Announcements
Conservation of Energy
23 February 2012
Definition of Energy
• Energy is difficult to define exactly.
• One definition is related to the ability of an
object to do useful work (like driving a nail
into a board.)
– You can use kinetic energy to do that (a
hammer.)
– You can use potential energy to do it (drop a brick
from a height onto the nail.)
– Internal energy is harder to use, but it can be
done (within limits.)
• Homework 12 is due at 6 PM Today.
• Next Lecture covers sections 4-5 of
Chapter 8.
Work
• Work is a method of energy transfer, from
one type to another.
• It can represent transfer from potential to
kinetic, or from one type of potential to
another.
• It is one of the methods for transfer of
energy into or out of a system.
B
Potential Energy
• This is energy that is stored in some way,
such as by raising an object in a
gravitational field or compressing a spring.
• PEg = mgh (zero of h is arbitrary)
• PEs = ½ k x2 (x is distance from
equilibrium position)
A
Will the people labeled A and B above
necessarily agree on the potential energy of
the ball that B has thrown off of the building?
Will they agree on how much it has
changed? Will they agree on the kinetic
energy the ball has?
Relationship of Potential Energy
to the Force
Pre-class Quiz
3. For the curve shown below, how many
points of stable equilibrium are there?
potential energy curve

a)
b)
c)
d)
1
2
3
Not enough information is given to know how
many there are.
Energy Transformation
Animations
Today’s Quiz
• Potential to kinetic conversion
• Kinetic to internal conversion
• Potential and kinetic interconversion
Change in how quizzes are to
be done
• The quizzes as now administered are not meeting
the objectives we had for them
– A large part of the objective was to give you practice
laying out your problems so that they would be
readable.
– The time constraint on the quiz defeats that purpose.
• Starting next week we will hand out the quizzes at
the end of class Tuesday and collect them at the
beginning of class Thursday.
– They will be at the same difficulty level as the current
quizzes.
Conservation of Energy
• If no dissipative forces are present, the
total mechanical energy (PE + KE) of a
system is conserved.
• This means (PE + KE)initial = (PE + KE)final
A Roller Coaster
A
15 m
10 m
B
•
Find the speed of the car at the points labeled A and B. Assume it
has a mass of 50 kg and it starts with a speed of 0 from the top of
the hill. Ignore friction.
At start: KE = 0, PE = (50 kg)(10 m/s2)(15 m)=7500 J
At A: KE = ½ m v2, PE = (50 kg)(10 m/s2)(10 m)=5000 J
Using conservation of energy:
½ (50 kg) v2 + 5000 J = 7500 J or (25 kg) v2 = 2500 J  v = 10 m/s
At B: KE = ½ m v2, PE = 0 J so (25 kg) v2 = 7500 J  v = 17.3 m/s
Shooting an arrow
Pre-class quiz
• An arrow is shot into the air. Which of the
options below describes the energy
transformations that occur?
1. Two blocks are tied together with a rope.
They are sliding as shown in the figure. If
friction is present, will the block M slide
faster or slower than if there was no
friction?
(a) Work, elastic PE, KE, grav PE, KE
(b) Work, KE, elastic PE, KE
(c) KE, grav PE, work
(d) Elastic PE, grav PE, KE
(e) None of these
a.
 b.
c.
d.
faster
slower
the same
you can't tell
87% got it right
Problem Solving Strategy – Conservation of
Mechanical Energy for an Isolated System with No
Non-conservative Forces
•Conceptualize
– Form a mental representation
– Imagine what types of energy are changing in the system
•Categorize
– Define the system
– It may consist of more than one object and may or may not
include springs or other sources of storing potential
energy.
– Determine if any energy transfers occur across the
boundary of your system.
• If there are transfers, use Esystem = T
• If there are no transfers, use Esystem = 0
– Determine if there are any non-conservative forces acting.
Problem-Solving Strategy, 2
•Analyze
– Choose configurations to represent initial and final
configuration of the system.
– For each object that changes elevation, identify the zero
configuration for gravitational potential energy.
– For each object on a spring, the zero configuration for
elastic potential energy is when the object is in equilibrium.
– If more than one conservative force is acting within the
system, write an expression for the potential energy
associated with each force.
– Write expressions for total initial mechanical energy and
total final mechanical energy.
– Set them equal to each other.
• If not, use the principle of conservation of mechanical energy.
Problem-Solving Strategy, 3
•Finalize
– Make sure your results are consistent with
your mental representation.
– Make sure the values are reasonable and
consistent with everyday experience.
Example – Spring Loaded Gun
•Conceptualize
– The projectile starts from rest.
– It speeds up as the spring
pushes upward on it.
– As it leaves the gun, gravity
slows it down.
•Categorize
– System is projectile, gun, and
Earth
– Model as an isolated system
with no non-conservative
forces acting
Example – Spring Gun, cont.
•Analyze
– Projectile starts from rest, so Ki = 0.
– Choose zero for gravitational potential energy
where projectile leaves the spring.
– Elastic potential energy will also be 0 here.
– After the gun is fired, the projectile rises to a
maximum height, where its kinetic energy is 0.
•Finalize
– Did the answer make sense?
– Note the inclusion of two types of potential
energy.
Clicker Quiz
Example – Spring Gun, final
•The energy of the gun-projectileEarth system is initially zero.
•The popgun is loaded by means of
an external agent doing work on the
system to push the spring downward.
•After the popgun is loaded, elastic
potential energy is stored in the
spring and the gravitational potential
energy is lower because the projectile
is below the zero height.
•As the projectile passes through the
zero height, all the energy of the
system is kinetic.
•At the maximum height, all the
energy is gravitational potential.
Two methods to solve the
problem
A spring loaded dart gun is used to shoot a dart straight up in the air. The
dart reaches a maximum height of 24 m. The same dart is shot straight up
a second time from the same gun, but this time the spring is compressed
only half as far before firing. How far up does the dart go this time,
neglecting friction and assuming an ideal spring?
• The problem of the two masses can be
solved either by Newton’s Laws and
kinematics or by conservation of energy.
A) 48 m
B) 24 m
C) 12 m
D) 6 m
E) 3 m
What is v after mass m
has fallen a distance d?
Newton’s Laws
Newton’s Laws
N
T
T
m
mg
M
Mg
Diver vs. Slider
Conservation of Energy
M
m
d
v
M
Two people are on the top of a tower in a water park. One
dives off the tower into the pool and the other takes a
slide in the frictionless slide from the top of the tower into
the same pool. Which one arrives first? Which one is
moving faster when they arrive?
Arrives First?
d m
v
Going Faster?
(A)
Diver
Diver
(B)
Both at the
same time
Both have the
same speed
(C)
Slider
Slider
(D)
Diver
Slider
(E)
Diver
Both have the
same speed