arellano (aa39398) – Homework 11 – weathers – (17101)
1
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001
10.0 points
A spring with a springconstant 2
.
9 N
/
cm
is compressed 36 cm and released. The 7 kg
mass skids down the frictional incline of height
27 cm and inclined at a 17
◦
angle.
The acceleration of gravity is 9
.
8 m
/
s
2
.
The path is frictionless except for a dis
tance of 0
.
8 m along the incline which has a
coefficient of friction of 0
.
5
.
7 kg
17
◦
μ
= 0
.
5
0
.
8 m
27 cm
36 cm
k
= 2
.
9 N
/
cm
v
f
Figure:
Not drawn to scale.
What is the final velocity
v
f
of the mass?
Correct answer: 1
.
77868 m
/
s.
Explanation:
Let :
g
= 9
.
8 m
/
s
2
=
,
k
= 2
.
9 N
/
cm = 290 N
/
m
,
x
= 36 cm = 0
.
36 m
,
μ
= 0
.
5
,
ℓ
= 0
.
8 m
,
h
= 0
.
27 m
,
m
= 7 kg
,
and
θ
= 17
◦
,
Consider the kinetic energy of the mass.
The mass receives its initial kinetic energy
from the potential energy of the spring
K
i
=
U
spring
=
1
2
k x
2
(1)
=
1
2
(290 N
/
m) (0
.
36 m)
2
= 18
.
792 J
.
It gains kinetic energy because of the potential
energy lost in moving down the incline
K
gained
=
U
lost
=
m g h
(2)
= (7 kg) (9
.
8 m
/
s
2
) (0
.
27 m)
= 18
.
522 J
.
and loses kinetic energy by doing work on the
frictional surface
K
lost
=
W
fr
=
μ m g ℓ
cos
θ
(3)
= (0
.
5) (7 kg) (9
.
8 m
/
s
2
)
×
(0
.
8 m) cos(17
◦
)
= 26
.
241 J
.
Since energy is concerved, the final kinetic
energy is
K
f
=
U
s
+
U
l
−
W
fr
= (18
.
792 J) + (18
.
522 J)
−
(26
.
241 J)
= 11
.
073 J
.
However, the final kinetic energy is
K
f
=
1
2
m v
2
.
(3)
Multiplying by 2 and dividing by
m
gives us
2
K
f
m
=
v
2
,
so
v
=
radicalbigg
2
K
f
m
=
radicalBigg
2 (11
.
073 J)
(7 kg)
= 1
.
77868 m
/
s
.
Alternate Explanation:
The potential
energy at the top of the hill will be converted
into kinetic energy at the bottom of the hill
minus energy lost due to the nonconservative
friction force.
The potential energy at the top of the hill
consists of the gravitational potential energy
plus the potential energy contained in the
compressed spring.
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arellano (aa39398) – Homework 11 – weathers – (17101)
2
Combining Eqs. 1, 2, 3, and 4, we have
v
=
radicalBigg
2
m
bracketleftbigg
m g h
+
1
2
k x
2
−
μ m g ℓ
cos
θ
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 Fall '10
 Weathers
 Kinetic Energy, Potential Energy, Work, Arellano

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