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Demonstrations for Class 11  Mechanical Oscillations
Most small mechanical oscillations can be approximated as simple harmonic oscillations
and can be modeled as a spring.
You remember that for a spring, the restoring force is
kx
F

=
, where
k
is known as the "restoring constant".
The force is directly proportional
to how far you stretch the spring.
That lets it be simple harmonic.
Simple harmonic oscillations are described by a differential equation.
For a spring and
mass, neglecting friction and gravity,
0
d
d
2
2
=
+
x
m
k
t
x
, and the solution for
x
is a sinusoid,
)
cos(
φ
ϖ
+
=
t
A
x
,
where
A
is the maximum displacement, or the Amplitude,
is the
angular frequency, and
is a phase offset called the phase constant.
The total energy for an oscillating mass is simply the sum of its potential energy at any
time (from compressing or extending the spring) and its kinetic energy at any time (from
the motion of the mass itself):
2
2
2
1
2
1
kx
mv
E
+
=
.
Taking
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This note was uploaded on 09/17/2008 for the course PHYS 1200 taught by Professor Stoler during the Spring '06 term at Rensselaer Polytechnic Institute.
 Spring '06
 Stoler
 Force

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