Physics For Scientists And Engineers 6E - part 48
SECTION 7.4 • Work Done by a Varying Force
189
Example 7.4 Calculating Total Work Done from a Graph
A force acting on a particle varies with x, as shown in
Figure 7.8. Calculate the work done by the force as the parti-
cle moves from x # 0 to x # 6.0 m.
Solution The work done by the force is equal to the area
under the curve from x
A
#
0 to x
C
#
6.0 m. This area is
equal to the area of the rectangular section from ! to "
plus the area of the triangular section from " to #. The
area of the rectangle is (5.0 N)(4.0 m) # 20 J, and the area
of the triangle is
. Therefore, the to-
tal work done by the force on the particle is
25 J.
1
2
(5.0
N)(2.0
m) # 5.0
J
Example 7.5 Work Done by the Sun on a Probe
Graphical Solution The negative sign in the equation for
the force indicates that the probe is attracted to the Sun.
Because the probe is moving away from the Sun, we expect
to obtain a negative value for the work done on it. A spread-
sheet or other numerical means can be used to generate a
graph like that in Figure 7.9b. Each small square of the grid
corresponds to an area (0.05 N)(0.1 * 10
11
m) # 5 * 10
8
J.
The work done is equal to the shaded area in Figure 7.9b. Be-
cause there are approximately 60 squares shaded, the total
The interplanetary probe shown in Figure 7.9a is attracted
to the Sun by a force given by
in SI units, where x is the Sun-probe separation distance.
Graphically and analytically determine how much work is
done by the Sun on the probe as the probe
–
Sun separation
changes from 1.5 * 10
11
m to 2.3 * 10
11
m.
F # &
1.3 * 10
22
x
2
1
2
3
4
5
6
x(m)
0
5
F
x
(N)
#
!
"
Figure 7.8 (Example 7.4) The force acting on a particle is con-
stant for the first 4.0 m of motion and then decreases linearly
with x from x
B
#
4.0 m to x
C
#
6.0 m. The net work done by
this force is the area under the curve.
If the size of the displacements is allowed to approach zero, the number of terms in
the sum increases without limit but the value of the sum approaches a definite value
equal to the area bounded by the F
x
curve and the x axis:
Therefore, we can express the work done by F
x
as the particle moves from x
i
to x
f
as
(7.7)
This equation reduces to Equation 7.1 when the component F
x
#
F cos ! is constant.
If more than one force acts on a system and the system can be modeled as a particle, the
total work done on the system is just the work done by the net force. If we express the
net force in the x direction as +F
x
, then the total work, or net work, done as the particle
moves from x
i
to x
f
is
(7.8)
If the system cannot be modeled as a particle (for example, if the system consists of
multiple particles that can move with respect to each other), we cannot use Equation
7.8. This is because different forces on the system may move through different dis-
placements. In this case, we must evaluate the work done by each force separately and
then add the works algebraically.
#
W # W
net
#
$
x
f
x
i
%
#
F
x
&
dx
W #
$
x
f
x
i
F
x
dx
lim
"
x :0
#
x
f
x
i
F
x
"
x #
$
x
f
x
i
F
x
dx
(a)
F
x
Area =
∆A = F
x
∆x
F
x
x
x
f
x
i
∆x
(b)
F
x
x
x
f
x
i
Work
Figure 7.7 (a) The work done by
the force component F
x
for the
small displacement "x is F
x
"
x,
which equals the area of the shaded
rectangle. The total work done for
the displacement from x
i
to x
f
is ap-
proximately equal to the sum of the
areas of all the rectangles. (b) The
work done by the component F
x
of
the varying force as the particle
moves from x
i
to x
f
is exactly equal to
the area under this curve.