Physics 106 Lesson #5
Work and Energy
Dr. Andrew Tomasch
2405 Randall Lab
atomasch@umich.edu
Newton’s Laws: Review
• First Law: Objects continue their state of
motion (rest or constant velocity) unless acted
upon by a net external force.
• Second Law: The action of a net external force
on an object is to cause its momentum to
change with time. For objects with a constant
mass this can be written as F = ma.
• Third Law: Any object which exerts a force on
another object experiences an equal and
opposite force from the object it acts upon.
Newton’s Laws: Summary
F  0
v  constant (or zero)
Newton Said:
p
 F  t   F t  p
Always True
• Newton’s 1st law:
Inertia
Rocket Cars:
F  ma
True if Mass is Constant
FA B   FB  A
• Newton’s 2nd law:
Force and Momentum
• Newton’s 3rd law:
Action and Reaction
Newton’s Second Law
Solves Galileo’s Puzzle!
F  ma
F  mg  ma
A Falling Object…
m
F  mg
a g
ag !
All objects fall with the same
downward acceleration g !
Our Hero!
Review: Linear Momentum
and Newton’s Second Law
• The Second Law in Newton’s own words:
“ The change in the quantity of motion is
proportional to the motive force impressed
and is made in the direction of the line in
which that force is impressed.”
("quantity of motion")
F 
t
"quantity of motion"  linear momentum:
p  mv
p
 F  t
Review: Impulse and
Momentum
If we multiply both sides of
Newton’s Second Law by time, we
get another way to say it: “The
impulse (force x time) delivered by
the net force equals the change in
an object’s momentum” This is our rocket
This is a
vector
equation
J  F t  p
This is also called the Impulse -- Momentum Theorem
car experiment
Force X Time =
Change in Momentum
Impulse and Momentum: Rocket Car Data
The final
momentum
is constant
to ~ 10%
The units for
impulse and
momentum are
kg-m/s or N-s
 1 
 1 
vfinal  F t 
J

m
m
 final 
 final 
The slope of the line is the car's
final momentum, which equals the
impulse J (force x time) delivered
by the CO2 "rocket engine".
Rocket Science: Impulse
Delivered by a Rocket Engine
Force
(N)
Thrust vs. Time
Area Under Curve = Impulse
In Newton-Seconds (N-s)
Impulse = Force x Time
Time (s)
We can measure the impulse of our CO2 rocket engines this way.
Impulse in Football
• In football, the
defensive player
applies a force for a
given amount of time
to stop the momentum
of the offensive player
who has the ball.
• An object with
momentum can be
stopped if a force is
applied against it for a
given amount of time.
p
F
p  J  ( F )(t )
+y
Rocket Science: Thrust
FThrust
vexhaust
Newton's Second Law :
P
t
Newton's Third Law :
F
Demo: Water Rocket
Ejected Momentum/Time Backward (Action)
Equals Thrust Force on Rocket Forward (Reaction) :
FThrust
 P 
  
  vex
 t  exhaust
 m 
 t  yˆ
ex
Thrust = Exhaust Speed x (Ejected Mass/Time)
Cannot be described with F = ma !
The Story So Far. . .
• Newton’s laws of motion provide a complete description
of mechanics. Using Newton’s laws we can:
– predict solar eclipses for millennia to come
– design suspension bridges
– understand the fate of distant galaxy clusters
• What about quantities that do not change during motion?
– energy and momentum are conserved in certain circumstances
– these conservation laws transcend Newtonian mechanics and lie
at the very core of our understanding of nature
– conservation laws enable us to solve many problems without a
detailed understanding of the forces that act  simplification
Newton’s Second Law: Another
Trick-Momentum Conservation!
Fext
p

t
If no net external force
acts on the system it
is isolated and linear
momentum is conserved
p
Fext  0 
0
t
p  constant  conserved!
Rocket Propulsion:
Momentum Conservation
• Define the system: Demonstration:
Cannon Recoil
rocket + fuel
• Initial momentum of the
system is zero
• Momentum of the system
is conserved because the
forces are internal
procket i  pfuel i  procket f  pfuel f  0
 procket f   pfuel f
The forward momentum
of the rocket must equal
the rearward momentum
of the fuel!
procket
We are
ignoring
gravity
pfuel
Image courtesy NASA
Rocket Propulsion:
Momentum Conservation
For our CO2 rocket cars:
pexhaust  pcar
The CO2 canisters
eject 4 grams of CO2
procket
 mexhaust vexhaust  pcar
 vexhaust 
pcar
mexhaust
We are
ignoring
gravity
1.1 kg-m/s

0.004 kg
 275 m/s  674 mi/hr
pfuel
Nearly Supersonic!
The speed of sound in air is 340
m/s→(275 m/s)/(340 m/s) = Mach 0.8
Image courtesy NASA
Work and Energy
Energy ≡ the ability to do work
Work is a scalar
• We call many activities “work”
• Work in physics has a concise definition
• Definition: Work is the action of a force through
a distance
Example: You exert a horizontal
force F to push an object an object
through a horizontal displacement d
F
W Fd
d
Work and Energy
• Most generally, work is the product of
the component of force parallel to the
displacement and the magnitude of the
Work can be positive or negative.
displacement:
Positive work is done if the force
acts in the direction of motion.
Negative work is done if the force
acts in opposition to the motion.
F
θ
F║
d
W F d
Work has units of N-m ≡ Joules ≡ J
A force which acts
perpendicular to the
direction of motion
does zero work.
Kinetic Energy and the
Work-Energy Theorem
• kinetic energy means
“energy of motion”
1
2
KE  mv
2
Caution
Quiz
Ahead
Wnet  KE
Wnet
1 2
1 2
 mvfinal  mvinitial
2
2
The Work- Energy Theorem
• The work done by a net
force is equal to the change
in the kinetic energy of the
object on which the net
force acts
• Net work is required for an
object to change speed
• We can either determine the
net force and calculate the
work it does or calculate the
work done be each force
and add it up to get the net
work done
Concept Test #1
Caution
Quiz
Ahead
Which statement about work is true?
A. Only the component of a force parallel
or anti-parallel to the displacement
does work.
B. The component of a force
perpendicular to the displacement
does positive work.
C. Only the net force on an object does
any work.
Concept Test #2
When you do positive work on an
object, its kinetic energy
A.
B.
C.
D.
decreases.
increases.
remains the same.
need more information about
how the work was done
Work by Conservative Forces
•
•
The work a conservative force does on an object in
moving it from a point A to a point B is path
independent . It depends only on the path’s end
points.
For conservative
Conservative Forces:
forces it is possible
to define a potential
D
1. Gravity
energy function
E
2. Elastic (spring) Force
3. Electromagnetic Force
The work done by
gravity is the same
for path A-B-C-D-E
as it is for path A-E
B
A
C
Work and Conservative Forces
• Alternate definition: a conservative
force does no work on an object
moving around a closed path.
Caution
Quiz
Ahead
The work done by
gravity on the car as
is travels in a closed
loop from start,
around the track and
back to start is zero.
Concept Test #3
After hiking all day, you return to the trail head
from which you departed that morning. The work
that gravity has done on you during your hike is:
A) Positive
B) Zero
C) Negative
Gravity is a conservative
force
no work done
moving in a closed loop.
Work Done by the Gravitational Force
PEgrav  mgh
Wgravity  mg (h0  hf )
Wgravity  PE0  PEf
Wgravity  PE
h0
mg
hf
mg
PE=0
An object loses gravitational
potential energy and gains an
equal amount of kinetic energy
as it falls toward the Earth. The
sum of gravitational potential
energy and kinetic energy never
changes → total energy is
conserved (neglecting air!).
Converting PE to KE
•
m
•
h
A ball dropped from rest
that falls a height h
looses PE and gains KE
With no air resistance, the
gain in KE is equal to the
loss in PE:
KE  PE
•
No friction
or air drag
Energy can be converted
from one form to another
A Bob-Sled Run
Comments on Gravitational
Potential Energy
• Gravitational potential energy is energy that an
object possesses by virtue of its position in a
gravitational field.
• We measure the height h with respect to some
arbitrary reference point. Therefore, define your
choice for the zero potential energy level first.
• There is no intrinsic significance to the absolute
value of an object's gravitational potential energy.
It's differences in gravitational potential energy that
matter.
• Gravitational potential energy can be thought of as
an energy reserve that you can draw on and convert
into kinetic energy and as a form of energy into
which kinetic energy can be converted.
Total Energy Conservation
• The principle of conservation of energy:
Energy can neither be created nor destroyed,
but can only be converted from one form to
Wnc ≡ Work done by
another.
•KE ≡ Cash
• Work-Energy:
nonconservative
forces other than
gravity → Friction
•PE ≡ Deposits
•Wfriction ≡ Taxes!
KEi  PEi  Wnc  KEf  PEf
The Work-Energy
The Work-Energy Theorem is
energy conservation and turns “Balance Sheet”
mechanics into accounting.
Demo: Downhill Duck