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mg
()
(
)
,
(14.123)
zt
=
C ωt δ
sin
+
−
k
the amplitude
C
and phase
δ
are given by Equation (14.114) with the modifica
tion that
z
is replaced by
0
z gk
. The influence of gravity has the effect of
increasing the oscillation amplitude and advancing the phase angle corresponding
to the initial displacement.
+
14.3.2 Pendulum Motion
Suppose that an object of mass
m
under the influence of gravity is attached to a
massless rod of length
L
hanging from a fixed point coinciding with the origin as
shown in Figure 14.10. We assume that the rod is able to pivot freely about its
fixed end and that the mass is able to move in the
x

z
plane. Let
I
be the moment
of inertia of the object with respect to the
y
axis (about which the mass rotates). If
all of the mass is concentrated at a single point, then
I
=
mL
2
.
()
r
represent the position of the object. Gravity pulls downward on the
object with the force
m
g
, exerting a torque
Let
()
τ
given by
()
()
τ r
t
=×
t
m
g
.
(14.124)
()
The resulting angular acceleration
α
is
()
()
τ
r
t
t
I
×
m
g
()
α
t
=
=
.
(14.125)
I
()
()
Since
α
are always perpendicular to the
x

z
plane in which the pendu
lum rotates, we can write them as scalar quantities
τ
and
()
()
τ t
and
α t
. Equation
(14.125) can then be written as
mgL
()
()
()
α t θ t
=
= −
sin
θ t
,
(14.126)
I
()
where
θ t
is the counterclockwise angle between the pendulum and the negative
z
axis.
()
θ t
due to
the presence of the sine function. We can, however, transform the equation into a
form that can be solved by replacing
Equation (14.126) cannot be solved analytically for the function
()
sin
θ t
with the first term of its power
series:
mgL
()
()
θ t
=−
θ t
.
(14.127)
I
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