Mat 303 Calculus Iv Homework 7 Worksheet With Answers - State University Of New York At Stony Brook - 2013 Page 5
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6MAT 303 Spring 2013
Calculus IV with Applications
3.4.16. A mass m
3 is attached to both a spring with spring constant k
63 and
a dashpot with damping constant c
30. The mass is set in motion with initial posi-
tion x
2 and initial velocity v
2. Find the position function x t and determine
0
0
whether the motion is overdamped, critically damped, or underdamped. If it is under-
pt
damped, write the position function in the form x t
Ce
t
cos
. Also, find
1
1
the undamped position function u t
C
t
cos
that would result if the mass
0
0
0
on the spring were set in motion with the same initial position and velocity but wth the
dashpot disconnected (c
0).
k
c
30
2
2
Solution: We first find that
21, and p
5. Then p
25,
0
m
2m
2 3
2
which is larger than
, so the motion is overdamped. The roots of the DE are r
0
2
2
p
p
2, or r
7 and r
5
3. Hence, the general solution is
0
7t
3t
7t
3t
x t
c
e
c
e
,
v t
x t
7c
e
3c
e
.
1
2
1
2
Setting t
0 and matching the initial conditions, x 0
c
c
2 and v 0
7c
1
2
1
2. Solving for c
and c
, c
c
c
3c
2
, so
7c
3 2
2, and
4c
8, Then
2
1
2
2
1
1
1
1
c
2, so c
4, and the solution is
1
2
3t
7t
x t
4e
3e
.
Removing the damping, we get simple harmonic motion with the frequency
21,
0
so the general solution is
u t
A cos
t
B sin
t,
u t
A
sin
t
B
cos
t.
0
0
0
0
0
0
2
Then u 0
A
2 and u 0
B
2, so A
2 and B
. Then u t
0
21
B
4
22
1
2
2
C cos
t
C
A
B
4
2
, and
arctan
arctan
.
0
21
21
A
21
3.4.18. A mass m
2 is attached to both a spring with spring constant k
50 and a
dashpot with damping constant c
12. The mass is set in motion with initial position
x
0 and initial velocity v
8. Find the position function x t and determine
0
0
whether the motion is overdamped, critically damped, or underdamped. If it is under-
pt
damped, write the position function in the form x t
Ce
t
cos
. Also, find
1
1
the undamped position function u t
C
cos
t
that would result if the mass
0
0
0
on the spring were set in motion with the same initial position and velocity but wth the
dashpot disconnected (c
0).
2
k
50
c
12
Solution: We have that
25, so
5, and p
3, so the motion
0
0
m
2
2m
2 2
2
2
is underdamped. The pseudofrequency is
p
25
9
16
4, so the
1
0
general solution is
3t
3t
x t
c
e
cos 4t
c
e
sin 4t,
1
2
3t
3t
v t
c
e
3 cos 4t
4 sin 4t
c
e
3 sin 4t
4 cos 4t
2
1
5
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