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Energy and Work
Energy - the ability of a
body or system of bodies to
perform work.
A body is given energy when
a force does work on it.
But What is Work?
A force does work on a body
(and changes its energy) when it
causes a displacement.
If a force causes no
displacement, it does no work.
Riddle Me This
If a man holds a 50 kg box at arms
length for 2 hours as he stands still, how
much work does he do on the box?
ZERO
NONE
Counterintuitive Results
 There is no work done by a force
if it causes no displacement.
 Forces perpendicular to
displacement, such as the normal
force, can do no work.
 Likewise, centripetal forces
never do work.
Calculating Work
Work is the dot product of
force and displacement.
Work is a scalar resulting
from the interaction of two
vectors.
Vector Multiplication
There are three ways to multiply
vectors:
•Scalar Multiplication
•Dot Product
•Cross Product
Scalar Multiplication
•Magnitude of vector changes.
•Direction of vector does not change.


F  ma
If m = 5 kg
a = 10 m·s-1
F = 50 N
Dot Product
Note that the dot
 
W  A  B product of two vectors
gives a scalar .
 
A  B  AB cos



 is the angle between A and B.

A
θ

B
Dot Product
Geometrically, the dot product is the
projection of one vector on a second
vector multiplied by the magnitude of
the second vector.

A
θ
A cos

B
Calculating Work
 
W  F  s  Fs cos

F
θ
F cos 

s
W   F ( x)dx
Which does more work?
Two forces are acting on the box
shown causing it to move across the
floor. Which force does more work?
F2
θ
F1
Vectors and Work
F
Vectors and Work
F
s
W=F•s
W = F s cos 0o
W=Fs
Maximum positive work
Vectors and Work
F
Vectors and Work
F

s
W=F•s
W = F s cos 
Only the component of force aligned
with displacement does work. Work is
less.
Vectors and Work
F
Vectors and Work
F

s
W=F•s
W = F s cos 180o
W=-Fs
Maximum negative work.
Gravity often does
negative work.
When the load goes up, gravity does negative
work and the crane does positive work.
When the load goes down, gravity does
positive work and the crane does negative
work.
F
mg
Positive, zero, or negative work?
A box is being moved with a velocity v
by a force P (parallel to v) along a
level floor. The normal force is FN, the
frictional force is fk, and the weight of
the box is mg.
Decide which forces do positive, zero,
or negative work.
Positive, zero, or negative work?
v
FN
fk
P
mg
s
Units of Work
 
W  F  s  Fs cos
J = N·m
J=
2
-2
kg·m ·s
Work and variable force
The area under the
curve of a graph of
force vs
displacement gives
the work done by
the force.
F(x)
xb
W = x F(x) dx
a
xa
xb
x