Part II: The Answers
454
Answers
701–800
30
5
100
600
N
m
N m
= ( )
⋅ =
k
k
Now compute the work, making sure to convert the distances into meters. Because
you’re computing the work while moving the spring from 5 centimeters beyond its
natural length to 10 centimeters beyond its natural length, the limits of integration
are from 5
100
to 10
100
.
W
x dx
x
=
=
=
−
(
)
=
∫ 600
300
300 0 1
005
2 25
5 100
10 100
2
0 05
0 10
2
2
.
.
( . ) ( . )
. J J
725.
400 ft·lb
Let n be the number of subintervals of length Δ x, and let x i
* be a sample point in the ith
subinterval [x i–1 , x i ]. Let x i
* be the distance from the middle of the chain.
Notice that when you’re lifting the end of the chain to the top, only the bottom half of
the chain moves. A section of chain that is Δ x in length and that is x i
* feet from the
middle of the chain weighs 4Δ x pounds and will move approximately 2x i
* feet.
Therefore, the work required is
W
x
x
x dx
x
n
i
i
n
=
( )
=
=
=
⋅
→∞ =
∑
∫
lim
(
)
*
2
4
8
4
400
1
0
10
2
0
10
∆
ft lb
454
Answers
701–800
30
5
100
600
N
m
N m
= ( )
⋅ =
k
k
Now compute the work, making sure to convert the distances into meters. Because
you’re computing the work while moving the spring from 5 centimeters beyond its
natural length to 10 centimeters beyond its natural length, the limits of integration
are from 5
100
to 10
100
.
W
x dx
x
=
=
=
−
(
)
=
∫ 600
300
300 0 1
005
2 25
5 100
10 100
2
0 05
0 10
2
2
.
.
( . ) ( . )
. J J
725.
400 ft·lb
Let n be the number of subintervals of length Δ x, and let x i
* be a sample point in the ith
subinterval [x i–1 , x i ]. Let x i
* be the distance from the middle of the chain.
Notice that when you’re lifting the end of the chain to the top, only the bottom half of
the chain moves. A section of chain that is Δ x in length and that is x i
* feet from the
middle of the chain weighs 4Δ x pounds and will move approximately 2x i
* feet.
Therefore, the work required is
W
x
x
x dx
x
n
i
i
n
=
( )
=
=
=
⋅
→∞ =
∑
∫
lim
(
)
*
2
4
8
4
400
1
0
10
2
0
10
∆
ft lb