Download Potential Energy and Conservation of Energy

Last Chapter: Kinetic energy and work
Chapter 8
Potential Energy and
Conservation of Energy
Five foci in this chapter
8-1 Potential Energy
1: Potential Energy
2: Conservation of Mechanical Energy
3: Reading a Potential Energy Curve
4: Work Done on a System by an External Force
5: Conservation of Energy
8.1 Potential energy
Potential energy is energy that can be associated with
the configuration (arrangement) of a system of objects
that exert forces on one another.
Some forms of potential energy:
1.  Gravitational Potential Energy
Ø  Interaction: gravity
Ø  Configuration: relative position between massive objects
2.  Elastic Potential Energy
•  Interaction: EM interaction between molecules
•  Configuration: relative position among molecules
3.  Nuclear Potential Energy: binding nucleons together
•  Interaction: Strong interaction between nucleons
•  Configuration: relative position among nucleons
8.1 Work and potential energy:
Start from an example
v
Spring force F
Change in
potential
energy
Case-(a)
Case-(b)
0à+
=
+
+à0
=
-
Work = FŸd
F
+
Always have
opposite sign
8.1 Work and potential energy
The change in potential energy, ΔU, (gravitational,
elastic, etc) is defined as being equal to the negative
of the work done on the object by the force
(gravitational, elastic, etc)
Do we care about the kinetic energy in this workenergy process?
No: If only the potential energy changes = If we
are looking at two states that have the same speed.
Yes: Regarding the total energy in which the
kinetic energy changes
8.1 Work and potential energy:
An exception
Change in
potential
energy
Case-(a)
Case-(b)
0à+
=
+
+à0
=
-
Work = FkŸd
-
-
NOT always have
opposite sign
v
Friction
8.1 Conservative and non-conservative forces
A popular scenario in reality
1. A system consists of two or more objects.
2. A force acts between a particle-like object in the system and the rest of the system.
3. When the system configuration changes, the force does work (call it W1) on the
object-1 à Transferring energy between the kinetic energy of the object, K, and
some other types of energy of the system.
4. When the configuration change is reversed, the force reverses the energy transfer,
doing work W2 in the process.
8.1 Conservative and non-conservative forces
In a situation in which
(1)  W1= -W2 is always true
(2)  the other type of energy is a potential
energy
à Then, the force is said to be a
conservative force.
Or, the work done by the force in moving
a particle between two points is
independent of the path taken:
•  gravitational force: mghcosϕ
•  spring force: kxi2/2-kxf2/2
•  magnetic force
•  electric force
A force that is not
conservative is called a
non-conservative force:
• 
• 
• 
The kinetic frictional force
Air/fluid drag force
non-elastic scattering force
8.1 Conservative force: One important property
The net work done by a
conservative force on a
particle moving around
any closed path is zero:
Work done from a to b along
path 1: Wab,1
Work done from b back to a
along path 2 as Wba,2.
Conservative force:
Wba,2 = - Wab,2
Path independent:
Wab,1 = Wab,2
Wab,1 + Wba,2 = - Wab,2 + Wab,2 = 0
Sample problem, power of the concept
Calculations: Let us choose the dashed path in Fig. 8-5b; it
consists of two straight segments. Along the horizontal
segment, the angle φ is a constant 90°. Even though we do
not know the displacement along that horizontal segment,
the work Wh done there is
Along the vertical segment, the displacement d is 0.80 m
and, with and both downward, the angle is a constant = 0°.
Thus, for the work Wv done along the vertical part of the
dashed path,
The total work done on the cheese by Fg as the cheese
moves from point a to point b along the dashed path is then
This is also the work done as the cheese slides along the
track from a to b.
Calculations: Let us choose the dashed path in Fig. 8-5b; it
consists of two straight segments. Along the horizontal
segment, the angle φ is a constant 90°. Even though we do
not know the displacement along that horizontal segment,
the work Wh done there is
Along the vertical segment, the displacement d is 0.80 m
and, with and both downward, the angle is a constant = 0°.
Thus, for the work Wv done along the vertical part of the
dashed path,
The total work done on the cheese by Fg as the cheese
moves from point a to point b along the dashed path is then
This is also the work done as the cheese slides along the
track from a to b.
Calculations: Let us choose the dashed path in Fig. 8-5b; it
consists of two straight segments. Along the horizontal
segment, the angle φ is a constant 90°. Even though we do
not know the displacement along that horizontal segment,
the work Wh done there is
Along the vertical segment, the displacement d is 0.80 m
and, with and both downward, the angle is a constant = 0°.
Thus, for the work Wv done along the vertical part of the
dashed path,
The total work done on the cheese by Fg as the cheese
moves from point a to point b along the dashed path is then
This is also the work done as the cheese slides along the
track from a to b.
Sample problem: Let’s try this together
2 kg
0.8 m
α
And we need this
α
to write down the calculations
Q&A:
8.1: Determining Potential Energy values:
For the most general case, in which the force may vary with
position, we may write the work W:
We can use it to derive “Potential Energy values”
corresponding different forces!
8.1: Determining Potential Energy values:
Gravitational Potential Energy
In a particle–Earth system: A particle with mass m moving vertically
along a y axis (upward:+), from point yi to point yf, the gravitational
force does work on it. The change in the gravitational potential energy
of the system is:
yf
yi
mg
Reference position
The gravitational potential energy associated with a particle–Earth
system depends only on the vertical position y (or height) of the
particle relative to the reference position y (= 0), not on the horizontal
position. So,
! !
W = F • d = Fd cos φ
8.1: Determining Potential Energy values:
Elastic Potential Energy
In a block-spring system: the block is moving on the end of a spring of
spring constant k. As the block moves from point xi to point xf , the
spring force Fx =- kx does work on the block. The corresponding change
in the elastic potential energy of the system is
Reference point
If the reference configuration is when the spring is at its relaxed length,
and the block is at xi = 0.
Sample problem: gravitational potential energy
Reference point with U = 0
yi
5m
yf
March 24
Potential energy: Energy associated with the configuration
(arrangement) of a system of objects that exert forces on one
another.
Conservative force: The net work done by a conservative
force on a particle moving around any closed path is zero.
Or path-independent.
We can use it to derive “Potential Energy values”
corresponding different forces!
yf
yi
mg
8.2: Conservation of Mechanical Energy
The mechanical energy Emec of a system is the sum of its potential energy U and
the kinetic energy K of the objects within it:
Principle of Conservation of (Mechanical) Energy:
In an isolated system where only conservative forces cause energy
changes, the kinetic energy and potential energy can change, but
their sum, the mechanical energy Emec of the system, cannot change.
With the absence of
Potential Energy
With the absence of
Kinetic Energy
With both Potential Energy
and Kinetic Energy
8.2: Conservation of Mechanical Energy: An example
Potential energy:
Reference point for zero U
•  (a) and (e): all the energy
is kinetic energy.
•  (c) and (g): all the energy
is potential energy.
•  (b), (d), ( f ), and (h), half
the energy is kinetic
energy and half is potential
energy.
Warning: If the swinging
involves a frictional force
then Emec is conserved, and
eventually the pendulum will
stop.
Discussion: Conservation of mechanical energy and
work done by friction force
Friction keeps
them together
Earth
Sample problem: water slide
Possible ways you might think of
(1) Newton’s Laws: Motion, Force,
Acceleration, …
-  Will it work?
-  No: Force, track, …
(2) Energy conservation: U, K
-  Will it work?
-  Yes
•  Two forces: (a) mg does
work, (b) normal force
doesn’t do work
The power of the
“Mechanical Energy
Conservation Law”
Some limitations:
•  Time taken to reach the
bottom of the slide?
•  Risk analysis: It needs
acceleration along the
track?
•  …
8.3: Reading a Potential Energy Curve: potential energy, force, kinetic
energy, motion
1) 
2) 
3) 
4) 
5) 
You should be able to
From a particle's potential energy as a function of its position
x à Find the force on the particle.
On a graph of potential energy versus x & a line for a
particle's mechanical energy à Determine the particle's
kinetic energy for any given value of x.
Use a potential-energy graph & the conservation of
mechanical energy à Relate the energy values at one position
to those at another position.
Three special states of motion on a potential-energy graph:
•  Stable equilibrium
•  Unstable equilibrium
•  Neutral equilibrium
On a potential-energy graph, identify any turning points and
any regions where the particle is not allowed because of
energy requirements.
8.3: Reading a Potential Energy Curve: Potential Energy & Force
Negative slope!
F(x) = −
A plot of U(x): the potential energy function of
a system containing a particle confined to move
along an x axis. (Since there is no friction, the
mechanical energy is conserved.)
ΔU(x) Δx→0
dU(x)
⎯ ⎯⎯
→ F(x) = −
Δx
dx
A plot of the force F(x) acting on the
particle, derived from the potential energy
plot by taking its slope at various points.
8.3: Reading a Potential Energy Curve: Potential Energy & Force
Negative slope!
F(x) = −
m
ΔU(x) Δx→0
dU(x)
⎯ ⎯⎯
→ F(x) = −
Δx
dx
U(y)
y
Earth
y
Negative force à Attractive
Positive slope à Repulsive
Q: Example of both
attractive and repulsive?
8.3: Reading a Potential Energy Curve: Potential & kinetic energy, motion
Emec
Emec=K+U=constant
8.3: Potential Energy Curve: Equilibrium Points (assuming K=0 on the line)
Case-3: An particle at any point to the
right of x5
•  The system’s mechanical energy is
equal to its potential energy, K=0 à it
must be stationary.
A particle at such a position is said to be in
neutral equilibrium.
Case-1: An particle at x2 or x4
•  It will be stuck there, cannot move
left or right on its own because to
do so would require a negative
kinetic energy.
•  If we push it slightly left or right, a
restoring force appears that moves
it back to x2 or x4.
A particle at such a position is said to
be in stable equilibrium.
Case-2: An particle at x3
When the particle at x3 has K =0.
•  If the particle is located exactly there,
the force on it is also zero, and the
particle remains stationary.
•  If it is displaced even slightly in either
direction, a nonzero force pushes it
farther in the same direction, and the
particle continues to move.
A particle at such a position is said to be in
unstable equilibrium.
March 27
Emech = U(x)+K(x)
Useful in describing processes in subatomic world: Fission, Particle decay
Sample problem: reading a potential energy graph
ΔEmec=ΔK+ΔU=0
Turning point:
ΔK=ΔEmec-ΔU=0
a'
K1=9 J
à
Turning point: 7+9=16
b’
a
c’
b
c
a/a’=b/b’=c/c’
Positive forceà
Repelling
Principle of Conservation of Energy
In an isolated system where only conservative forces
cause energy changes, the kinetic energy and potential
energy can change, but their sum, the mechanical
energy Emec. of the system, remains as a constant.
Emec=K+U=constant
How about a non-isolated system?
–
A system subject to external forces
8.4: Work done on a System by an External Force
Work is energy transferred to or from a system by means of an
external force acting on that system.
8.4: Work done on a System by an External Force
1. When FRICTION is NOT involved:
Attention:
The conservative
gravitational force: internal
force: ΔEmec=ΔK+ΔU=0
8.4: Work done on a System by an External Force
2. When FRICTION is involved:
Net external force
except frictions
Next slide
To include
conservative
force
Gravity does not do work
Fdx − fk dx = ma ∗ dx
d
d
d
∫ F dx − ∫ f
0
k
0
dx =
∫ ma ∗ dx
0
d
dv
Fd = fk d + ∫ m ∗ dx =
dt
0
d
∫
0
vd
fk dx + ∫ m
v0
dx
∗ dv
dt
vd
Fd = fk d + ∫ mv ∗ dv
v0
1
1
1
Fd = fk d + (mvd2 − mv02 ) = fk d + ( mvd2 − mv02 )
2
2
2
Fd = fk d + ΔK
Gravity (conservative force)
does work
(F + mgsin θ )dx − fk dx = ma ∗ dx
v
Fd + (mgsin θ )d − fk d =
∫ mv dv
v0
Fd + mgh − fk d = ΔK
mg
h
Fd = W = ΔK − mgh + fk d
θ
∵ΔU = mg(y f − yi ) = mg(0 − h) = −mgh
∴W = ΔK + ΔU + fk d
Through experiment, the change in thermal
energy is:
W=ΔEmec+ΔEth (8-33)
Kinetic friction only! Not static friction.
March 29
ΔEth = fsd ?
Static friction does work to change Ek or U
Sample problem: change of thermal energy
F
m
m
d
How could this
happen?
How all types of energies vary in the process?
• 
• 
• 
• 
• 
Force F // d
Work done by F to the crate-floor system: +
Change in the kinetic energy of the crate: Change in the potential energy of the crate: 0
W=ΔK+ΔU+ΔEth à (+)=(-)+(0)+ΔEth
àΔEth>0 :Thermal energy changes due to
friction!
(8-34)
8.5: Conservation of Energy: with the presence of non-conservative
external forces & internal structure
Law of Conservation of Energy
The total energy E of a system can change only by amounts of
energy that are transferred to or from the system.
(8.35)
W: work done to the system by external force(s)
ΔEmec: change in the mechanical energy of the system
ΔEth: change in the thermal energy of the system
ΔEint: change in any other type of internal energy of the
system, such as spin, vibration, material change, …
8.5: Conservation of Energy: with the presence of non-conservative
external forces & internal structure
Law of Conservation of Energy
No energy or material
exchanges with environment!
For an isolated system the total energy E cannot change.
(8.36)
In an isolated system, we can relate the total energy at
one instant to the total energy at another instant without
considering the energies at intermediate times.
Total mechanical ΔEmec = Emec,2 – Emec,1
energy example: Emec,2 = Emec,1 - ΔEth – ΔEint
With external force Emec,2 = Emec,1 - ΔEth – ΔEint +W
Sample problem: energy, friction, spring, and tamales
From conservation of energy
Emec,2 = Emec,1 - ΔEth – ΔEint
Forces:
(1)  The normal force on the package from the floor à
does NO work on the package.
(2)  The gravitational force on the package à does
NO work.
(3)  As the spring is compressed, a spring force à
does “-” work on the package.
(4)  The spring force also pushes against a rigid wall
à does NO work.
(5)  There is friction between the package and the
floor à Increases their thermal energies.
System:
Package–spring–floor–wall à An isolated system.
8.5: Conservation of Energy: Attention !!
An external force can change the kinetic energy or potential
energy of an object or a system without doing work on the
system! That is, without transferring energy to/from the
system.
Change in potential energy
8.5: Conservation of Energy: Attention!!
Should not treat the skater as a simple point.
Instead,
the internal “structure” needs to be taken into account.
What actually did the work are internal force(s)!
8.5: Conservation of Energy: Power
Power P is the rate at which energy is transferred by a force from one
type to another. If an amount of energy E is transferred in an amount of
time t, the average power due to the force is
Pavg
ΔE
=
Δt
and the instantaneous power due to the force is
dE
P=
dt
Obvious because of: