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portillo (gdp347) – oldhomework 35 – Turner – (58185)
1
This printout should have 11 questions.
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beFore answering.
001
(part 1 oF 2) 10.0 points
The motion oF a piston in an auto engine is
simple harmonic. The piston travels back and
Forth over a distance oF 26 cm, and the piston
has a mass oF 2
.
1 kg.
7817 rpm
26 cm
What is the maximum speed oF the piston
when the engine is running at 7817 rpm?
Correct answer: 106
.
417 m
/
s.
Explanation:
Let :
A
=
d
2
=
26 cm
2
= 0
.
13 m
,
and
f
= 7817 rpm
ω
= 2
π f
= 2
π
(7817 rpm)
(60 s
/
min)
= 818
.
594 rad
/
s
.
±rom conservation oF energy,
K
max
=
U
max
, so
1
2
mv
2
=
1
2
k A
2
.
This yields
v
A
=
r
k
m
,
(1)
where
A
is the maximum displacement. In
this case, the displacement is halF oF the dis
tance that the piston travels.
±rom the reFerence circle, the Frequency oF
SHM equals
f
=
1
T
=
2
π
ω
=
1
2
π
r
k
m
=
1
2
π
v
A
,
so
v
= 2
π f
d
2
=
ω A
= (818
.
594 rad
/
s) (0
.
13 m)
=
106
.
417 m
/
s
,
where
ω
= 2
π f
and
A
=
d
2
.
Remember
to
convert
the
Frequency
7817 rpm to Hz by converting minutes to
seconds by dividing by 60 s.
002
(part 2 oF 2) 10.0 points
What is the maximum Force acting on the
piston when the engine is running at the same
rate?
Correct answer: 1
.
82936
×
10
5
N.
Explanation:
Using Eq. 1, we have
k
=
m
v
2
A
2
=
m
(
ω A
)
2
A
2
=
mω
2
= (2
.
1 kg) (818
.
594 rad
/
s)
2
= 1
.
4072
×
10
6
m
/
s
2
.
b
V
F
b
=
k A
=
mω
2
A
= (2
.
1 kg) (818
.
594 rad
/
s)
2
(0
.
13 m)
=
1
.
82936
×
10
5
N
.
003
10.0 points
The mass oF the deuterium molecule D
2
is
twice that oF the hydrogen molecule H
2
.
IF the vibrational Frequency oF H
2
is 1
.
36
×
10
14
Hz, what is the vibrational Frequency
oF D
2
, assuming that the “spring constant”
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2
of attracting forces is the same for the two
species?
Correct answer: 9
.
61665
×
10
13
Hz.
Explanation:
Let :
M
D
= 2
M
H
.
The angular frequencies depend only on
spring constant and mass:
ω
=
r
k
M
∝
r
1
M
ω
D
=
R
k
M
D
ω
H
=
R
k
M
H
The spring constants
k
are the same, so
ω
D
ω
H
=
R
M
H
M
D
=
R
M
H
2
M
H
=
r
1
2
=
1
√
2
.
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This note was uploaded on 01/25/2012 for the course CHEM 302 taught by Professor Mccord during the Spring '10 term at University of Texas at Austin.
 Spring '10
 McCord

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