Goals:
Lecture 15
• Chapter 11
 Employ the dot product
 Employ conservative and non-conservative forces
 Use the concept of power (i.e., energy per time)
• Chapter 12
 Extend the particle model to rigid-bodies
 Understand the equilibrium of an extended object.
 Understand rigid object rotation about a fixed axis.
 Employ “conservation of angular momentum” concept
Assignment:
 HW7 due March 10th
 For Thursday: Read Chapter 12, Sections 7-11
do not concern yourself with the integration process in
regards to “center of mass” or “moment of inertia”
Physics 207: Lecture 15, Pg 1
Scalar Product (or Dot Product)
 
 
A  B  A B cosq
 Useful for finding parallel components
A  î = Ax
îî=1
îĵ=0
A
ĵ
q Ay
î Ax
 Calculation can be made in terms of
components.
A  B = (Ax )(Bx) + (Ay )(By ) + (Az )(Bz )
Calculation also in terms of magnitudes and relative angles.
A  B ≡ | A | | B | cos q
You choose the way that works best for you!
Physics 207: Lecture 15, Pg 2
Work in terms of the dot product
Ingredients: Force ( F ), displacement (  r )
Work, W, of a constant force F
acts through a displacement  r :
F




W  F cosq r  F  r
q
r


F  dr
Looks just like a Dot Product!
If the path is curved dW 

and
r


W   F  dr


F  dr at each point
f

ri
Physics 207: Lecture 15, Pg 4
Energy and Work
Work, W, is the process of energy transfer in which a
force component parallel to the path acts over a
distance; individually it effects a change in energy
of the “system”.
1. K or Kinetic Energy
2. U or Potential Energy (Conservative)
and if there are losses (e.g., friction, non-conservative)
3. ETh Thermal Energy
Positive W if energy transferred to a system
Physics 207: Lecture 15, Pg 6
A child slides down a playground slide at
constant speed. The energy transformation
is
A.
B.
C.
D.
E.
UK
U  ETh
KU
K ETh
There is no transformation because
energy is conserved.
Physics 207: Lecture 15, Pg 7
Exercise
Work in the presence of friction and non-contact forces
 A box is pulled up a rough (m > 0) incline by a rope-pulley-
weight arrangement as shown below.
 How many forces (including non-contact ones) are
doing work on the box ?
 Of these which are positive and which are negative?
 State the system (here, just the box)
 Use a Free Body Diagram
 Compare force and path
v
A. 2
B. 3
C. 4
D. 5
Physics 207: Lecture 15, Pg 8
Work and Varying Forces (1D)
Area = Fx x
F is increasing
Here W = F · r
becomes dW = Fx dx
 Consider a varying force F(x)
Fx
xf
W
x
x
  Fx ( x ) dx
xi
Finish
Start
F
F
q = 0°
x
Work has units of energy and is a scalar!
Physics 207: Lecture 15, Pg 9
•
Example: Hooke’s Law Spring (xi equilibrium)
How much will the spring compress (i.e. x = xf - xi) to bring
the box to a stop (i.e., v = 0 ) if the object is moving initially at a
constant velocity (vi) on frictionless surface as shown below
with xi = xeq , the equilibrium position of the spring?
xf
Wbox
ti vi
  Fx ( x ) dx
xi
xf
m
Wbox
  - k ( x  xeq ) dx
xi
spring at an equilibrium position
x
Wbox
V=0
t
F
Wbox
m
2
 - 2 k ( x f  xi )  2 k 0  K
1
2
1
2
- 2 k x  2 m0  2 mvi
1
spring compressed
xf
 - 2 k ( x  xi ) |
xi
1
2
1
2
1
Physics 207: Lecture 15, Pg 10
2
Work signs
ti vi
Notice that the spring force is
opposite the displacement
m
spring at an equilibrium position
x
For the mass m, work is negative
V=0
t
F
For the spring, work is positive
m
spring compressed
They are opposite, and equal (spring is conservative)
Physics 207: Lecture 15, Pg 11
Conservative Forces & Potential Energy
 For any conservative force F we can define a potential energy
function U in the following way:
W =
 F ·dr ≡ - U
The work done by a conservative force is equal and opposite to
the change in the potential energy function. r
U
f
ri
f
Ui
Physics 207: Lecture 15, Pg 12
Conservative Forces and Potential Energy
 So we can also describe work and changes in
potential energy (for conservative forces)
U = - W
 Recalling (if 1D)
W = Fx x
 Combining these two,
U = - Fx x
 Letting small quantities go to infinitesimals,
dU = - Fx dx
 Or,
Fx = -dU / dx
Physics 207: Lecture 15, Pg 13
Exercise
Work Done by Gravity
 An frictionless track is at an angle of 30° with respect to the
horizontal. A cart (mass 1 kg) is released from rest. It slides
1 meter downwards along the track bounces and then
slides upwards to its original position.
 How much total work is done by gravity on the cart when it
reaches its original position? (g = 10 m/s2)
30°
(A) 5 J
(B) 10 J
(C) 20 J
h = 1 m sin 30°
= 0.5 m
(D) 0 J
Physics 207: Lecture 15, Pg 14
A Non-Conservative Force
Path 2
Path 1
Since path2 distance >path1 distance the puck will be traveling
slower at the end of path 2.
Work done by a non-conservative force irreversibly removes
energy out of the “system”.
Here WNC = Efinal - Einitial < 0  and reflects Ethermal
Physics 207: Lecture 15, Pg 19
Work & Power:
 Two cars go up a hill, a Corvette and a ordinary
Chevy Malibu. Both cars have the same mass.
 Assuming identical friction, both engines do the same
amount of work to get up the hill.
 Are the cars essentially the same ?
 NO. The Corvette can get up the hill quicker
 It has a more powerful engine.
Physics 207: Lecture 15, Pg 20
Work & Power:
 Power is the rate at which work is done.
Average
Power:
P 
W
t
Instantaneous
Power:
P
dW
Units (SI) are
Watts (W):
1 W = 1 J / 1s
dt
Example:
 A person, mass 80.0 kg, runs up 2 floors (8.0 m). If
they climb it in 5.0 sec, what is the average power
used?
 Pavg = F h / t = mgh / t = 80.0 x 9.80 x 8.0 / 5.0 W
 P = 1250 W
Physics 207: Lecture 15, Pg 21
Work & Power:
 Power is also,
P 
W
t

Fx x
t
 P  Fx v x
 If force constant, W= F x = F ( v0 t + ½ at2 )
and
P = W / t = F (v0 + at)
Physics 207: Lecture 15, Pg 22
Exercise
Work & Power
 Starting from rest, a car drives up a hill at constant
acceleration and then quickly stops at the top.
(Hint: What does constant acceleration imply?)
 The instantaneous power delivered by the engine during
this drive looks like which of the following,
A. Top
time
B. Middle
C. Bottom
time
time
Physics 207: Lecture 15, Pg 23
Chap. 12: Rotational Dynamics
 Up until now rotation has been only in terms of circular motion
with ac = v2 / R and | aT | = d| v | / dt
 Rotation is common in the world around us.
 Many ideas developed for translational motion are transferable.
Physics 207: Lecture 15, Pg 24
Rotational Variables
 Rotation about a fixed axis:
 Consider a disk rotating about
an axis through its center:
q

 Recall :

dq
dt

2
T
(rad/s)  vTangential /R
(Analogous to the linear case v 
dx
)
dt
Physics 207: Lecture 15, Pg 25
Rotational Variables...
At a point a distance R away from the axis of rotation, the
tangential motion:
v=R
 x (arc)
= qR
x
 vT (tangential) =  R
R
q
=R
 aT
  constant

2

(angular accelation in rad/s )
  0  t
(angular v elocity in rad/s)
1
q  q 0  0 t  t 2 (angular position in rad)
2
Physics 207: Lecture 15, Pg 26
Lecture 15
Assignment:
 HW7 due March 10th
 For Thursday: Read Chapter 12, Sections 7-11
Do not concern yourself with the integration process in
regards to “center of mass” or “moment of inertia”
Physics 207: Lecture 15, Pg 28