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6.055J / 2.038J The Art of Approximation in Science and Engineering
Spring 2008
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5 Proportional reasoning
k
x
= 0
m
x
27
So imagine a mass
m
connected to the wall by a spring with spring constant
k
. If disturbed, the mass oscillates. The period of the system is the time for
the mass to make a round trip through the equilibrium position.
Extend the spring by a distance
x
0
; this displacement is the amplitude. To see
how it affects the period, make an approximation, which will be an example
of throwing away information (the topic of
Part 3
). The approximation is to pretend that
the pendulum moves with a constant speed
v
. Then the period is
distance
,
T
∼
speed
v
and the distance that the mass travels in one period is
4
x
0
. Ignore the factor of 4:
.
T
∼
x
v
0
Proportional reasoning helps us estimate
v
by an energy argument. The initial potential
energy is
PE
∼
kx
2
0
or
PE
∝
x
2
0
.
The maximum kinetic energy, which we use as a proxy for the typical kinetic energy, is the
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 Spring '08
 SanjoyMahajan

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