Transcript Energy - Edublogs

Work, Energy, Power, and
Machines
Energy
Energy: the currency of the universe.
Just like money, it comes in many
forms!
Everything that is accomplished has to
be “paid for” with some form of energy.
Energy can’t be created or destroyed, but
it can be transformed from one kind
into another and it can be transferred
from one object to another.
• Doing WORK is one way to transfer
energy from one object to another.
Work = Force x displacement
W = Fd
• Unit for work is Newton x meter. One
Newton-meter is also called a Joule, J.
Work = Force x displacement
• Work is not done unless there is a
displacement.
• If you hold an object a long time, you
may get tired, but NO work was done.
• If you push against a solid wall for
hours, there is still NO work done.
• Energy and Work have no direction
associated with them and are therefore
scalar quantities, not vectors.
YEAH!!
• For work to be done, the displacement
of the object must be along the same
direction as the applied force. They
must be parallel.
• If the force and the displacement are
perpendicular to each other, NO work is
done by the force.
• For example, in lifting a book, the force
exerted by your hands is upward and the
displacement is upward- work is done.
• Similarly, in lowering a book, the force
exerted by your hands is still upward, and
the displacement is downward.
• The force and the displacement are STILL
parallel, so work is still done.
• But since they are in opposite directions,
now it is NEGATIVE work.
• On the other hand, while carrying a
book down the hallway, the force from
your hands is vertical, and the
displacement of the book is horizontal.
• Therefore, NO work is done by your
hands.
• Since the book is obviously moving,
what force IS doing work???
The static friction force between your
hands and the book is acting parallel to
the displacement and IS doing work!
Work = Force x distance
Your Force
Vertical component of d
• So,….while
climbing stairs or
walking up an
incline, only the
vertical
component of
the displacement
is used to
calculate the
work done in
moving the
object from the
bottom to the
top.
Horizontal component of d
Example
How much work is done to carry a 5
kg cat to the top of a ramp that is
7 meters long and 3 meters tall?
W = Force x displacement
Force = weight of the cat
Which is parallel to the weight- the
length of the ramp or the height?
d = height NOT length
W = mg x h
W = 5 x 10 x 3
W = 150 J
3m
How much work do you do to
carry a 30 kg cat from one
side of the room to the other
if the room is 10 meters
long?
ZERO, because your Force is
vertical, but the displacement
is horizontal.
Example
Displacement = 20 m
A boy pushes a
lawnmower 20 meters
across the yard. If he
pushed with a force of
200 N and the angle
between the handle
and the ground was
50 degrees, how
much work did he do?
F cos q
q
F
W = (F cos q )d
W = (200 cos 50) 20
W = 2571 J
Watch for those “key words”
NOTE: If while pushing an object, it is
moving at a constant velocity,
the NET force must be zero.
So….. Your applied force must be exactly
equal to any resistant forces like friction.
A 5.0 kg box is pulled 6m across a rough horizontal floor
(m = 0.4) with a force of 80N at an angle of 35 degrees
above the horizontal. What is the work done by EACH
force exerted on it? What is the NET work done?
Does the gravitational force do any work?
Normal
NO! It is perpendicular to the displacement.
Does the Normal force do any work?
FA
No! It is perpendicular to the displacement.
q
f
Does the applied Force do any work?
Yes, but ONLY the horizontal component!
WF = Fcosq x d = 80cos 35 x 6 = 393.19J
mg
Does friction do any work?
Yes, but first, what is the normal force? It’s NOT mg!
Normal = mg – Fsinq
Wf = -f x d = -mNd = -m(mg – Fsinq)d = -90.53J
What is the NET work done?
393.19 – 90.53 = 302.66 J
• Energy and Work have no direction
associated with them and are therefore
scalar quantities, not vectors.
YEAH!!
• Power is the rate at which
work is done- how fast you
do work.
Power = work / time
P=W/t
• You may be able to do a lot
of work, but if it takes you a
long time, you are not very
powerful.
• The faster you can do work,
the more powerful you are.
• The unit for power is Joule / seconds
which is also called a Watt, W
(just like the rating for light bulbs)
In the US, we usually measure power
developed in motors in “horsepower”
1 hp = 746 W
Example
A power lifter picks up a 80 kg barbell above his
head a distance of 2 meters in 0.5 seconds.
How powerful was he?
P=W/t
W = Fd
W = mg x h
W = 80 x 10 x 2 = 1600 J
P = 1600 / 0.5
P = 3200 W
Another way of looking at Power:
work
power 
time
(force x displacement)
power =
time
 displacement 
power  force x 



time
power  force x velocity
Power = Force x velocity
Kinds of Energy
Kinetic Energy
the energy of motion
K = ½ mv2
Potential Energy
Stored energy
It is called potential
energy because it
has the potential to
do work.
• Example 1: Spring potential energy in
the stretched string of a bow or spring
or rubber band. SPE = ½ kx2
• Example 2: Chemical potential energy
in fuels- gasoline, propane, batteries,
food!
• Example 3: Gravitational potential
energy- stored in an object due to its
position from a chosen reference point.
Gravitational potential energy
GPE = weight x
height
GPE = mgh
Since you can
measure height
from more than
one reference
point, it is
important to
specify the
location from
which you are
measuring.
• The GPE may be negative. For
example, if your reference point is the
top of a cliff and the object is at its
base, its “height” would be negative, so
mgh would also be negative.
• The GPE only depends on the weight
and the height, not on the path that it
took to get to that height.
Work and Energy
Often, some force must do work
to give an object potential or
kinetic energy.
You push a wagon and it starts
moving. You do work to
stretch a spring and you
transform your work energy
into spring potential energy.
Or, you lift an object to a certain
height- you transfer your
energy into the object in the
form of gravitational potential
energy.
Work = Force x distance =
change in energy
Example of
Work = change in energy
How much more distance is required to stop if a
car is going twice as fast (all other things
remaining the same)?
Fd = D½ mv2
The work done by the forces acting = the change in the kinetic energy
With TWICE the speed, the car has
FOUR times the kinetic energy.
Therefore it takes FOUR times the stopping
distance.
(What FORCE is doing the work??)
The Work-Kinetic Energy
Theorem
NET Work done = D Kinetic Energy
Wnet = ½ mv2f – ½ mv2o
Example
A 500kg car moving at 15m/s skids 20m to a stop.
How much kinetic energy did the car lose?
DK = ½ mvf2 – ½ mvo2
DK = -½ (500)152
DK = -56250J
What force was applied to stop the car?
F·d = DK
F = DK / d
F = -56250 / 20
F = -2812.5N
Example
A 500kg car moving at 15m/s slows to 10m/s.
How much kinetic energy did the car lose?
DK = ½ mvf2 – ½ mvo2
DK = ½ (500)102 - ½ (500)152
DK = -31250J
What force was applied to stop the car if the
distance moved was 12m?
F·d = DK
F = DK / d
F = -31250 / 12
F = -2604N
Example
A 500kg car moving on a flat road at 15m/s skids
to a stop.
How much kinetic energy did the car lose?
DK = ½ mvf2 – ½ mvo2
DK = -½ (500)152
DK = -56250J
How far did the car skid if the effective coefficient
of friction was 0.6?
Stopping force = friction = mN = mmg
F·d = DK
(mmg)·d = DK
d = DK / mmg
d = 56250 / (0.6 · 500 · 10) = 18.75m
Back to Power…
Since Power = Work / time and
Net work = DK…
Power = DK / time
In fact, Power can be calculated in many
ways since Power = Energy / time, and
there are MANY forms of energy!
Graphs
• If you graph the applied force vs. the
position, you can find how much work
was done by the force.
Work = Fd = “area under the curve”.
Total Work = 2N x 2m + 3N x 4m
= 16 J
Area UNDER the x-axis is
NEGATIVE work = - 1N x 2m
Force, N
F
Position, m
d
Net work = 16J – 2J = 14J
Back to the Work-Kinetic Energy
Theorem…
According to that theorem,
net work done = the change in the kinetic energy
Wnet = DK
But, if the work can be found by taking the “area
under the curve”, then it is also true that
Area under the curve = DK = ½ mvf2 – ½ mvo2
so that the area can be used to predict the final
velocity of an object given its initial velocity and
its mass.
For example…
Suppose from the previous graph
(Area = Wnet = 14J), the object upon which
the forces were exerted had a mass of 3kg
and an initial velocity of 4m/s. What would
be its final velocity?
Area under the curve = ½ mvf2 – ½ mvo2
14 = ½ 3vf2 – ½ 3(4)2
vf = 5.0 m/s
The Spring Force
If you hang an object
from a spring, the
gravitational force
pulls down on the
object and the spring
force pulls up.
The Spring Force
The spring force is
given by
Fspring = kx
Where x is the amount
that the spring
stretched and k is the
“spring constant”
which describes how
stiff the spring is
The Spring Force
If the mass is hanging at
rest, then
Fspring = mg
Or
kx = mg
(this is called “Hooke’s Law)
The easiest way to
determine the spring
constant k is to hang a
known mass from the
spring and measure how
far the spring stretches!
k = mg / x
Graphing the Spring Force
Suppose a certain spring had a spring constant k =
30 N/m.
Graphing spring force vs. displacement:
On horizontal axis- the displacement of the spring: x
On vertical axis- the spring force = kx = 30x
What would the graph look like?
Fs = kx
In “function” language: f(x) = 30x
Spring Force vs. Displacement Fs = 30x
Fs
How could you use the graph
To determine the work done by
The spring from some x1 to x2?
Take the AREA under the curve!
x1
x2
x
Analytically…
The work done by the spring is given by
Ws = ½ kxf2 – ½ kxo2
where x is the distance the spring is
stretched or compressed
(Which would yield the same result as taking
the area under the curve!)
I love mrs. BRown
Mechanical Energy
Mechanical Energy = Kinetic Energy + Potential Energy
E
= ½ mv2
+
mgh
“Conservative” forces - mechanical energy is
conserved if these are the only forces
acting on an object.
The two main conservative forces are:
Gravity, spring forces
“Non-conservative” forces - mechanical
energy is NOT conserved if these forces
are acting on an object.
Forces like kinetic friction, air resistance
Conservation of Mechanical Energy
If there is no kinetic friction or air resistance, then the
total energy of an object remains the same.
If the object loses kinetic energy, it gains potential
energy.
If it loses potential energy, it gains kinetic energy.
For example: tossing a ball upward
Conservation of Mechanical Energy
The ball starts with kinetic energy…
Which changes to potential energy….
Which changes back to kinetic energy
What about the
energy when it is
not at the top or
bottom?
PE = mgh
Energybottom = Energytop
½ mvb2 = mght
E = ½ mv2 + mgh
K = ½ mv2
K = ½ mv2
Examples
•
•
•
•
•
dropping an object
box sliding down an incline
tossing a ball upwards
a pendulum swinging back and forth
A block attached to a spring oscillating
back and forth
If there is kinetic friction or air resistance,
energy will not be conserved.
Energy will be lost in the form of heat.
The DIFFERENCE between the
original energy and the final energy
is the amount of energy lost due to heat.
Original energy – final energy = heat loss
Sometimes, mechanical energy is
actually INCREASED!
For example: A bomb sitting on the floor
explodes.
Initially:
½ mv2 = 0 mgh = 0 ½ kx2 = 0 E = 0
After the explosion, there’s lots of kinetic
and gravitational potential energy!!
Did we break the laws of the universe
and create energy???
Of course not! NO ONE, NO ONE, NO
ONE can break the laws!
The mechanical energy that now
appears came from the chemical
potential energy stored within the
bomb itself!
According to the Law of Conservation of
Energy, energy cannot be created or
destroyed. The total amount of mechanical
energy in a system remains constant when
there are no NONCONSERVATIVE forces
doing work, but one form of energy may be
transformed into another as conditions
change.
K1
E1 = E2
+ GPE1 = K2 + GPE2
½ mv12 + mgh1 = ½ mv22 + mgh2
The starting point in using Conservation of
Energy to solve for an unknown is to
locate a position where the total
mechanical energy IS known and use that
value as Eo. (i.e. you know all of these
quantities: the height, the velocity, the
distance the spring is stretched)
Simple Machines and Efficiency
Machine: A device that HELPS do work.
A machine cannot produce more WORK
ENERGY than the energy you put into it,
but it can make your work easier to do.
• Some common “simple
machines” include levers,
pulleys, wheels and
axles, and inclined planes
• Ideally, with no friction,
the work energy you get
out of a machine equals
the work energy you put
into it.
Ideally:
Work in = work out
Work = Force x distance
The work you put into a machine is called
EFFORT work.
The work you get out of the machine- is called
RESISTANCE work, so ideally
Effort Work = Resistance Work
Feffortdeffort = Fresistancedresistance
(if there’s no NON-conservative forces!)
Levers
Effort force
Effort force
Effort force
• The RESISTANCE force is the weight
of the load being lifted.
• Which arrangement will require the least
EFFORT force?
• How do you “pay” for a small effort force?
Inclined Planes
Weight =
Resistance Force
Height =
Resistance
Distance
Which arrangement will require the least
EFFORT force?
How do you “pay” for a smaller effort force?
Two pulleys with a belt
A motor is attached to one of the
pulleys so that as it turns, the belt
causes the second pulley to turn.
Which pulley should the motor be
attached to so that it requires the
least effort force from the motor?
To have the least effort force, the
effort distance must be the
greatest. In this case the effort
distance is the number of turns
around. Which pulley will have to
go around more times?
This is the pulley that the motor
should be attached to for the least
effort force.
Pulleys
pulley simulation
Efficiency
No machine is perfect. That is reflected in
the “efficiency” of the machine. In the real
world, the efficiency will always be less
that 100%. It is found by
Energy out work out (resis tan ce)
Efficiency 

Energy in
work in (effort)
A man pushes a 50 kg box up a 25 m long incline that is 8
meters high by applying a force of 200 N.
What is the effort (input) work?
Weffort = Feffortdeffort
W = 200N x 25 m = 5000J
What is the resistance (output) work?
Wresistance = Frdr
Energy out work out (resis tan ce)
Efficiency 

Energy
in
work in (effort)
W = mg x h
W = 50 x 10 x 8 = 4000J
What is the efficiency of the incline?
Efficiency = 4000 J / 5000 J = .80 = 80%