Transcript Energy - Rolla Public Schools

Chapters 10/11
Work, Power,
Energy, Simple
Machines
10.1 Energy and Work
• Some objects, because of their
– Composition
– Position
– movement
Possess the ability to cause
change, or to do Work.
Anything that has energy has
the ability to do work.
In this chapter, we focus on
Mechanical Energy only….
Old Man on the Mountain
(before and after)
A. Energy of Things in Motion
• Called Kinetic Energy… here’s the
derivation…starting with an
acceleration equation…
v1  v  2ad
2
2
0
v1  v  2ad
2
2
0
2
add - V to both sides
v1  v  2ad
2
2
0
v1  v  2ad
2
2
0
Substitute F/m for a
fd
v v  2
m
2
1
2
0
fd
v v  2
m
2
1
2
0
Multiply by ½ m
1 2 1 2
mv1  mv 0  fd
2
2
1 2
mv  fd
2
Let’s look at each side of this equation, one side at a time….
• Left side contains
terms that describe
energy of a system
1 2
K E  mv
2
…where the change in velocity is due to
work being done.
Kinetic Energy
Kinetic energy is the energy of motion. By
definition kinetic energy is given by:
KE = ½ m v 2 Derive the
The equation shows that . . .
• the more mass a body has
• or the faster it’s moving
unit for
Energy,
the
Joule!!!!
. . . the more kinetic energy it has.
K is proportional to v2, so doubling the speed quadruples kinetic
energy, and tripling the speed makes it nine times greater.
The formula for
kinetic energy,
KE = ½ m v 2
shows that its units
are:
kg · (m/s)2
= kg · m 2 / s 2
= (kg · m / s 2 ) m
= N·m
= Joule
SI Kinetic Energy Units
So the SI unit
for kinetic
energy is the
Joule, just as it
is for work.
The Joule is
the SI unit for
all types of
energy.
Sample Calculations….
• What is the kinetic energy of a 75.0 kg warthog
sliding down a muddy hill at 35.0 m/s?
• What is the kinetic energy of a 50.0 kg anvil
after free-falling for 3.0 seconds?
Mechanical Work
Right Side of
out earlier
equation
implies that a
force, applied
through a
distance, causes
changes in KE
1 2
mv  fd
2
Work-Energy Theorem
Looking at both sides of the equation…..
K E  fd
Simply says that by doing work on
a system, you increase the kinetic
energy
Work is done when…..
• Work done against a force,
including friction, or gravity
– (no net work is done however)
• Work done to change speed
(momentum)
– (net work is done)
Work is only done by a force on an
object if the force causes the object
to move in the direction of the force.
Objects that are at rest may
have many forces acting on them,
but no work is done
if there is no movement.
Work
The simplest definition for the amount of work a force does on an object
is magnitude of the force times the distance over which it’s applied:
W=Fd
This formula applies when:
• the force is constant
• the force is in the same direction as the displacement of the object
F
d
Work Example
A 50 N horizontal force is applied to a 15 kg crate of
BHM over a distance of 10 m. The amount of work
this force does is
W = 50 N · 10 m = 500 N · m = 500 J
In this problem, work is done to change the kinetic energy of the box….
Big Heavy Mass
50 N
10 m
Negative Work
A force that acts opposite to the direction of motion of
an object does negative work. Suppose the BHM
skids across the floor until friction brings it to a stop.
The displacement is to the right, but the force of
friction is to the left. Therefore, the amount of work
friction does is -140 J.
v
BHM
fk = 20 N
7m
When zero work is done
As the crate slides horizontally, the normal force and weight do no
work at all, because they are perpendicular to the displacement. If the
BHM were moving vertically, such as in an elevator, then each force
would be doing work. Moving up in an elevator, the normal force
would do positive work, and the weight would do negative work.
Another case when zero work is done is when the displacement is
zero. Think about a weight lifter holding a 200 lb barbell over her
head. Even though the force applied is 200 lb, and work was done in
getting over her head, no work is done just holding it over her head.
N
BHM
mg
7m
Work done in lifting an object
• If you lift an object at constant velocity,
there is no net force acting on the
object….therefore there is no net work done
on the object.
• However, there is work done, but not on the
object, but against gravity
Net Work
The net work done on an object is the sum of all the work done on it
by the individual forces acting on it. Net Work is a scalar, so we can
simply add work up. The applied force does +200 J of work; friction
does -80 J of work; and the normal force and weight do zero work.
So, Wnet = 200 J - 80 J + 0 + 0 = 120 J
Note that (Fnet ) (distance) = (30 N) (4 m) = 120 J.
Therefore, Wnet = Fnet d
N
BHM
fk = 20 N
mg
FA = 50 N
4m
Net Work done????
• Is work done in…
–Lifting a bowling ball???
–Carrying a bowling ball
across the room???
–Sliding a bowling ball along a
table top???
If the force and displacement are not
in the exact same direction, then
work = Fd(cosq),
where q is the angle between the force
direction and displacement direction.
F =40 N
35
d = 3.0 m
The work done in moving the block 3.0 m
to the right by the 40 N force at an angle
of 35 to the horizontal is ...
W = Fd(cos q) = (40N)(3.0 m)(cos 35) = 98 J
B. Energy of Position
• Called Potential Energy
Potential Energy
energy of position or condition
Ug = m g h
The equation shows that . . .
• the more mass a body has
• or the stronger the gravitational field it’s in
• or the higher up it is
. . . the more gravitational potential
energy it has.
SI Potential Energy Units
From the equation
Ug = m g h
the units of gravitational
potential energy must be:
=m·g·h
= kg · (m/s2) ·m
= (kg · m/s2) ·m
= N·m
= J
What a surprise!!!!!
This shows the SI
unit for potential
energy is still the
Joule, as it is for
work and all other
types of energy.
Reference point for U is arbitrary
Example: A 190 kg mountain
goat is perched precariously
atop a 220 m mountain ledge.
How much gravitational
potential energy does it have?
Ug = mgh = (190kg) (9.8m/s2) (220m) = 410 000J
This is how much energy the goat has with respect to the
ground below. It would be different if we had chosen a
different reference point.
Conservation and Exchange of Energy
Law of Conservation of Energy
In Conservation of Energy, the total
mechanical energy remains constant
In any isolated system of objects interacting
only through conservative forces, the total
mechanical energy of the system remains
constant.
Law of Conservation of Energy
Energy may neither be created, nor destroyed,
but is transformed from one form to another.
Example:
kinetic energy of
flowing water is
converted into
electrical energy
using magnets.
Energy is Conserved
• Conservation of Energy is different from
Energy Conservation, the latter being about
using energy wisely
• Don’t we create energy at a power plant?
– That would be cool…but, no, we simply
transform energy at our power plants, from one
form to another
• (fossil fuel energy or nuclear energy or potential
energy of water to electrical energy)
• Doesn’t the sun create energy?
– Nope—it exchanges mass for energy
• E=mc2
Energy Exchange
• Though the total energy of a system is
constant, the form of the energy can change
• A simple example is that of a simple
pendulum, in which a continual exchange
goes on between kinetic and potential
pivot
energy
K.E. = 0; P. E. = mgh
h
K.E. = 0; P. E. = mgh
height reference
P.E. = 0; K.E. = mgh
Perpetual Motion
• Why won’t the pendulum swing forever?
• It’s hard to design a system free of energy paths
• The pendulum slows down by several mechanisms
– Friction at the contact point: requires force to
oppose; force acts through distance  work is
done
– Air resistance: must push through air with a
force (through a distance)  work is done
– Gets some air swirling: puts kinetic energy into
air (not really fair to separate these last two)
• Perpetual motion means no loss of energy
– solar system orbits come very close
Law of Conservation of Energy
PE = mgh
KE = 0
The law says that
energy must be
conserved.
On top of the shelf,
the ball has PE.
h
Since it is not
moving, it has NO
kinetic energy.
Law of Conservation of Energy
PE = mgh
KE = 0
If the ball rolls off
the shelf, the
potential energy
becomes kinetic
energy
PE = 0
KE = ½ mv2
h
Law of Conservation of Energy
Since the energy at the top MUST equal the
energy at the bottom…
PEtop + KEtop = PEbottom + KEbottom
Notice that the MASS can cancel!
Example 1
A large chunk of ice with mass
15.0 kg falls from a roof 8.00
m above the ground.
a) Find the KE of the ice when it reaches
the ground.
b) What is the velocity of the ice when it
reaches the ground?
Where is
the ball
the
fastest?
Why?
Energy at A?
Energy at B?
Energy at C?
3.0 kg ball
Calculate the energy
values for A-K
Bouncing Ball
• Superball has gravitational potential energy
• Drop the ball and this becomes kinetic
energy
• Ball hits ground and compresses (force times
distance), storing energy in the spring
• Ball releases this mechanically stored energy
and it goes back into kinetic form (bounces
up)
• Inefficiencies in “spring” end up heating the
ball and the floor, and stirring the air a bit
• In the end, all is heat
Power,
by definition, is
the rate of doing work
.
P=W/t
Unit=????
Power
• US Customary units are generally hp (horsepower)
– Need a conversion factor
ft lb
1 hp  550
 746 W
s
– Can define units of work or energy in terms of
units of power:
• kilowatt hours (kWh) are often used in
electric bills
• This is a unit of energy, not power
Simple Machines
Ordinary machines are typically complicated combinations of
simple machines. There are six types of simple machines:
Simple Machine
Example / description
• Lever
crowbar
• Incline Plane
ramp
• Wedge
chisel, knife
• Screw
drill bit, screw (combo of a wedge & incline
plane)
• Pulley
• Wheel & Axle
wheel spins on its axle
door knob, tricycle wheel (wheel & axle spin
together)
Simple Machines: Force & Work
A machine is an apparatus that changes the magnitude or
direction of a force.
Machines often make jobs easier for us
by reducing the amount of force we must
apply.
However, simple machines do not
reduce the amount of work we do! The
force we apply might be smaller, but we
must apply that force over a greater
distance.
Force / Distance Tradeoff
Suppose a 300 lb crate of silly string has to be loaded onto a 1.3 m
high silly string delivery truck. Too heavy to lift, a silly string truck
loader uses a handy-dandy, frictionless, ramp, which is at a 30º
incline. With the ramp the worker only needs to apply a 150 lb force
(since sin 30º = ½). A little trig gives us the length of the ramp: 2.6
m. With the ramp, the worker applies half the force over twice the
distance. Without the ramp, he would apply twice the force over half
the distance, in comparison to the ramp. In either case the work
done is the same!
continued on next slide
150 lb
300 lb
1.3 m
1.3 m
30º
Silly
String
Force / Distance Tradeoff
(cont.)
So why does the silly string truck loader bother with the ramp if he
does as much work with it as without it? In fact, if the ramp were
not frictionless, he would have done even more work with the ramp
than without it.
answer: Even though the work is the same or more, he simply
could not lift a 300 lb box straight up on his own. The simple
machine allowed him to apply a lesser force over a greater distance.
This is the “force / distance tradeoff.”
A simple machine allows a job to be done with a
smaller force, but the distance over which the force
is applied is greater. In a frictionless case, the
product of force and distance (work) is the same
with or without the machine.
Mechanical Advantage
Mechanical advantage is the ratio of the amount of force that must
be applied to do a job with a machine to the force that would be
required without the machine. The force with the machine is the
input force, Fin and the force required without the machine is the
force that, in effect, we’re getting out of the machine, Fout which is
often the weight of an object being lifted.
M.A. =
Fout
Fin
Note: a mechanical advantage has no units and is typically > 1.
Ideal vs. Actual Mechanical Advantage
When friction is present, as it always is to some extent, the actual
mechanical advantage of a machine is diminished from the ideal,
frictionless case.
Ideal mechanical advantage = I.M.A. = the mechanical advantage
of a machine in the absence of friction. Determined by comparing
physical attributes of the machine.
Actual mechanical advantage = A.M.A. = the mechanical
advantage of a machine in the presence of friction. Determined by
comparing the output force with the input force
I.M.A. > A.M.A, but if friction is negligible we don’t distinguish
between the two and just call it M.A.
Efficiency & Mechanical Advantage
Efficiency always comes out to be less than one. If eff > 1, then we
would get more work out of the machine than we put into it, which
would violate the conservation of energy. Another way to calculate
efficiency is by the formula:
A.M.A. To prove this, first remember that Wout (the
work we get out of the machine) is the same
eff =
I.M.A. as Fin × d when there is no friction, where d
is the distance over which Fin is applied. Also, Win is the Fin × d when
friction is present.
Fin w/ no friction
Fout / Fin w/ friction
A.M.A.
=
=
Fin w/ friction
Fout / Fin w/ no friction
I.M.A.
d Fin w/ no friction
Wout
In the last pulley problem,
=
= eff
=
Win
I.M.A. = 3, A.M.A. = 2.308.
d Fin w/ friction
Check the formula: eff = 2.308 / 3 = 76.9%, which is the same answer
we got by applying the definition of efficiency on the last slide.
Levers
• a lever (from French lever, "to raise",
c. f. alevant) is a rigid object that is
used with an appropriate fulcrum or
pivot point to multiply the mechanical
force that can be applied to another
object
Mechanical Advantage?
Crowbar (or pry bar)
Archimedes said “Give me a lever long enough,
and a place to stand and I can move the earth”
I.M.A. for a Lever
A lever magnifies an input force (so long as dF > do). Here’s why:
In equilibrium, the net torque on the lever is zero. So, the actionreaction pair to Fout (the force on the lever due to the rock) must
balance the torque produced by the applied force, Fin. This means
Fin·dF = Fout·do
Fout
Therefore, I.M.A. =
Fin
do = distance from object to fulcrum
dF = distance from applied force to
fulcrum
dF
=
do
dF
Fout
d0
Fin
fulcrum
Inclined Plane
• A more common word for an inclined plane
is a ramp.
• It is a surface that is set at an angle.
• The smaller the angle of a ramp the less
effort is needed, but it will take a longer
distance to gain the same height.
• A screw is an inclined plane and a curved
ramp
• A wedge is a modification of an inclined
plane it is made of two inclined planes
Johnstown, PA inclined Plane
Archimede’s Screw
I.M.A. for an Incline Plane
I.M.A. = d / h
A more gradual the incline will have a greater mechanical
advantage. This is because when q is small, so is mg sin q (parallel
component of weight)(equal to the force necessary to push box up
hill).
d is very big, though, which means, with the ramp, we apply a
small force over a large distance, rather than a large force over a
small distance without it. In either case we do the same amount of
work (ignoring friction).
IMA of a screw
Determined by the
“pitch” of the
threads
Pulleys
• A pulley is a
grooved wheel,
called a sheave, and
a block.
• Used with a rope or
chain to change
direction or
magnitude of a
force.
• IMA = # of support
strands
What’s Wrong with this Cartoon???????
M.A. for a Single Pulley #1
With a single pulley the IMA is only 1.
The only purpose of this pulley is that it
allows you to lift something up by applying a
force down.
Fout
The AMA of this pulley would be less than
one, depending on how much friction is
present.
m
Fin
mg
Pulley systems, with multiple pulleys, can have
large mechanical advantages, depending on
how they’re connected.
M.A. for a Single Pulley #2
With a single pulley used in this
way the I.M.A. is 2.
Fin
F
F
m
mg
The reason for this is that there are
two supporting ropes.
The tradeoff is that you must pull
out twice as much rope as the
increase in height, e.g., to lift the
box 10 feet, you must pull 20 feet
of rope.
M.A: Pulley System #1
In this type of 2-pulley system the
I.M.A. = 3
Fin
F
The reason for this is that there are
three supporting ropes
F
F
m
mg
A 300 lb object could be lifted with a
100 lb force if there is no friction.
The tradeoff is that you must pull out
three times as much rope as the
increase in height, e.g., to lift the box 4
feet, you must pull 12 feet of rope.
I.M.A: Pulley System #2
1. Number of pulleys: 3, but this matters not
2. Number of supporting ropes: 3, and this does matter
3. I.M.A. = 3, since there are 3 supporting ropes
4. Force required to lift box if no friction: 20 N
5. If 2 m of rope is pulled, box goes up: 0.667 m
F
6. Potential energy of box 0.667 m up: 40 J
7 a. Work done by input force to lift box 0.667 m
up with no friction: 20 N · 2 m = 40 J
7 b. Work done lifting box 0.667 m straight up
without pulleys: 60 N · 0.667 m = 40 J
If the input force needed with friction is 26 N,
9. A.M.A. = (60 N) / (26 N) = 2.308 < I.M.A.
10. Work done by input force now is: 26 N · 2 m = 52 J
F
F
60 N
60 N
Fin
Efficiency
Note that in the last problem:
Work done using
Work done lifting
Potential energy
= at high point
pulleys (no friction) =
straight up
little force ×
big force ×
mgh
big distance
little distance
All three of the above quantities came out to be 40 J. When we had
to contend with friction, though, the rope still had to be pulled a “big
distance,” but the “little force” was a little bigger. This meant the
work done was greater: 52 J. The more efficient a machine is, the
closer the actual work comes to the ideal case in lifting: mgh.
Efficiency is defined as:
Wout
work done with no friction (often mgh)
eff =
= work actually done by input force
Win
In the last example eff = (40 J) / (52 J) = 0.769, or 76.9%. This means
about 77% of the energy expended actually went into lifting the box.
The other 13% was wasted as heat, thanks to friction.
Wheel & Axle
The axle and wheel move together here, as in a doorknob. Not all
wheel and axles are actually simple machines……..a wheel on a
little red wagon does NOT act as a simple machine.
I.M.A. = rin / rout
With a wheel and axle a small force can produce
great turning ability. (Imagine trying to turn a
doorknob without the knob.) Note that this
simple machine is almost exactly like the lever.
Using a bigger wheel and smaller axle is just like
moving the fulcrum of a lever closer to object
being lifted.
Human Body
as a Machine
The center of mass of the forearm w/ hand is
shown. Their combined weight is 4 lb.
Fbicep
tendon
bicep
40 lb
dumbbell
humerus
radius
ligament
4 lb
c.m.
4 cm
40 lb
Because the biceps attach so close to the elbow,
14 cm
the force it exerts must be great in order to
30 cm
match the torques of the forearm’s weight and
dumbbell: Fbicep(4 cm) = (4 lb)(14 cm) + (40 lb)(30 cm)
continued on next slide
Fbicep= 314 lb !
Human Body as
a Machine (cont.)
Fbicep
Let’s calculate the mechanical advantage
of this human lever:
Fout / Fin = (40 lb) / (314 lb) = 0.127
4 lb
4 cm
14 cm
30 cm
40 lb
Note that since the force the biceps
exert is less than the dumbbell’s
weight, the mechanical advantage is
less than one. This may seem pretty
rotten. It wouldn’t be so poor if the
biceps didn’t attach so close to the
elbow. If our biceps attached at the
wrist, we would be super duper strong,
but we wouldn’t be very agile!