REPRESENTING%20SIMPLE%20HARMONIC%20MOTION

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REPRESENTING SIMPLE HARMONIC MOTION
simple
not simple
http//hyperphysics.phy-astr.gsu.edu/hbase/imgmec/
shm.gif
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Simple Harmonic Motion
y(t)
Watch as time evolves
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A
-A
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Position (cm)
Velocity (cm/s)
Acceleration (cm/s2)
time (s)
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These representations of the position of a simple
harmonic oscillator as a function of time are all
equivalent - there are 2 arbitrary constants in
each. Note that A, f, Bp and Bq are REAL C and
D are COMPLEX. x(t) is real-valued variable in
all cases.
A
B
C
D
Engrave these on your soul - and know how to
derive the relationships among A f Bp Bq C
and D .
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Example initial conditions
m 0.01 kg k 36 Nm-1. At t 0, m is
displaced 50mm to the right and is moving to the
right at 1.7 ms-1. Express the motion in form
A form B
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A particle executes simple harmonic motion. When
the velocity of the particle is a maximum which
one of the following gives the correct values of
potential energy and acceleration of the
particle. (a)potential energy is maximum and
acceleration is maximum. (b)potential energy is
maximum and acceleration is zero. (c)potential
energy is minimum and acceleration is
maximum. (d)potential energy is minimum and
acceleration is zero.
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A particle executes simple harmonic motion. When
the velocity of the particle is a maximum which
one of the following gives the correct values of
potential energy and acceleration of the
particle. (a)potential energy is maximum and
acceleration is maximum. (b)potential energy is
maximum and acceleration is zero. (c)potential
energy is minimum and acceleration is
maximum. (d)potential energy is minimum and
acceleration is zero. Answer (d). When velocity
is maximum displacement is zero so potential
energy and acceleration are both zero.
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• A mass vibrates on the end of the spring. The
  mass is replaced with another mass and the
  frequency of oscillation doubles. The mass was
  changed by a factor of
• 1/4 (b) ½ (c) 2 (d) 4

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• A mass vibrates on the end of the spring. The
  mass is replaced with another mass and the
  frequency of oscillation doubles. The mass was
  changed by a factor of
• 1/4 (b) 1/2 (c) 2 (d) 4
• Answer (a). Since the frequency has increased the
  mass must have decreased. Frequency is inversely
  proportional to the square root of mass, so to
  double frequency the mass must
• change by a factor of 1/4.

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• A mass vibrates on the end of the spring. The
  mass is replaced with another mass and the
  frequency of oscillation doubles. The maximum
  acceleration of the mass
• remains the same.
• is halved.
• is doubled.
• is quadrupled.

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• A mass vibrates on the end of the spring. The
  mass is replaced with another mass and the
  frequency of oscillation doubles. The maximum
  acceleration of the mass
• remains the same.
• is halved.
• is doubled.
• is quadrupled.
• Answer (d). Acceleration is proportional to
  frequency squared. If frequency is doubled than
  acceleration is quadrupled.

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A particle oscillates on the end of a spring and
its position as a function of time is shown
below. At the moment when the mass is at the
point P it has (a) positive velocity and positive
acceleration (b) positive velocity and negative
acceleration (c) negative velocity and negative
acceleration (d) negative velocity and positive
acceleration
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A particle oscillates on the end of a spring and
its position as a function of time is shown
below. At the moment when the mass is at the
point P it has (a) positive velocity and positive
acceleration (b) positive velocity and negative
acceleration (c) negative velocity and negative
acceleration (d) negative velocity and positive
acceleration Answer (b). The slope is
positive so velocity is positive. Since the slope
is getting smaller with time the acceleration is
negative.
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Complex numbers
Argand diagram
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Eulers relation
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Consistency argument
If these represent the same thing, then the
assumed Euler relationship says
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PHASOR
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Adding complex numbers is easy in rectangular form
b
a
c
d
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Multiplication and division of complex numbers is
easy in polar form
z
q
f
w
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Another important idea is the COMPLEX CONJUGATE
of a complex number. To form the c.c., change i
-gt -i
z
b
f
a
The product of a complex number and its complex
conjugate is REAL. We say zz equals mod z
squared
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And finally, rationalizing complex numbers, or
what to do when there's an i in the denominator?
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Using complex numbers initial conditions. Same
example as before, but now use the "C" and "D"
forms
m 0.01 kg k 36 Nm-1. At t 0, m is
displaced 50mm to the right and is moving to the
right at 1.7 ms-1. Express the motion in form
C form D
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