ph2a_quiz1

ph2a_quiz1 - QUIZ
1

 


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1

 
 Two
pendula
pivot
about
an
axis
A
and
are
constrained
to
remain
in
planes
 perpendicular
to
that
axis.
Furthermore
the
pivot
points
remain
fixed.
Each
has
a
 length
L
and
a
mass
m.
The
angles
from
the
vertical
are
θ1
and
θ2.
(In
the
picture
 θ1<0
and
θ2>0).
The
two
pendula
are
coupled
by
a
weak
torsional
spring
that
obeys
 Hooke's
Law
with
torsional
spring
constant
κ
 τ 2 = −κ (θ 2 − θ1 ) = −τ 1 
 where
τ2
is
the
torque
on
mass
2
and
τ1
is
the
torque
on
mass
1.
 (a)
[2.5
pts]
Write
down
the
linear
coupled
differential
equations
describing
the
 motion
of
the
pendula
when
the
angles
 θ i 1 .
 (b)
[2.5
pts]
Find
the
two
normal‐mode
frequencies
(eigenfrequencies).
 (c)
[2.5
pts]
Find
the
two
normal‐mode
functions
(eigenfunctions).

Each
 eignefunction
can
be
written
as
a
vector
amplitude
times
a
phasor
(complex
 exponential).
The
overall
amplitude
and
phase
(or
complex
amplitude)
for
each
 eigenfunction
can
remain
undetermined.
 (d)
[2.5
pts]
Determine
the
time
evolution
of
each
pendulum
with
the
initial
 conditions
 θ1 ( 0 ) = 0, θ 2 ( 0 ) = A ,
with
both
starting
at
rest,
and
sketch
the
two
 functions
for
at
least
two
beat
cycles.
You
can
assume
that
the
spring
constant
 1 κ = mgL .
Please
no
graphing
calculators
allowed!
 8 ...
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This note was uploaded on 01/28/2011 for the course PH 2 taught by Professor Dudko during the Spring '09 term at UCSD.

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