Phys. 7268
Assignment 4
Due: 2/24/11
Problem 1
Identification of a Dimensionless Stress Parameter.
To see how a dimensionless stress parameter like the
Rayleigh number might be discovered from known dynamical equations, consider the evolution equation for a damped
driven pendulum,
m
¨
θ
+
α
˙
θ
+
C
sin(
θ
) =
A
sin(
ω
0
t
)
,
(1)
where
m
is the mass of the pendulum,
θ
(
t
) is the angle of the pendulum with respect to the vertical (so
θ
= 0 means
the particle hangs down vertically underneath the fulcrum),
α
is the damping coefficient,
A
is the amplitude of the
external driving,
ω
0
is the frequency of the external driving, and overhead dots denotes differentiation with respect to
time, e.g.,
¨
θ
=
d
2
θ/dt
2
. One parameter in this equation can be eliminated immediately by dividing both sides of Eq.
(1) by
m
, which redefines
m
= 1 and the other parameters to the values
α/m
,
C/m
, and
A/m
. A second parameter
can be eliminated by defining a new dimensionless time coordinate
¯
t
by the transformation:
t
=
c
t
¯
t,
where
c
t
is a new unit of time.
(a) Write down Eq. (1) in the new variable
¯
t
by substituting and using the chain rule of calculus.
(b) Describe the three possible ways to choose the value of
c
t
so as to eliminate a second coefficient in the new
equation by setting the coefficient equal to one. Discuss also the physical meaning of the three different choices
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 Spring '11
 Staff
 Parametric equation, Inequation, dimensionless stress parameter

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