Download The equations of motion

Physics 218
Chapter 15
Prof. Rupak Mahapatra
Physics 218, Lecture XXII
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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 we’ll finish:
– Momentum  Angular Momentum L
– Energy
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Rotational Kinetic Energy
2
½mv

KEtrans =
2
KErotate = ½Iw
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 didn’t 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
I=½MR2, 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 I=ml/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
}
Physics 218, Lecture XXIV
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. It’s 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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What’s Next
1.First we’ll “model” oscillations with
a mass on a spring
•You’ll see why we do this later
2.Then we’ll talk about what
happens as a function of time
3.Then we’ll 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 it’s equilibrium point
(I.e., x = 0) and let go?
What happens if we attach a mass, then stretch
the spring, and then let go?
k
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Questions
• What are the forces?
Hooke’s 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. It’s important that
you know how to reproduce what I’m 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?
k
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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
f is the “phase”
(phase just allows us to set t=0 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 t=0
– 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
• What is
• What is
• What is
points?
• What is
the
the
the
the
amplitude?
phase?
angular frequency?
velocity at the end
the velocity at the middle?
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