portillo (gdp347) – oldhomework 36 – Turner – (58185)
1
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001
(part 1 oF 2) 10.0 points
A 7
.
3 kg mass slides on a Frictionless surFace
and is attached to two springs with spring
constants 23 N
/
m and 64 N
/
m as shown in
the fgure.
23 N
/
m
64 N
/
m
7
.
3 kg
±ind the Frequency oF oscillation.
Correct answer: 0
.
549438 Hz.
Explanation:
Let :
k
1
= 23 N
/
m
,
k
2
= 64 N
/
m
,
and
m
= 7
.
3 kg
.
By Hooke’s law,
F
=

k x
=
m a
=
m
d
2
x
dt
2
and
d
2
x
dt
2
+
k
m
x
= 0
,
(1)
whose integral Form has a sine Function
x
(
t
) =
A
sin(
ω t
+
δ
)
,
where
ω
=
r
k
m
, the square root oF the coe²
cient oF
x
in Eq. 1.
Call the displacement oF the mass
x
, and
choose the positive direction to be to the right.
Then, the Forces From the springs on the mass
m
are to the leFt:
F
1
=

k
1
x
and
F
2
=

k
2
x
,
so that Force equilibrium is
F
=

k
parallel
x
F
1
+
F
2
=

k
parallel
x

k
1
x

k
2
x
=

k
parallel
x
k
parallel
=
k
1
+
k
2
.
ω
=
r
k
m
,
so
ω
parallel
=
r
k
parallel
m
=
r
k
1
+
k
2
m
and the Frequency oF oscillation is
f
=
ω
parallel
2
π
=
r
k
1
+
k
2
m
2
π
=
r
23 N
/
m + 64 N
/
m
7
.
3 kg
2
π
=
0
.
549438 Hz
.
002
(part 2 oF 2) 10.0 points
Now the two spring are coupled together as
shown in the fgure.
23 N
/
m
64 N
/
m
7
.
3 kg
±ind the period oF oscillation.
Correct answer: 4
.
12712 s.
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 Spring '10
 McCord
 Energy, Mass, Correct Answer, dt, k2

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