
540 Classical Mechanics
© 2010 Taylor & Francis Group, LLC
so that the length of the loaded string remains constant at D = (n + 1)d. The total mass of the string,
M= nm, also remains constant. Thus, as the number n of particles increases, the mass of an indi-
vidual particle decreases. Now as n becomes very large, we can replace the sine term in Equation
16.11 by the argument
sin
()
N
n
N
n
21
+
≅
+
and Equation 16.11 reduces to
ωω
ππ
ρ
π
ρ
ππ
ρ
N
N
n
S
md
N
n
SN
SN
D
N
D
S
=
+
=
+
=
+
== =
2
21
0
()
NNω
1
(16.17)
where the linear density
md
/and
1
() . (16.18)
ω
1
can be considered to be the fundamental frequency. The other different normal frequen-
cies can then be reg ...