Download Part II

Ch. 6, Work & Energy, Continued
Summary So Far
•
Work (constant force):
W = F||d = Fd cosθ
• Work-Energy Theorem:
Wnet = (½)m(v2)2 - (½)m(v1)2  KE
Total work done by ALL forces!
• Kinetic Energy: l
KE  (½)mv2
Potential Energy
A mass can have a Potential Energy due
to its environment
Potential Energy (PE) 
An energy associated with the position or
configuration of a mass.
Examples of Potential Energy:
A wound-up spring
A stretched elastic band
An object at some height above the ground
Gravitational Potential Energy
• When an object of mass m follows any
path that moves through a vertical
distance h, the work done by the
gravitational force is always equal to
W = mgh
• So, we say that an object near the Earth’s
surface has a Potential Energy (PE) that
depends only on the object’s height, h
• The PE is a property of the Earth-object system
• Potential Energy (PE) 
Energy associated with the
position or configuration of a mass.
Potential Work Done!
• Example:
Gravitational Potential
Energy: PEgrav  mgy
• y = distance above Earth.
• m has the potential to do
work mgy when it falls
(W = Fy, F = mg)
Gravitational Potential Energy
For constant speed:
ΣFy = Fext – mg = 0
So,
Wext = Fext hcosθ = mghcos(0)
= mgh = mg(y2 – y1)
Work-Energy Theorem
Wnet = KE
 (½)[m(v2)2 - m(v1)2]
(1)
In raising a mass m to a height h, the work done by the
external force is mgh. So we define the gravitational potential
energy at a height y above some reference point (y1) as
(PE)grav = mgh
• Consider a problem in which the height of a
mass above the Earth changes from y1 to y2:
Change in Gravitational PE is:
(PE)grav = mg(y2 - y1)
Work done on the mass: W = (PE)grav
y = distance above Earth
• Where we choose y = 0 is arbitrary,
since we take the difference in 2 y’s
in (PE)grav
Of course, this
Potential energy can be converted to
kinetic energy if the object is dropped.
PE is a property of a system as a whole, not
just of the object (it depends on external forces).
If PEgrav = mgy, from where do we measure y?
It turns out not to matter!
As long as we are consistent about where we
choose y = 0 that choice won’t matter because only
changes in potential energy can be measured.
Example: PE Changes for a Roller Coaster
A roller-coaster car, mass m = 1000 kg, moves
from point 1 to point 2 & then to point 3.
∆PE depends only
on differences in
height.
a. Calculate the gravitational potential energy at points
2 & 3 relative to point 1. (That is, take y = 0 at point 1.)
b. Calculate the change in potential energy when the
car goes from point 2 to point 3.
c. Repeat parts a. & b., but take the reference point
(y = 0) at point 3.
Many Other Types of Potential Energy
Besides Gravitational Exist!
Consider an Ideal Spring
An Ideal Spring, is characterized by a
spring constant k, which is a measure
of it’s “stiffness”. The restoring force
of the spring on the hand is:
L
Fs = - kx
(Fs >0, x <0; Fs <0, x >0)
This is known as Hooke’s “Law”
(but, it isn’t really a law!)
It can be shown that the work done by the person is:
W = (½)kx2  (PE)elastic
We use this as the definition of Elastic Potential Energy
Elastic Potential Energy
(PE)elastic ≡ (½)kx2
Relaxed Spring
Work to compress spring distance x:
W = (½)kx2  (PE)elastic
The spring stores potential energy!
When the spring is released, it
transfers it’s potential energy
PEe = (½)kx2
to mass in the form of kinetic energy
KE = (½)mv2
The applied Force Fapp is equal & opposite to the force
Fs exerted by block on the spring: Fs = - Fapp = -kx
Force Exerted by a
Spring on a Block
x > 0, Fs < 0
x = 0, Fs = 0
x < 0, Fs > 0
The spring force Fs
varies with the block
position x relative to
equilibrium at x = 0.
Fs = -kx.
Spring constant k > 0
Fs(x) vs. x
x=0

 Relaxed Spring
Spring constant k
W

x
In (a), the work to compress
 the spring a distance x:
W = (½)kx2
W=
(½)kx2
WWWSo, the spring stores potential
W
energy in this amount.
In (b), the spring does work on
the ball, converting it’s stored
potential energy into
kinetic energy.
W
W
W
Elastic PE
PEelastic = (½)kx2
KE = 0
PEelastic = 0
KE = (½)mv2
Measuring k for a Spring
Hang the spring vertically.
Attach an object of mass m
To the lower end. The spring
stretches a distance d.
At equilibrium,
Newton’s 2nd Law says
∑Fy = 0.
So, mg – kd = 0, mg = kd
Knowing m & measuring d,
 k = (mg/d)
Example: d = 2.0 cm, m = 0.55 kg
 k = 270 N/m
• In a problem in which compression or
stretching distance of spring changes
from x1 to x2, The change in PE is:
(PE)elastic = (½)k(x2)2 - (½)k(x1)2
• The work done is:
W = - (PE)elastic
The PE belongs to the system, not
to individual objects.
Conservative Forces
Conservative Forces
•Conservative Force  The work
done by that force depends only on
initial & final conditions & not on
path taken between the initial &
final positions of the mass.
 A PE CAN be defined
for conservative forces
•Non-Conservative Force  The
work done by that force depends on
the path taken between the initial &
final positions of the mass.
 A PE CAN’T be defined
for non-conservative forces
• The most common example of a nonconservative force is
FRICTION
Definition: A force is conservative if & only if the
work done by that force on an object moving from
one point to another depends ONLY on the initial
& final positions of the object, & is independent of
the particular path taken. Example: gravity.
Gravitational PE Again!
• The work done by the gravitational
force as the object moves from its
initial position to its final position is
Independent of the
path taken!
• The potential energy is related to the
work done by the force on the object
as the object moves from one location
to another.
• Because of this property, the
gravitational force is called a
Conservative Force.
Conservative Force: Another definition: A force is
conservative if the net work done by the force on
an object moving around any closed path is zero.
Potential Energy
• The relationship between work & PE:
ΔPE = PEf – PEi = - W
• W is a scalar, so PE is also a scalar
• The Gravitational PE of an object
when it is at a height y is PE = mgy
Applies only to objects near the Earth’s surface
• Potential Energy, PE is stored energy
– The energy can be recovered by letting
the object fall back down to its initial
height, gaining kinetic energy
Potential Energy:
Can only be defined for Conservative Forces!
In other words, if a force
is
Conservative,
a PE CAN be defined.
But, if a force is
Non-Conservative,
a PE CANNOT be
defined!!
If friction is present, the work done depends not only
on the starting & ending points, but also on the path taken.
Friction is a non-conservative force!
Friction is non-conservative!!!
The work done depends on the path!
• If several forces act, (conservative & nonconservative), the total work done is:
Wnet = WC + WNC
WC ≡ work done by conservative forces
WNC ≡ work done by non-conservative forces
• The work energy theorem still holds:
Wnet = WC + WNC = KE
• For conservative forces (by the definition of PE):
WC = -PE
 KE = -PE + WNC
or: WNC = KE + PE
 In general,
WNC = KE + PE
• The total work done by all
non-conservative forces ≡
The total change in KE +
The total change in PE
Mechanical Energy & its Conservation
GENERALLY: In any process, total
energy is neither created nor
destroyed.
• Energy can be transformed from one form to
another & from one object to another, but the
Total Amount Remains Constant.
 Law of Conservation
of Total Energy
• In general, for mechanical systems, we found:
WNC = KE + PE
• For the Very Special Case of
Conservative Forces Only
 WNC = 0 = KE + PE = 0
 The Principle of Conservation of
Mechanical Energy
• Please Note!! This is NOT (quite) the same
as the Law of Conservation of Total Energy! It
is a very special case of this law (where all
forces are conservative)
• So, for conservative forces ONLY! In any process
KE + PE = 0
Conservation of Mechanical Energy
• It is convenient to define the Mechanical Energy:
E  KE + PE
 In any process (conservative forces!):
E = 0 = KE + PE
Or,
E = KE + PE = Constant
≡ Conservation of Mechanical Energy
Conservation of Mechanical Energy
• In any process with conservative forces ONLY! 
Or,
E = 0 = KE + PE
E = KE + PE = Constant
• In any process (conservative forces!), the
sum of the KE & the PE is unchanged: That
is, the mechanical energy may change from
PE to KE or from KE to PE, but
Their Sum Remains Constant.
Principle of Conservation of
Mechanical Energy:
If only conservative forces are doing
work, the total mechanical energy of
a system neither increases nor
decreases in any process. It stays
constant—it is conserved.
Conservation of Mechanical Energy:
 KE + PE = 0
Or E = KE + PE = Constant
• This is valid for conservative forces ONLY
(gravity, spring, etc.)
• Suppose that, initially:
E = KE1 + PE1, & finally: E = KE2+ PE2.
• But, E = Constant,
so
 KE1 + PE1 = KE2+ PE2
A very powerful method of
calculation!!
Conservation of Mechanical Energy

KE + PE = 0
or
E = KE + PE = Constant
• For gravitational PE: (PE)grav = mgy
E = KE1 + PE1 = KE2+ PE2
 (½)m(v1)2 + mgy1 = (½)m(v2)2 + mgy2
y1 = Initial height, v1 = Initial velocity
y2 = Final height, v2 = Final velocity
all PE
KE1 + PE1 = KE2 + PE2
= KE3 + PE3
PE1 = mgh, KE1 = 0
but their sum remains constant!
half KE
half PE
all KE
KE3 + PE3 = KE2 + PE2
= KE1 + PE1
KE1 + PE1 = KE2 + PE2
0 + mgh = (½)mv2 + 0
v2 = 2gh
PE2 = 0
KE2 = (½)mv2
Example: Falling Rock
Energy “buckets” are not real!!
• Speeds at y2 = 0.0, & y3 = 1.0 m?
Mechanical Energy
Conservation!
KE1 + PE1 = KE2 + PE2
(½)m(v1)2 + mgy1 = (½)m(v2)2 + mgy2 = (½)m(v3)2 + mgy3
(Mass cancels!)
• y1 = 3.0 m, v1 = 0, y2 = 1.0 m, v2 = ?, y3 = 0.0, v3 = ?
• Results: v2 = 6.3 m/s, v3 = 7.7 m/s
NOTE!!
• Always use KE1 + PE1 = KE2 + PE2 = KE3 + PE3
• NEVER KE3 = PE3!!!!
In general, KE3 ≠ PE3!!!
This is a very common error! WHY????
• Speeds at y2 = 0.0, & y3 = 1.0 m?
Mechanical Energy
Conservation!
Cartoon Version!
(½)m(v1)2 + mgy1 = (½)m(v2)2 + mgy2
= (½)m(v3)2 + mgy3 (Mass cancels!)
y1 = 3.0 m, v1 = 0, y2 = 1.0 m,
v2 = ? , y3 = 0.0 m, v3 = ?
Results: v2 = 6.3 m/s, v3 = 7.7 m/s
v3PE
= ?only
part PE
part KE v1 = 0
y2 = 1.0 my1 = 3.0 m
v2 = ?
y3 = 0
KE only
Example: Roller Coaster
• Mechanical energy conservation! (Frictionless!)
 (½)m(v1)2 + mgy1 = (½)m(v2)2 + mgy2 (Mass cancels!)
Only height differences matter!
• Speed at the bottom?
y1 = 40 m, v1 = 0
y2 = 0 m, v2 = ?
Find: v2 = 28 m/s
• What is y when
v3 = 14 m/s?
Use: (½)m(v2)2 + 0
= (½)m(v3)2 + mgy3
Find:
Horizontal distance doesn’t matter!
Height of hill = 40 m. Car starts from rest at
top. Calculate: a. Speed of the car at bottom
of hill. b. Height at which it will have half
this speed. Take y = 0 at bottom of hill.
1
In general,
KE3 ≠ PE3!!!
3
2
y3 = 30 m
A very common error!
WHY????
NOTE!! Always use KE1 + PE1 = KE2 + PE2
= KE3 + PE3 Never
KE3 = PE3 !
Conceptual Example : Speeds on 2 Water Slides
• Who is traveling faster at
v = 0, y = h
the bottom?
• Who reaches the
bottom first?
Frictionless
water slides!
Both start here!
• Demonstration!
y=0
v=?
Both get to the
bottom here!