational
    kinetic energy into account

5
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

6
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
7
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?

8
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?

9
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

10
Same Problem Forces
  • Same problem but with Forces

11
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
12
(No Transcript)
13
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

14
Examples
  • Lots of stuff Vibrates or Oscillates
  • Radio Waves
  • Guitar Strings
  • Atoms
  • Clocks, etc
  • In some sense, the Moon oscillates around the
    Earth

15
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

16
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

17
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?

18
Questions
  • What are the forces?
  • Hookes Law F -kx
  • Does this equation describe our motion?
  • x x0 v0t ½at2

19
The forces
No force
Force in x direction
Force in x direction
20
More Detail
Time
21
Some Terms
Amplitude Max distance Period Time it takes
to get back to here
22
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

23
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)

24
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?

25
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)

26
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

27
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?

28
Check
  • This looks like a cosine. Makes sense

Spring and Mass Paper which tells us what happens
as a function of time
29
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

30
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?
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