Physics I
Class 11
11-1

Work

Work is a measure of the energy that a force puts into (+) or takes away from (–) an object as it moves.

We will see that work is a useful way to solve problems where the force on an object is a known function of position.

Example: the force of an object connected to an ideal spring: where

F
 x
  k
 x
is the displacement from equilibrium. (Hooke’s Law)
11-2

Work for Constant Force
W
W



F

F

 d
 d cos(

)
11-3

Vector Dot Product
F
 d
If you know lengths and angle:
If you know components:
If in the same direction:
If in opposite directions:
If at right angles:
W


F
 d cos(

)
W

W

W

F x

F d
 d x


F
 d

F y d y
 
F z d z

W

0
11-4

Work for Variable Force
W
  x x
F x dx i f
(This is the version for one dimension.)
11-5

Work-Energy Theorem

Net work is done on an object by the net force:
W net
 f  x x i
F net dx

Kinetic energy defined for an object:
K
 1
2 m
 v
2  1
2 m
 v x
2  v y
2 

Work-Energy Theorem: (without proof) v z
2

K f

K i

W net
11-6

Class #11
Take-Away Concepts
1.
W


F

 d


F
 d cos(

) (constant force)
W
 f  x x i
F dx (variable force, 1D)
2.
Vector dot product defined.
3.
Kinetic Energy:
K
 1
2 m
4.
Work-Energy Theorem:
K f
 v
2


K i
1
2 m

 v x
W net
2  v y
2  v z
2

5.
Positive net work means an object’s K.E. increases (speeds up).
6.
Negative net work means an object’s K.E. decreases (slows down).
7.
Zero work means an object’s K.E. stays constant (constant speed).
11-7

Activity #11
Work-Energy Theorem
Objective of the Activity:
1.
Use LoggerPro to study the Work-Energy Theorem for a one-dimensional case.
11-8

Class #11 Optional Material
Work-Energy Theorem Proof
W net
 f
 x x i
F net dx
 f
 x x i m a dx
 m f
 x x i a dx
 m f
 x x i d d v dx t
Now we work on the integrand using calculus rules: d v
 d t d v d x d x d t
 d v v d x

1 d
2 d x
Putting that back into the integral:
W net

1
2 m
 x x f i d d x dx

1
2 m
 v f
2  v i
2


K f

K i
11-9