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Consider the massspring systemwhere dry frictionis present between the block
and the horizontalsurface. The frictionalforce has a constant magnitude
µ
mg
(
is the coefficientoffriction) and alwaysopposes the motion. The differential
equation for the motion of the block can beexpressedas
µ
k
m
y
y
y
=−
−
µ
g

y

where
y
ismeasured fromthe positionwhere the spring is unstretched. If the block
is releasedfromrest at
y
=
y
0
, verifybynumerical integrationthat thenextpositive
peak valueof
y
is
y
0
−
4
µ
mg
/
k
(this relationship can be derived analytically). Use
80665 m/s
2
and
y
0
=
k
=
3000 N/m,
m
=
6 kg,
µ
=
0
.
5,
g
=
9
.
0
.
1m.
18.
Integrate the following problemsfrom
x
=
0to20 and plot
y
vs.
x
:
(a)
y
+
.
5(
y
2
−
1)
y
+
=
=
y
(0)
=
0
y
0
y
(0)
1
0
(b)
y
=
y
(0)
y
cos 2
x
y
(0)
=
0
=
1
These differentialequations arise in nonlinear vibration analysis.
19.
The solution of the problem
1
x
y
+
y
+
y
(0)
y
y
(0)
=
1
=
0
is the Bessel function
J
0
(
x
)
.
Use numerical integration to compute
J
0
(5) and com
pare the result with
−
0
.
17760, the valuelistedinmathematicaltables.
Hint
: to
10
−
12
.
avoid singularityat
x
=
0, start the integrationat
x
=
20.
Consider the initial value problem
y
=
y
(0)
16
.
81
y
y
(0)
=
1
.
0
=−
4
.
1
(a) Derive the analytical solution. (b) Do youanticipate difficulties in numerical
solution of this problem? (c) Try numerical integration from
x
=
0to8 to see if
your concerns were justified.
21.
2
R
i
2
R
i
1
R
E
(
t
)
i
1
L
C
i
2
Kirchoff'sequationsfor the circuit shown are
L
di
1
dt
+
Ri
1
+
2
R
(
i
1
+
i
2
)
=
E
(
t
)
(a)
q
2
C
+
Ri
2
+
2
R
(
i
2
+
i
1
)
=
E
(
t
)
(b)
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