P16_025.PDF

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25. When displaced from equilibrium, the magnitude of the net force exerted by the springs is
|k
1
x
+
k
2
x|
acting in a direction so as to return the block to its equilibrium position (x = 0). Since the acceleration
a
=
d
2
x/dt
2
, Newton’s second law yields
m
d
2
x
=
−k
1
x
−
k
2
x .
dt
2
Substituting
x
=
x
m
cos(ωt +
φ)
and simplifying, we find
ω
2
=
k
1
+
k
2
m
where
ω
is in radians per unit time. Since there are 2π radians in a cycle, and frequency
f
measures
cycles per second, we obtain
1
k
1
+
k
2
ω
=
.
f
=
2π
2π
m
The single springs each acting alone would produce simple harmonic motions of frequency
f
1
=
1
2π
k
1
m
and
f
2
=
1
2π
k
2
,
m
2
2
f
1
+
f
2
.
respectively. Comparing these expressions, it is clear that
f
=

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