—
1
—
Work
When a body moves a distance
d
along a straight line as a result of being acted on by a force of
constant magnitude
F
in the direction of motion, then the work
W
done by the force is
W
=
Fd
.
We will generalize the definition of work to the case of variable force. Given a force function
F
(
x
)
defined and continuous at each point
x
of the straight line segment
[
a, b
]
, we will define the work
W
done by this variable force in moving a particle along the
x
axis from the point
x
=
a
to the
point
x
=
b
.
Partition the interval
[
a, b
]
into
n
subintervals with the same length
Δ
x
. Choose an arbitrary point
c
k
in the
k
’th interval
[
x
k

1
, x
k
]
. Approximate the work
Δ
W
k
done by the force from the point
x
=
x
k

1
to the point
x
=
x
k
by
Δ
W
k
≈
F
(
c
k
)Δ
x.
We approximate the total work by summing
from
1
to
n
, so
W
≈
n
k
=1
F
(
c
k
)Δ
x.
This is a Riemann sum for
F
(
x
)
. When
Δ
x
→
0
, the sum approaches the definite integral of
F
(
x
)
from
a
to
b
. Therefore, we are motivated to define the work
W
done by the force
F
(
x
)
in moving
a particle from
a
to
b
to be
b
a
F
(
x
)
dx .
Problem.
An electric elevator with a motor at the top has a multistrand cable weighing
2
kg/m.
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 Fall '07
 BERMAN
 Force, 2 kg, 60 meters, 0 meters, 36000 Nm

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