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Nodes occur where cos(
kx
) = 0 or
kx
=
n
π
+
π
/2, where
n
is an integer (including zero).
Since
k
= 1.0
π
m
–1
, this means
()
1
2
(1.00 m)
xn
=+
. Thus, the smallest value of
x
which
corresponds to a node is
x
= 0.500 m (
n
=0).
(e) The second smallest value of
x
which corresponds to a node is
x
= 1.50 m (
n
=1).
(f) The third smallest value of
x
which corresponds to a node is
x
= 2.50 m (
n
=2).
(g) The displacement is a maximum where cos(
kx
) =
±
1. This means
kx
=
n
π
, where
n
is
an integer. Thus,
x
=
n
(1.00 m). The smallest value of
x
which corresponds to an anti
node (maximum) is
x
= 0 (
n
=0).
(h) The second smallest value of
x
which corresponds to an antinode (maximum) is
1.00 m
x
=
(
n
=1).
(i) The third smallest value of
x
which corresponds to an antinode (maximum) is
2.00 m
x
=
(
n
=2).
55. (a) The angular frequency is
ω
= 8.00
π
/2 = 4.00
π
rad/s, so the frequency is
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 Spring '09
 NAJAFZADEH
 mechanics

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