2. November 22, 2022
Energy
Energy and Mechanical Energy
Work
Kinetic Energy
Work and Kinetic Energy
The Scalar Product of Two Vectors
3. November 22, 2022
Why Energy?
Why do we need a concept of energy?
The energy approach to describing motion is
particularly useful when Newton’s Laws are
difficult or impossible to use
Energy is a scalar quantity. It does not have a
direction associated with it
4. November 22, 2022
What is Energy?
Energy is a property of the state of a system,
not a property of individual objects: we have to
broaden our view.
Some forms of energy:
Mechanical:
Kinetic energy (associated with motion, within system)
Potential energy (associated with position, within system)
Chemical
Electromagnetic
Nuclear
Energy is conserved. It can be transferred from
one object to another or change in form, but
cannot be created or destroyed
5. November 22, 2022
Kinetic Energy
Kinetic Energy is energy associated with the
state of motion of an object
For an object moving with a speed of v
SI unit: joule (J)
1 joule = 1 J = 1 kg m2/s2
2
2
1
mv
KE
7. November 22, 2022
Work W
Start with Work “W”
Work provides a link between force and energy
Work done on an object is transferred to/from it
If W > 0, energy added: “transferred to the
object”
If W < 0, energy taken away: “transferred from
the object”
x
F
mv
mv x
2
0
2
2
1
2
1
8. November 22, 2022
Definition of Work W
The work, W, done by a constant force on an
object is defined as the product of the component
of the force along the direction of displacement
and the magnitude of the displacement
F is the magnitude of the force
Δ x is the magnitude of the
object’s displacement
q is the angle between
x
F
W
)
cos
( q
and
F x
9. November 22, 2022
Work Unit
This gives no information about
the time it took for the displacement to occur
the velocity or acceleration of the object
Work is a scalar quantity
SI Unit
Newton • meter = Joule
N • m = J
J = kg • m2 / s2 = ( kg • m / s2 ) • m
x
F
W
)
cos
( q
x
F
mv
mv
)
cos
(
2
1
2
1 2
0
2
q
10. November 22, 2022
Work: + or -?
Work can be positive, negative, or zero. The
sign of the work depends on the direction of
the force relative to the displacement
Work positive: W > 0 if 90°> q > 0°
Work negative: W < 0 if 180°> q > 90°
Work zero: W = 0 if q = 90°
Work maximum if q = 0°
Work minimum if q = 180°
x
F
W
)
cos
( q
11. November 22, 2022
Example: When Work is Zero
A man carries a bucket of water
horizontally at constant velocity.
The force does no work on the
bucket
Displacement is horizontal
Force is vertical
cos 90° = 0
x
F
W
)
cos
( q
12. November 22, 2022
Example: Work Can Be
Positive or Negative
Work is positive when lifting
the box
Work would be negative if
lowering the box
The force would still be upward,
but the displacement would be
downward
13. November 22, 2022
Work Done by a Constant Force
The work W done on a system by
an agent exerting a constant force
on the system is the product of
the magnitude F of the force, the
magnitude Δr of the displacement
of the point of application of the
force, and cosθ, where θ is the
angle between the force and
displacement vectors:
q
cos
r
F
r
F
W
F
II
F
III
r
F
I
r
F
IV
r
r
0
I
W
q
cos
r
F
WIV
r
F
WIII
r
F
WII
14. November 22, 2022
Work and Force
An Eskimo pulls a sled as shown. The total mass
of the sled is 50.0 kg, and he exerts a force of
1.20 × 102 N on the sled by pulling on the rope.
How much work does he do on the sled if θ =
30°and he pulls the sled 5.0 m ?
J
m
N
x
F
W
2
2
10
2
.
5
)
0
.
5
)(
30
)(cos
10
20
.
1
(
)
cos
(
q
15. November 22, 2022
Work Done by Multiple Forces
If more than one force acts on an object, then
the total work is equal to the algebraic sum of
the work done by the individual forces
Remember work is a scalar, so
this is the algebraic sum
net by individual forces
W W
r
F
W
W
W
W F
N
g
net
)
cos
( q
16. November 22, 2022
Work and Multiple Forces
Suppose µk = 0.200, How much work done on
the sled by friction, and the net work if θ = 30°
and he pulls the sled 5.0 m ?
q
q
sin
0
sin
,
F
mg
N
F
mg
N
F y
net
J
m
N
s
m
kg
x
F
mg
x
N
x
f
x
f
W
k
k
k
k
fric
2
2
2
10
3
.
4
)
0
.
5
)(
30
sin
10
2
.
1
/
8
.
9
0
.
50
)(
200
.
0
(
)
sin
(
)
180
cos
(
q
J
J
J
W
W
W
W
W g
N
fric
F
net
0
.
90
0
0
10
3
.
4
10
2
.
5 2
2
17. November 22, 2022
Kinetic Energy
Kinetic energy associated with the motion of
an object
Scalar quantity with the same unit as work
Work is related to kinetic energy
2
2
1
mv
KE
x
F
mv
mv net
2
0
2
2
1
2
1
net f i
W KE KE KE
19. November 22, 2022
Work-Kinetic Energy Theorem
When work is done by a net force on an object
and the only change in the object is its speed,
the work done is equal to the change in the
object’s kinetic energy
Speed will increase if work is positive
Speed will decrease if work is negative
net f i
W KE KE KE
2
0
2
2
1
2
1
mv
mv
Wnet
20. November 22, 2022
Work and Kinetic Energy
The driver of a 1.00103 kg car traveling on the interstate
at 35.0 m/s slam on his brakes to avoid hitting a second
vehicle in front of him, which had come to rest because of
congestion ahead. After the breaks are applied, a constant
friction force of 8.00103 N acts on the car. Ignore air
resistance. (a) At what minimum distance should the brakes
be applied to avoid a collision with the other vehicle? (b) If
the distance between the vehicles is initially only 30.0 m, at
what speed would the collisions occur?
21. November 22, 2022
Work and Kinetic Energy
(a) We know
Find the minimum necessary stopping distance
N
f
kg
m
v
s
m
v k
3
3
0 10
00
.
8
,
10
00
.
1
,
0
,
/
0
.
35
2
3
3
)
/
0
.
35
)(
10
00
.
1
(
2
1
)
10
00
.
8
( s
m
kg
x
N
2
2
2
1
2
1
i
f
fric
N
g
fric
net mv
mv
W
W
W
W
W
2
0
2
1
0 mv
x
fk
m
x 6
.
76
22. November 22, 2022
Work and Kinetic Energy
(b) We know
Find the speed at impact.
Write down the work-energy theorem:
N
f
kg
m
v
s
m
v k
3
3
0 10
00
.
8
,
10
00
.
1
,
0
,
/
0
.
35
N
f
kg
m
s
m
v
m
x k
3
3
0 10
00
.
8
,
10
00
.
1
,
/
0
.
35
,
0
.
30
x
f
m
v
v k
f
2
2
0
2
2
2
2
1
2
1
i
f
k
fric
net mv
mv
x
f
W
W
2
2
3
3
2
2
/
745
)
30
)(
10
00
.
8
)(
10
00
.
1
2
(
)
/
35
( s
m
m
N
kg
s
m
vf
s
m
vf /
3
.
27
23. Work Done By a Spring
Spring force
November 22, 2022
x
F kx
26. Fig. 7.9, p. 173
0
lim
f
f
i
i
x
x
x x
x
x
x
F x F dx
max
0
2
1
2
f f
i i
x x
x
x x
x
W F dx kxdx
kxdx kx
2 2
1 1
2 2
f
i
x
i f
x
W kxdx kx kx
Work done by
spring on block
27. Measuring Spring Constant
Start with spring at its
natural equilibrium length.
Hang a mass on spring and
let it hang to distance d
(stationary)
From
so can get spring constant.
November 22, 2022
0
x
F kx mg
mg
k
d
28. November 22, 2022
Scalar (Dot) Product of 2 Vectors
The scalar product of
two vectors is written
as
It is also called the dot
product
q is the angle
between A and B
Applied to work, this
means
A B
A B cos q
A B
cos
W F r q
F r
29. November 22, 2022
Dot Product
The dot product says
something about how parallel
two vectors are.
The dot product (scalar
product) of two vectors can be
thought of as the projection of
one onto the direction of the
other.
Components
A
x
A
A
i
A
AB
B
A
q
q
cos
ˆ
cos
B
z
z
y
y
x
x B
A
B
A
B
A
B
A
q
B
A )
cos
( q
)
cos
( q
B
A
30. November 22, 2022
Projection of a Vector: Dot Product
The dot product says
something about how parallel
two vectors are.
The dot product (scalar
product) of two vectors can be
thought of as the projection of
one onto the direction of the
other.
Components
A
B
z
z
y
y
x
x B
A
B
A
B
A
B
A
p/2
Projection is zero
x
A
A
i
A
AB
B
A
q
q
cos
ˆ
cos
1
ˆ
ˆ
;
1
ˆ
ˆ
;
1
ˆ
ˆ
0
ˆ
ˆ
;
0
ˆ
ˆ
;
0
ˆ
ˆ
k
k
j
j
i
i
k
j
k
i
j
i
31. November 22, 2022
Derivation
How do we show that ?
Start with
Then
But
So
z
z
y
y
x
x B
A
B
A
B
A
B
A
k
B
j
B
i
B
B
k
A
j
A
i
A
A
z
y
x
z
y
x
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
)
ˆ
ˆ
ˆ
(
ˆ
)
ˆ
ˆ
ˆ
(
ˆ
)
ˆ
ˆ
ˆ
(
ˆ
)
ˆ
ˆ
ˆ
(
)
ˆ
ˆ
ˆ
(
k
B
j
B
i
B
k
A
k
B
j
B
i
B
j
A
k
B
j
B
i
B
i
A
k
B
j
B
i
B
k
A
j
A
i
A
B
A
z
y
x
z
z
y
x
y
z
y
x
x
z
y
x
z
y
x
1
ˆ
ˆ
;
1
ˆ
ˆ
;
1
ˆ
ˆ
0
ˆ
ˆ
;
0
ˆ
ˆ
;
0
ˆ
ˆ
k
k
j
j
i
i
k
j
k
i
j
i
z
z
y
y
x
x
z
z
y
y
x
x
B
A
B
A
B
A
k
B
k
A
j
B
j
A
i
B
i
A
B
A
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
32. November 22, 2022
Scalar Product
The vectors
Determine the scalar product
Find the angle θ between these two vectors
ˆ
2
ˆ
and
ˆ
3
ˆ
2 j
i
B
j
i
A
?
B
A
3
.
60
65
4
cos
65
4
5
13
4
cos
5
2
)
1
(
13
3
2
1
2
2
2
2
2
2
2
2
q
q
AB
B
A
B
B
B
A
A
A y
x
y
x
4
6
-2
2
3
(-1)
2
y
y
x
x B
A
B
A
B
A
