Physics 106P: Lecture 13 Notes

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• How to Get into Physics Grad School Seminar
• Learn all about the process, what to do at every
  step of the way, and gain insights from John
  Stack and Lance Cooper, who are both on the
  admissions committee for grad school at UIUC.
  This seminar is open to any one from freshman to
  seniors interested in Physics grad school.
• Wednesday, November 14
• 6pm-8pm
• 464 Loomis
• FREE PAPA DEL'S PIZZA WILL BE SERVED!

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On average, about how much solar energy hits
each square meter of Earth every second ?

• 1 Joule
• 10 Joules
• 100 Joules
• 1,000 Joules
• 10,000 Joules

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Physics 211 Lecture 24Todays Agenda

• Introduction to Simple Harmonic Motion
• Horizontal spring mass
• The meaning of all these sines and cosines
• Vertical spring mass
• The energy approach
• The simple pendulum
• The rod pendulum

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Simple Harmonic Motion (SHM)
Horizontal Spring

• We know that if we stretch a spring with a mass
  on the end and let it go, the mass will oscillate
  back and forth (if there is no friction).
• This oscillation is called Simple Harmonic
  Motion, and is actually very easy to understand...

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SHM Dynamics

• At any given instant we know that F ma must be
  true.
• But in this case F -kx and
  ma
• So -kx ma

a differential equation for x(t)!
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SHM Dynamics...
define
Where w is the angular frequency of motion
Try the solution x A cos(?t)
This works, so it must be a solution!
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SHM Dynamics...
Movie (shm)
Shadow

• But wait a minute...what does angular frequency ?
  have to do with moving back forth in a straight
  line ??

• y R cos ? R cos (?t)

y
1
1
1
2
2
3
3
?
0
x
4
-1
4
6
6
5
5
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SHM Solution

• We just showed that (which
  came from F ma) has the solution x A
  cos(?t) .
• This is not a unique solution, though. x A
  sin(?t) is also a solution.
• The most general solution is a linear combination
  of these two solutions! x B
  sin(?t) C cos(?t)

ok
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Derivation
We want to use the most general solution

• x A cos(?t ?) is equivalent to x B
  sin(?t) C cos(?t)

x A cos(?t ?)
A cos(?t) cos? - A sin(?t) sin?
So we can use x A cos(?t ?) as the most
general solution!
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SHM Solution...

• Drawing of A cos(?t )
• A amplitude of oscillation

T 2?/?
A
?
??
?
??
-??
A
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SHM Solution...

• Drawing of A cos(?t ?)

?
?
??
?
??
-??
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SHM Solution...

• Drawing of A cos(?t - ?/2)

? ??/2
A
?
??
?
??
-??
A sin(?t)!
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Lecture 24, Act 1Simple Harmonic Motion

• If you added the two sinusoidal waves shown in
  the top plot, what would the result look like?

(a)
(b)
(c)
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Lecture 24, Act 1 Solution

• Recall your trig identities

So
Where

• The sum of two or more sines or cosines having
  the same frequency is just another sine or
  cosine with the same frequency.
• The answer is (b).

Prove this with Excel
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What about Vertical Springs?
Vertical Spring

• We already know that for a vertical spring
• if y is measured from
  the equilibrium position
• The force of the spring is the negative
• derivative of this function
• So this will be just like the horizontal
  case-ky ma

j
k
y 0
F -ky
Which has solution y A cos(?t ?)
where
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SHM So Far

• The most general solution is x A cos(?t ?)
• where A amplitude
• ? angular frequency
• ? phase
• For a mass on a spring
• The frequency does not depend on the amplitude!!!
• We will see that this is true of all simple
  harmonic motion!
• The oscillation occurs around the equilibrium
  point where the net force is zero!

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Cosmic Rays
Energies from 106 1020 eV
p
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Cloud chamber demo- alcohol
p
About 200 ms per square meter per second at sea
level. (lots of neutrinos too)
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Fig. 2. Aitoff projection of the celestial sphere
in galactic coordinates with circles of radius
3.1degrees centered at the arrival directions
of the 27 cosmic rays with highest energy
detected by the Pierre Auger Observatory
The Pierre Auger Collaboration et al.,
Science 318, 938 -943 (2007)
Published by AAAS
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The Simple Pendulum
Simple Pendulum

• A pendulum is made by suspending a mass m at the
  end of a string of length L. Find the angular
  frequency of oscillation for small
  displacements.

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Aside sin ? and cos ? for small ?

• A Taylor expansion of sin ? and cos ? about ? 0
  gives

and
So for ? ltlt 1,
and
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The Simple Pendulum...

• Recall that the torque due to gravity about the
  rotation (z) axis is ? -mgd.
• d Lsin ? ? L? for small ? so ? -mg
  L?
• But ? I????I??mL2

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Lecture 24, Act 2Simple Harmonic Motion

• You are sitting on a swing. A friend gives you a
  small push and you start swinging back forth
  with period T1.
• Suppose you were standing on the swing rather
  than sitting. When given a small push you start
  swinging back forth with period T2.
• Which of the following is true

(a) T1 T2 (b) T1 gt T2 (c) T1 lt T2
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Lecture 24, Act 2 Solution

• We have shown that for a simple pendulum

Since

• If we make a pendulum shorter, it oscillates
  faster (smaller period)

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Lecture 24, Act 2 Solution
Standing up raises the CM of the swing, making it
shorter!
Since L1 gt L2 we see that T1 gt T2 .
L2
L1
T1
T2
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The Rod Pendulum

• A pendulum is made by suspending a thin rod of
  length L and mass m at one end. Find the angular
  frequency of oscillation for small displacements.

z
?
x
CM
L
mg
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The Rod Pendulum...

• The torque about the rotation (z) axis is ?
  -mgd -mg(L/2)sinq ? -mg(L/2)q for small q
• In this case
• So ? I???becomes

z
d
I
L/2
?
x
CM
L
d
mg
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Lecture 24, Act 3Period
Physical Pendulum

• What length do we make the simple pendulum so
  that it has the same period as the rod pendulum?

(a) (b) (c)
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Lecture 24, Act 3Solution
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Recap of todays lecture

• Introduction to Simple Harmonic Motion
• Horizontal spring mass
• The meaning of all these sines and cosines
• Vertical spring mass
• The energy approach
• The simple pendulum
• The rod pendulum