Consider an object that is pulled a distance d at

constant velocity by an applied force FA as in the diagram.


Constant v
F

N
FA

F
A



F fk
FAx
d

F
g
System Box, Earth
What force actually pulls
the box across the floor?
Work W  - the transfer of energy
into or out of a system
by means of an external force


- The area under the Fext vs. x graph,

where Fext is the external force that
transfers energy

Fext

FAy
Fg  FN  FAy
FAx  F fk

FAx  constant, positive

FAx

d
W  Fextd cos

x
For constant Fext
Work Done by a Variable Force
Example 1
 


The force acting upon a box is given by F  x   3.0 x  2.0 where F is in Newtons.


How much work is done by the force in moving the box from x  1.0 m to x  3.0 m?

F N 
Method 1
12
W  Area under the curve
10
W  A03  A01
8
6
4
2
1
2
3

x m 
1


W  3 m 2 N   3 m 9 N 
2


1


 1 m 2 N   1 m 3 N 
2


W  16 J
Work Done by a Variable Force
Example 1
 


The force acting upon a box is given by F  x   3.0 x  2.0 where F is in Newtons.


How much work is done by the force in moving the box from x  1.0 m to x  3.0 m?
Method 2


W  Area under the curve of Fext vs. d
W  13 Fdx
W  13 3.0 x  2.0 dx
3
3 2

W   x  2 x
2
1
3 2
 3 2

W   3  23   1  21
2
 2

W  16 J
Work Done by a Variable Force
Example 2


The force acting upon a box is given by F  4.0 x iˆ  2 y  ˆj  5kˆ where F is in Newtons.

How much work is done by the force in moving the box from ri  2iˆ  3 ˆj  1kˆ

to r  4iˆ  2 ˆj  3kˆ?
f
 
W   F  dr


W   Fxiˆ  Fy ˆj  Fz kˆ  dx iˆ  dy ˆj  dz kˆ

W   Fx dx  Fy dy  Fz dz

W   Fx dx   Fy dy   Fz dz

Work Done by a Variable Force
Example 2


The force acting upon a box is given by F  4.0 x iˆ  2 y  ˆj  5kˆ where F is in Newtons.

How much work is done by the force in moving the box from ri  2iˆ  3 ˆj  1kˆ

to r  4iˆ  2 ˆj  3kˆ?
f
W   Fx dx   Fy dy   Fz dz
4 x 
2 y  dy  5 dz
W
4
2
W
   
2 4
2x 2
2
dx  3
3
1
2 2
y 3
 5 z 
W  2 4  22
  2

2
2
W  24   5  20
W  49 J
3
1
2

 32  53  5 1