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CHAPTER 6. WORK, KINETIC ENERGY AND POTENTIAL ENERGY
0.0
5.0
10.0
15.0
x, m
0.00
1.00
2.00
3.00
4.00
Fx, N
Figure 6.1:
Force F
x
, which depends on position x; see Example 4.
The force does 4950 J of work.
4. A particle is subject to a force F
x
that varies with position as in Fig. 6.1. Find
the work done by the force on the body as it moves (a) from x = 0 to x = 5.0 m,
(b) from x = 5.0 m to x = 10 m and (c) from x = 10 m to x = 15 m. (d) What is the
total work done by the force over the distance x = 0 to x = 15 m?
[Ser4 7-23]
(a) Here the force is not the same all through the object's motion, so we can't use the
simple formula W = F
x
x. We must use the more general expression for the work done when
a particle moves along a straight line,
W =
x
f
x
i
F
x
dx .
Of course, this is just the "area under the curve" of F
x
vs. x from x
i
to x
f
.
In part (a) we want this "area" evaluated from x = 0 to x = 5.0 m. From the figure, we
see that this is just half of a rectangle of base 5.0 m and height 3.0 N. So the work done is
W =
1
2
(3.0 N)(5.0 m) = 7.5 J .
(Of course, when we evaluate the "area", we just keep the units which go along with the
base and the height; here they were meters and newtons, the product of which is a joule.)
So the work done by the force for this displacement is 7.5 J.
(b) The region under the curve from x = 5.0 m to x = 10.0 m is a full rectangle of base
5.0 m and height 3.0 N. The work done for this movement of the particle is
W = (3.0 N)(5.0 m) = 15. J
(c) For the movement from x = 10.0 m to x = 15.0 m the region under the curve is a half
rectangle of base 5.0 m and height 3.0 N. The work done is
W =
1
2
(3.0 N)(5.0 m) = 7.5 J .