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## Physics 218 Chapter 15

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### Physics 218 Chapter 15 Prof. Rupak Mahapatra Physics 218, Lecture XXII * Physics 218, Lecture XXII * Checklist for Today Midterm 3 average 61 Collect your exams from ... – PowerPoint PPT presentation

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Title: Physics 218 Chapter 15

1
Physics 218Chapter 15
• Prof. Rupak Mahapatra

2
Checklist for Today
• Midterm 3 average 61
• 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
• 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?