Wednesday, December 2, 1998
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Transcript of Wednesday, December 2, 1998
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Wednesday, December 2, 1998
Chapter 13: Simple Harmonic Motionvelocity vs positioncircular motionperiods of springs
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There will be two more meetings of Lab:- Thursday, December 3 (Lab 10)
- Thursday, December 10 (Lab Final=Party)MLK CenterPizza & Beverages providedOrders taken in lab tomorrow5:00 - 7:00 pm
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Now, sketch a plotof the height of theblock above the flooras a function of time.
h
What kind ofmathematicalfunctions (withwhich you’refamiliar) result insuch a pattern?
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Height of Block vs Time
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Time (s)
Hei
gh
t (m
)
Equilibriumposition
Amplitude
period
Amplitude
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This type of oscillatory behavior is known as
An object in simple harmonic motion displaysan acceleration that is proportional to thedisplacement and in the opposite direction.
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Now, sketch a plotof the velocity of theblock as a function oftime as it goes throughits oscillating motion.
h
How is this plotrelated to that forthe height?
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Velocity of Block vs Time
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Time (s)
Vel
oci
ty (
m/s
)
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Height of Block vs Time
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Time (s)
Hei
gh
t (m
) Height vs Time
Velocity of Block vs Time
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Time (s)
Vel
oci
ty (
m/s
)
Velocity vs Time
The wave for theheight is 1/4 periodbehind the wave forthe velocity!
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Using your sketch of velocity vs time, try to sketch the acceleration of the block as a function of time.
h
How is this plotrelated to thevelocity plot?
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Acceleration of Block vs Time
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Time (s)
Acc
eler
atio
n (
m/s
/s)
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Acceleration of Block vs Time
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Time (s)
Accele
rati
on
(m
/s/s
)
Acceleration vs Time
Velocity of Block vs Time
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Time (s)
Vel
oci
ty (
m/s
)
Velocity vs Time
The wave for thevelocity is 1/4 periodbehind the wave forthe velocity!
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Okay, so we’ve seen that objects in simpleharmonic motion can be described bytrigonometric functions (sines and cosines).
We know that these trigonometric functionscomplete one cycle (peak-to-peak) overwhat angular displacement?
0 1
That is to say,what is 1-0?
2
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So what if we observe our spring to oscillatewith a period of 3 seconds. How would wewrite our function for the height of our blockversus time?
Take a stabat it!t = 0 s t = 3 s
3 s
At
cos 23
s
FHG
IKJ
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t = 0 s t = 3 s
3 s
At
cos 23
s
FHG
IKJ
What is the Period of this wave?
What is the Frequency of this wave?
3 seconds
1/3 s
ATt A f tcos cos
22
FHGIKJ b g
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ATt A f tcos cos
22
FHGIKJ b g
Now…dust off the cobwebs…
Where have we seenthis quantity before? 2f
Circular motion!Angular Frequency!!!
f
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Now…you’ve got to be thinking to yourself...
What doesa spring have
to do withcircularmotion?
Well…I’m gonnatell ya!
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Here’s mytennis ballon a stringagain.
We’relookingdown on theplane of motion.
What if I were now to shine a spotlight on this system from the right side and look at the motion of the shadow on a wall to the left?
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L
I G
H
T
!
What will a graphof the height of the shadow on the wall look like?
Just like the position vs timefor the mass ona spring!
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Height of Block vs Time
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Time (s)
Hei
gh
t (m
)Height of Shadow vs Time
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So, the trigonometric functions describeBOTH circular motion (as we saw in Chapter 7)and the motion of a mass on a spring! WOW!
On what quantities did the period of ourtennis ball moving in a circle at the endof a string depend?
its speed and the radius of the circle.
Tr
v
2
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If we look at the motionof the shadow, what willbe the amplitude of theoscillation?
r
So we might surmise that the period for ourmass-spring system will also be related tothe amplitude of the motion!
TA
v
2
max
Where A representsthe amplitude of theoscillation.