Transcript Lecture 16.Conservat..

Conservation of Energy
Lecturer:
Professor Stephen T. Thornton
Reading Quiz
Mike performed 5 J
of work in 10 secs.
Joe did 3 J of work
in 5 secs. Who
produced the greater
power?
A) Mike produced
more power
B) Joe produced more
power
C) both produced the
same amount of
power
Reading Quiz
Mike performed 5 J
of work in 10 secs.
Joe did 3 J of work
in 5 secs. Who
produced the greater
power?
A) Mike produced
more power
B) Joe produced more
power
C) both produced the
same amount of
power
Because power = work / time, we see that Mike
produced 0.5 W and Joe produced 0.6 W of power.
Thus, even though Mike did more work, he
required twice the time to do the work, and
therefore his power output was lower.
Last Time
Conservative and nonconservative forces
Gravitational potential energy
Other kinds of potential energy
Conservation of mechanical energy
Today
Conservation of Energy
Escape velocity
Power
Potential energy diagrams
Potential Energy
A spring has potential
energy, called elastic
potential energy, when it
is compressed. The force
exerted by the spring
when compressed or
stretched is
where k is called the
spring constant, and
needs to be measured for
each spring.
Copyright © 2009 Pearson Education, Inc.
Then the potential energy of the spring is:
Copyright © 2009 Pearson Education, Inc.
Springs
The work required to
compress a spring is
1 2
W  kx
2
The potential energy
of a spring is
1 2
U  kx
2
Mass on Spring. When a mass
m sits at rest on a spring, the
spring is compressed by a
distance d from its undeformed
length. Suppose instead that the
mass is released from rest when
it barely touches the undeformed
spring. Find the distance D that
the spring is compressed before
it is able to stop the mass.
Conceptual Quiz
A truck, initially at rest, rolls
down a frictionless hill and
attains a speed of 20 m/s at the
bottom. To achieve a speed of
40 m/s at the bottom, how many
times higher must the hill be?
A) half the height
B) the same height
C)  2 times the height
D) twice the height
E) four times the height
Conceptual Quiz
A truck, initially at rest, rolls
down a frictionless hill and
attains a speed of 20 m/s at the
bottom. To achieve a speed of
40 m/s at the bottom, how many
times higher must the hill be?
Use energy conservation:
 initial energy: Ei = PEg = mgH
 final energy: Ef = KE =
Conservation of Energy:
Ei = mgH = Ef =
therefore:
gH =
1
2
1
2
1
2
mv
2
v2
So if v doubles, H quadruples!
mv2
A) half the height
B) the same height
C)  2 times the height
D) twice the height
E) four times the height
Conceptual Quiz
A box sliding on a frictionless flat
surface runs into a fixed spring,
which compresses a distance x to
stop the box. If the initial speed
of the box were doubled, how
much would the spring compress
in this case?
A) half as much
B) the same amount
C)  2 times as much
D) twice as much
E) four times as much
x
Conceptual Quiz
A box sliding on a frictionless flat
surface runs into a fixed spring,
which compresses a distance x to
stop the box. If the initial speed
of the box were doubled, how
much would the spring compress
in this case?
Use energy conservation:
1
initial energy: Ei = KE = 2 mv2
1
final energy: Ef = PEs = 2 kx2
Conservation of Energy:
1
1
2
Ei = 2 mv = Ef = 2 kx2
therefore: mv2 = kx2
So if v doubles, x doubles!
A) half as much
B) the same amount
C)  2 times as much
D) twice as much
E) four times as much
x
Conceptual Quiz
A cart starting from rest rolls down a hill
and at the bottom has a speed of 4 m/s. If
the cart were given an initial push, so its
initial speed at the top of the hill was 3 m/s,
what would be its speed at the bottom?
A) 4 m/s
B) 5 m/s
C) 6 m/s
D) 7 m/s
E) 25 m/s
Conceptual Quiz
A) 4 m/s
A cart starting from rest rolls down a hill
and at the bottom has a speed of 4 m/s. If
the cart were given an initial push, so its
initial speed at the top of the hill was 3 m/s,
what would be its speed at the bottom?
B) 5 m/s
C) 6 m/s
D) 7 m/s
E) 25 m/s
When starting from rest, the
cart’s PE is changed into KE:
DPE = DKE = 21 m(4)2
When starting from 3 m/s, the
final KE is:
KEf
= KEi + DKE
1
1
= 2 m(3)2 + 2 m(4)2
1
= 2 m(25)
1
= 2 m(5)2
Speed is not the same as kinetic energy
Conceptual Quiz:
Two unequal masses are hung from a string
that pass over an ideal pulley. What is true
about the gravitational potential energy U and
the kinetic energy K of the system after the
masses are released from rest?
A)
B)
C)
D)
E)
DU > 0 and DK < 0.
DU > 0 and DK > 0.
DU > 0 and DK = 0.
DU = 0 and DK = 0.
DU < 0 and DK > 0.
Answer: E
Initially the system is at rest. Let the
potential energy be zero at this point.
Therefore the total mechanical energy
is zero. If the system starts moving,
then DK > 0. Since E = 0, then DU <
0.
Conceptual Quiz
D) same speed
for all balls
A
B
C
Three balls of equal mass start from rest
and roll down different ramps. All ramps
have the same height. Which ball has the
greater speed at the bottom of its ramp?
Three balls of equal mass start from rest
and roll down different ramps. All ramps
have the same height. Which ball has the
greater speed at the bottom of its ramp?
A
B
C
D) same speed
for all balls
All of the balls have the same initial
gravitational PE, since they are all at the
same height (PE = mgh). Thus, when they
get to the bottom, they all have the same final
KE, and hence the same speed (KE = 1/2 mv2).
Follow-up: Which ball takes longer to get down the ramp?
Law of Conservation of Energy
We discussed Conservation of Mechanical
Energy last time. DK  DU  0
Nonconservative, or dissipative, forces associated
with:
Friction
Heat
Electrical energy
Chemical energy
and more
do not conserve mechanical energy. However,
when these forces are taken into account, the total
energy is still conserved:
D K + D U + [change in all other forms of energy] = 0
Law of Conservation of Energy
The law of conservation of energy is one of
the most important principles in physics.
The total energy is neither increased
nor decreased in any process. Energy
can be transformed from one form to
another, and transferred from one
object to another, but the total amount
remains constant.
Ball rolling on a frictionless track
Gravitational potential energy vs position for
the previous track. See also kinetic and total energy.
Height
Gravitational potential energy vs position for the
previous track. See also kinetic and total energy.
New total energy
Height
A Mass on a Spring
E
1 2
U  kx
2
K
U
Potential Energy Diagrams;
Stable and Unstable Equilibrium
This is a potential energy
diagram for a particle moving
under the influence of a
conservative force. Its
behavior will be determined
by its total energy.
With energy E1, the object oscillates between x3
and x2, called turning points. An object with energy
E2 has four turning points; an object with energy
E0 is in stable equilibrium. An object at x4 is in
unstable equilibrium.
Bath County, Virginia, pumped storage facility
electrical power plant.
Day – water flows down from upper reservoir
producing electricity.
Night – use power from other (nuclear) plants to pump
water back up.
Gravitational Potential Energy
Far from the surface of the Earth, the force
of gravity is not constant:
The work done on an object
moving in the Earth’s
gravitational field is given by:
Gravitational Potential Energy
Solving the integral gives:
GmM E
GmM E
W=
r2
r1
Because the value of the integral depends
only on the end points, the gravitational
force is conservative and we can define
gravitational potential energy:
GmM E
U (r ) = r
Gravitational Potential Energy
and Escape Velocity
If an object’s initial kinetic energy is equal to the
negative of the potential energy at the Earth’s
surface, its total energy will be zero. The velocity
at which this is true is called the escape velocity;
for Earth: E  0  K U ; K   U  GmM / rE
E
Think about this. E = 0 at Earth’s surface; E = 0 at r   .
At r   : v  0 , U = 0 and K = 0.
At Earth's surface:
GmM E
1 2
mvesc 
2
rE
vesc = 11.2 km/s
vesc
2GM E

rE
Power
Power measures how fast work is done.
Average power = P = W/t
dW
dt
Power is so important that it also has its own
unit. SI unit: watt
Instantaneous power  P 
1 watt = 1 W = 1 J/s = 1 joule/sec
1 horsepower = 1 hp = 746 watt
Power is also needed for
acceleration and for moving against
the force of friction.
The power can be written in terms of
the net force and the velocity:
dW
d
P=
= F × = F ×v
dt
dt
Lance Armstrong was tested and could ride
up the mountains in France during the Tour de
France generating about 500 watts of power
for 20 minutes. A typical college student
could only do this for 30 s. (Lance has a large
heart and low levels of lactic acid.)
Lance exerts 500 W x 1200 s = 600,000 J = W
Climbing: mgh = (70 kg)(9.8 m/s2 )h = W
energy; h = 875 m = 2900 ft.
This is why he won the Tour de France seven
consecutive years!
Conceptual Quiz
Paul and Kathleen start
from rest at the same time
on frictionless water slides
with different shapes. At
the bottom, whose velocity
is greater?
A) Paul
B) Kathleen
C) both the same
Conceptual Quiz
Paul and Kathleen start
from rest at the same time
on frictionless water slides
with different shapes. At
the bottom, whose velocity
is greater?
Conservation of Energy:
Because they both start
from the same height,
they have the same
velocity at the bottom.
A) Paul
B) Kathleen
C) both the same
Conceptual Quiz
Paul and Kathleen start
from rest at the same time
on frictionless water slides
with different shapes. Who
makes it to the bottom first?
A) Paul
B) Kathleen
C) both the same
Conceptual Quiz
Paul and Kathleen start from
rest at the same time on
frictionless water slides with
different shapes. Who makes
it to the bottom first?
Even though they both have
the same final velocity,
Kathleen is at a lower height
than Paul for most of her
ride. Thus, she always has a
larger velocity during her
ride and therefore arrives
earlier!
A) Paul
B) Kathleen
C) both the same
Space Shuttle. Early test flights for the
space shuttle used a “glider” (mass of 980 kg
including pilot). After a horizontal launch at
480 km/h at a height of 3500 m, the glider
eventually landed at a speed of 210 km/h.
(a) What would its landing speed have been
in the absence of air resistance?
(b) What was the average force of air
resistance exerted on it if it came in at a
constant glide angle of 12° to the Earth’s
surface?
Ski Lift Power. A ski area claims that its
lifts can move 47,000 people per hour. If the
average lift carries people about 200 m
(vertically) higher, estimate the maximum
total power needed.
Conceptual Quiz
You and your friend both solve a
problem involving a skier going
down a slope, starting from rest.
The two of you have chosen
different levels for y = 0 in this
problem. Which of the following
quantities will you and your
friend agree on?
1) skier’s PE
2) skier’s change in PE
A)
B)
C)
D)
E)
only 2
only 3
1, 2, and 3
only 1 and 3
only 2 and 3
3) skier’s final KE
Conceptual Quiz
You and your friend both solve a
problem involving a skier going
down a slope, starting from rest.
The two of you have chosen
different levels for y = 0 in this
problem. Which of the
following quantities will you and
your friend agree on?
1) skier’s PE
2) skier’s change in PE
A)
B)
C)
D)
E)
only 2
only 3
1, 2, and 3
only 1 and 3
only 2 and 3
3) skier’s final KE
The gravitational PE depends upon the reference
level, but the difference DPE does not! The work
done by gravity must be the same in the two
solutions, so DPE and DKE should be the same.
Follow-up: Does anything change physically by the choice of y = 0?
Conceptual Quiz
Which contributes
more to the cost of
your electric bill each
month, a 1500-Watt
hair dryer or a 600Watt microwave
oven?
A)
B)
C)
D)
hair dryer
microwave oven
both contribute equally
depends upon what you
cook in the oven
E) depends upon how long
each one is on
600 W
1500 W
Conceptual Quiz
Which contributes
more to the cost of
your electric bill each
month, a 1500-Watt
hair dryer or a 600Watt microwave
oven?
A) hair dryer
B) microwave oven
C) both contribute equally
D) depends upon what you
cook in the oven
E) depends upon how long
each one is on
We already saw that what you actually
pay for is energy. To find the energy
consumption of an appliance, you
must know more than just the power
rating—you have to know how long it
was running.
600 W
1500 W
Conceptual Quiz
How does the work required to
A) same amount of work
stretch a spring 2 cm compare
B) twice the work
with the work required to
C) four times the work
stretch it 1 cm?
D) eight times the work
Conceptual Quiz
How does the work required to
A) same amount of work
stretch a spring 2 cm compare
B) twice the work
with the work required to
C) four times the work
stretch it 1 cm?
D) eight times the work
The elastic potential energy is
1
2
kx2. So in the second case,
the elastic PE is four times greater than in the first case. Thus,
the work required to stretch the spring is also four times
greater.