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Alvarado, Patrick – Homework 12 – Due: Dec 14 2006, 10:00 am – Inst: Andrei Sirenko
1
This printout should have 19 questions.
Multiplechoice questions may continue on
the next column or page – fnd all choices
beFore answering.
The due time is Central
time.
001
(part 1 oF 2) 10 points
A block on a horizontal Frictionless plane is
attached to a spring, as shown below.
The
block oscillates along the
x
axis with simple
harmonic motion oF amplitude
A
.
m
k
0

A
+
A
v
x
0
Which oF the Following statements about
the block is correct?
1.
At
x
= 0, its velocity is zero.
2.
At
x
=
A
, its acceleration is zero.
3.
At
x
=
A
, its velocity is at a maximum.
4.
At
x
= 0, its acceleration is at a maxi
mum.
5.
At
x
=
A
, its displacement is at a maxi
mum.
correct
Explanation:
The block oscillates From maximum dis
placements at
x
=
A
and
x
=

A
. At those
points the velocity is momentarily zero.
002
(part 2 oF 2) 10 points
Which oF the Following statements about en
ergy is correct?
1.
The kinetic energy oF the block is at a
minimum at
x
= 0.
2.
The potential energy oF the spring is at a
minimum at
x
= 0.
correct
3.
The kinetic energy oF the block is at a
maximum at
x
=
A
.
4.
The potential energy oF the spring is at a
minimum at
x
=
A
.
5.
The kinetic energy oF the block is always
equal to the potential energy oF the spring.
Explanation:
±rom conservation oF energy,
v
= 0 at
x
=
±
A
so the kinetic energy is zero and the spring
potential energy is at its maximum. At
x
= 0,
the spring potential energy is 0 and the kinetic
energy is at its maximum.
keywords:
003
(part 1 oF 1) 10 points
A particle oscillates harmonically
x
=
A
sin(
ωt
+
φ
0
)
,
with amplitude
A
= 21 m, angular Frequency
ω
=
π
s

1
, and initial phase
φ
0
=
π
3
radians.
Every now and then, the particle’s kinetic
energy and potential energy happen to be
equal to each other,
K
=
U
.
When does this equality happen For the frst
time aFter
t
= 0?
1.
0.4167 s
correct
2.
0.1294 s
3.
0.5884 s
4.
0.2238 s
5.
0.7615 s
6.
0.9967 s
7.
0.5167 s
8.
0.8623 s
9.
0.6547 s
10.
0.3467 s
Explanation:
Basic Concepts:
x
=
A
cos(
ω t
+
φ
)
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View Full DocumentAlvarado, Patrick – Homework 12 – Due: Dec 14 2006, 10:00 am – Inst: Andrei Sirenko
2
K
=
1
2
mv
2
∝
sin
2
(
ω t
+
δ
)
U
=
1
2
mx
2
∝
= cos
2
(
ω t
+
δ
)
p
sin(
π t
+
π/
3) =
p
cos(
π t
+
π/
3)

sin(
π t
+
π/
3)

=

cos(
π t
+
π/
3)

.
We note that

sin(
x
)

=

cos(
x
)

when
x
=
(2
n
+ 1)
π
4
,
where
n
is an integer. Hence
(2
n
+ 1)
π
4
=
π t
+
π
3
(2
n
+ 1)
4
=
t
+
1
3
,
so
t
=
(2
n
+ 1)
4

1
3
=
0
.
416667 s
.
Note:
When
n
= 0, we obtain a negative
value for time.
The Frst positive answer is
when
n
= 1.
keywords:
004
(part 1 of 1) 10 points
The displacement in simple harmonic motion
is a maximum when
1.
the potential energy function has its max
imum value per cycle.
correct
2.
kinetic energy is maximum.
3.
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