PHYSICS 231 Lecture 38: Resonances, beats and review

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PHYSICS 231Lecture 38 Resonances, beats and
review

• Remco Zegers
• Question hours Thursday 1200-1300
  1715-1815
• Helproom

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standing waves in a rope
both ends fixed ?n2L/n or Ln?n/2
F tension in rope ? mass per unit length
f1 fundamental frequency
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Both ends open
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One end open, one end closed
even harmonics are missing!!!
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example
A simple flute is played by blowing air in on one
side and the other end is open. The length of the
tube can be varied manually (like a trombone).
What are the frequencies of the first two
possible harmonics if L0.5m? If the length is
made half of the original length, how will these
change v343m/s?
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example
A simple flute is played by blowing air in on one
side and the other end is closed. The length of
the tube can be varied manually (like a
trombone). What are the frequencies of the first
two possible harmonics if L0.5m? If the length
is made half of the original length, how will
these change v343m/s?
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harmonics
Generally speaking, many harmonics with different
intensities can be present at the same time.

L
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beats
Superposition of 2 waves with slightly different
frequency
DEMO
The amplitude changes as a function of time, so
the intensity of sound changes as a function of
time. The beat frequency (number of intensity
maxima/minima per second) fbeatfa-fb
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example
Someone is trying to tune a guitar. One of the
strings is supposed to have a frequency of 500
Hz. The person is using a tuning fork which
produces a sound of exactly this frequency, but
while sounding the fork and the playing the
guitar, hears a beat in the sound with a
frequency of 3 Hz (3 beat per second). a) What is
the real frequency of the guitar string? b) By
what fraction does the person need to change the
tension of the guitar string to tune it properly?
a) fbffork-fguitar 3500-fguitar
fguitar497 or 503 Hz
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Resonances
Realistically, oscillations are damped due to
frictional forces. However, we can drive the
oscillation via an external source. Example mass
on a spring natural frequency f1/(2?)?(k/m)
If the frequency of the driving force equals the
natural frequency large oscillations occur
Resonance
demo

• Resonances occur in many daily situations
• shock absorber in car
• playing basketball
• resonating lecture room!!

Famous example Tacoma bridge
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Review
Start with the sample problems on the web to see
how you stand on each chapter!!
see also recitations on Thursday (12-1 and
1715-1815) review on Friday any more???
today one problem each from ch.
2,3,4,5,6 Friday one problem each from 7,8,9 and
the rest 10,11,12,13,14 (2 each)
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chapter 2.
x(t)x(0)v(0)t½at2 v(t)v(0)at
A person throws 2 stones from the top of a
building with a speed of 20 m/s. One is thrown
up, and the other is thrown down. The first one
hits the street after 5 s. How much later does
the second one hit?
Stone thrown down x(t)x(0)v(0)t½at2h-20t-½(9.
8)t2 if t5, x0, so 0h-100-½(9.8)52 so
h222.5 m Stone thrown up x(t)x(0)v(0)t½at2h
20t-½(9.8)t2222.520t-½(9.8)t2 when it reaches
the ground, x0 so 0222.520t-½(9.8)t2 so
-4.9t220t222.50 youll find t-5 or t9.1 s
must be 9.1 s difference between the times that
the stones hit 9.1-54.1 s
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chapter 3
v0
A car is trying to jump over a 30m-wide river
using a ramp of 3 m high set at an angle of
300 with the horizontal. a) What is the minimum
velocity v0 required to cross the bridge? b)
What is the highest point of the car?
3m
30m
?300

• x(t)x(0)vx(0)t½at2v0tcos?0.866v0t
• must at least be 30m, so 300.866v0t and
  thus t34.6/v0
• y(t)y(0)vy(0)t-½gt2y(0)v0tsin?-4.9t23
  0.5v0t-4.9t2
• when it hits the ground
  y(t)030.5v0t-4.9t2
• use t34.6/v0 and find 030.534.6-4.9(34
  .6/v0)2
• solve for v0 and find v017 m/s (61.2 km/h)
• At highest point vertical component of
  velocity0
• vy(t)vy(0)atv0sin?-9.8t170.5-9.8t0
  t0.87 s
• y(0.87)30.5170.87-4.9(0.87)26.7 m