L40{ Work
Work
w
: product of force and distance moved
Let
f
(
x
) be the force acting on an object at each
point
x
in [
a; b
]. Then the
work
,
w
, needed to move
the object from
x
=
a
to
x
=
b
is:
w
= lim
n
!1
n
X
i
=1
f
(
x
i
)
x
w
=
R
b
a
f
(
x
)
dx
To calculate work of a non-constant force, you inte-
grate over the force function.
Unit
:
f
(
x
) in newtons,
d
in meters,
!
W
in joule(
J
)).
f
(
x
) in pounds,
d
in feet,
!
W
in foot-pound.
We will calculate work problems
1.
related to Hook’s Law, Spring Problems
2.
of a cable/rope problem
3.
required to Drain a Tank
1

Finding Work in conjunction with Hooke’s
Law
Hooke’s Law
states that the force required to hold
a spring stretched
0
x
0
units beyond
its natural length
is proportional to the amount/distance
x
stretched:
f
(
x
) =
kx
(
k
= spring constant)
ex.
A force of 10 lb is required to hold a spring
stretched 4in beyond its natural length. How much
work is done in stretching it from its natural length
to 6in beyond its natural length? (Use foot-pounds!)
Want to calculate:
Given:
2

ex.
Suppose 2
J