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Work
When you hold something, are you exerting a
force on the object? Yes.
When you hold something, are you doing
work?
If you set the object on a table, does the table
exert a force on the object? Yes. Does the
table do any work? No.
When you hold something, then, you are not
doing work either.
Work
What can you do that the table can’t do? You
can lift the object up - which is work!
We define the concept of WORK as the
exertion of force through a distance.
There is one more consideration, however. In
tether ball, the pole exerts a force on the
moving ball (via the rope). Does the pole
do work?
Work
The pole does NOT do work!
Does the player who hits the tether ball do
work? YES.
What is the difference between what the pole
does and what the player does?
Work
The difference is in the direction. The pole
was pulling on the tether ball perpendicular
to the motion of the tether ball. The player
was pushing on the tether ball in the same
direction as the motion.
In part two, however, we also had torque as
being force across a distance. What is the
difference?
Work and Torque
In applying torque, the direction of the force
had to be perpendicular to the distance.
This caused a turning force: t = r F sin(qrF)
In doing work, the direction of the force has to
be parallel (or anti-parallel) to the distance
moved. We write this this way:
Work = F s
where the dot indicates the cosine of the angle
between F and s: Work = F s cos(qFs) .
Work
Work = F s
Although F and s are vectors (with magnitude
and direction), Work is a scalar
(magnitude only).
Can we have positive and negative work? If
the Force and distance are parallel, the
amount of work is positive, but if the two
vectors are anti-parallel, then the work is
negative.
Energy
We can now define the concept of energy:
Energy is the capacity to do work (in ideal
circumstances).
We all know that we can do work: exert a
force through a distance. But to do that
requires food. Thus we convert the energy
in food into work. The same thing happens
when we burn coal to generate heat which
can be converted into electricity which can
be converted into lots of useful work.
Conservation of Energy
A Natural Law
Many such examples as we just saw lead us to
propose a natural law. Remember that a
natural law is a statement of how nature
seems to work - it is not “derived” from
anything more basic, it is observed to fit the
results of observations (experiments).
Energy can neither be created nor
destroyed (that is, energy is conserved).
However, it can be transformed from one
form into another.
Conservation of Energy
The equation that comes from this law of
conservation of energy is:
S Energiesinitially = S Energiesfinally .
Our job now is to find out how the amount of
energy in different forms relates to the
various parameters associated with that
form. That is, we need to derive formulas
for various kinds of energy.
Units
The units of energy (and work) are:
Nt*m = Joule.
A British unit of energy is the BTU (British
Thermal Unit). 1 BTU = 1,054 Joules
(This is the energy necessary to heat one pound of water 1 oF.)
Another unit of energy is the calorie.
1 calorie = 4.186 Joules
(This is the energy necessary to heat one gram of water 1 oC.)
However, the calorie we refer to when we eat is
really a kilocalorie = 4,186 Joules.
Units
The units of torque are: Nt*m = Nt*m.
Note that even though torque and energy both
have units of Nt*m, they are different
quantities, and so they have different formal
names. Energy units are in Joules, while
torque units are simply specified as Nt*m.
In the British system, the unit of torque is
simply called the foot-pound (ft-lb).
Positive and Negative
Can we have energies that are negative?
First, can we have negative money? Yes - it’s
called debt. You need to earn money to pay
off the debt and reach up to zero.
In the same way, some energies can be
negative - we need to gain some energy to
reach what we define as zero energy.
Forms of Energy
Kinetic Energy
Energy of motion, called Kinetic Energy:
should depend on mass and speed of object.
Your car has energy when it is moving.
The wind has energy when it is moving, and
we can convert this wind energy into
electric energy via windmills.
Potential Energies
Energy of position, called Potential Energy:
should depend on why that position has
energy.
The water stored behind a dam has energy due to it’s
height above the base of the dam. We can use this
to run a hydroelectric station. The energy in food
is due to the molecular binding of the atoms in the
food. The same is true for coal, oil and gas.
There is also energy stored in the nucleus of atoms
- nuclear energy.
Some other forms
• Heat: should depend on temperature, type
and amount of material. We burn coal to
get heat to turn water into steam and use the
steam pressure to get work (or electricity).
• Light: should depend on type and intensity
of the light.
• Sound: should depend on type and
intensity of the sound.
Kinetic Energy - derivation
If we let an object fall, it gains speed. It also
gains what we call kinetic energy. By the
Conservation of Energy law, the amount of
work going into the object (from gravity)
will equal the amount of energy the object
has (kinetic): F s cos(q) = mg h (1). But if
an object falls a distance h with an
acceleration of g, how fast is it going?
Kinetic Energy - derivation
KE(m,v) = mgh (The amount of kinetic energy,
which depends on the quantities mass and speed in
this case equals the amount of work done by
gravity, mgh).
From our motion equations, v = vo + gt and
h = ho + vot + (1/2)gt2 or in this case (ho=0,
vo=0): h = (1/2)gt2, or t = (2h/g)1/2 so v = gt =
g(2h/g)1/2, or v = (2hg)1/2, or h = (1/2)v2/g; thus
mgh = mg(1/2)v2/g = (1/2)mv2 = KE .
Kinetic Energy - formula
KE = (1/2)mv2 . Note that the kinetic energy
depends on m (the more mass the more
kinetic energy) and on v2 (if you double the
speed, you quadruple the kinetic energy).
Also note that the kinetic energy must always
be either zero or positive - it can’t be
negative. (This is like cash in the money
analogy.)
Kinetic Energy - considerations
You’ve probably heard the expression: “speed
kills”. This comes from the fact that KE
depends on the square of the speed. If you
double your speed, you quadruple the
amount of energy of the object. And
remember that energy is the capacity to do
work - for either good or bad. Uncontrolled
energy can exert large forces through
significant distances - which can be very
dangerous!
Kinetic Energy - considerations
Note that the difference
between (1 m/s)2 and (2 m/s)2 is 3 m2/s2,
whereas the difference
between (99 m/s)2 and (100 m/s)2 is 199 m2/s2.
What this indicates is that it takes more and more
energy to move faster and faster.
This explains why there is so little difference
between first and tenth in a speed race between
trained athletes!
Gravitational Potential Energy
If we let the force of gravity act on an object
as it moves, gravity is exerting a force
through a distance and may add or subtract
energy from the object. We can work with
this near the earth (where gravity is
constant) with PEgravity = mgh . Farther
from the earth’s surface, gravity changes,
and we need a different formula (from
calculus): PEgravity = -Gm1m2/r12 .
PEgravity Considerations
In the simpler formula near the earth’s
surface, PEgravity = mgh both m and g are
positive numbers, but h is a height
measured from some point that you
determine. It can be the ground, but doesn’t
have to be. Note that h can be either
positive or negative!
h and Dh
The formula for PEgravity really should be
DPEgr = mgDh instead of PEgr = mgh. We often
use h when we mean Dh, as in the case of “what is
your height?”. What we mean is, “what is the
difference in height between the top of your head
and the bottom of your feet?”. We often assume
that we are measuring from the floor, ground, or
some other standard position. If you use (or see)
PEgr = mgh, be sure to interpret it as DPEgr = mgDh,
and know where the “standard” position of h=0 is.
PEgravity Considerations
In the more general form,
PEgravity = -Gm1m2/r12 , the PE is always
negative, with the highest (least negative)
value being when PE=0 or r12 goes to
infinity.
Note in particular, that while the force of
gravity goes as r122, the PE of gravity goes
only as r12.
PE versus DPE
We have a similar case here as we did with the
near-earth PE: when should we use PE and
when should we use DPE ?
If we use PE = mgh, we are assuming that there
is some standard position where h=0 (usually
the ground, the floor, or table top).
If we use PE = -GM1m2/r12 , we are assuming
that the “standard r12” is a position that makes
PE = 0; to have this, the “standard r12” would
have to be infinity, since 1/infinity = 0.
Problems
Having KE and PEgravity, we can start solving
some problems using the Conservation of
Energy.
Problem: How high will a ball go if it is
thrown with a speed of 25 m/s?
We could solve this problem using Newton’s
Second Law and the equations for constant
acceleration, or we could use Conservation
of Energy.
Tossing a ball up
Let’s try this problem using Conservation of
Energy:
We recognize that we have kinetic energy (since we
have motion), and we recognize that we have
gravitational potential energy (since we have
gravity); also vi=25 m/s; hi=0 m (start from the
ground); vf=0 (highest point). These are all
related by the Conservation of Energy:
Tossing a Ball Up
S Energiesinitially = S Energiesfinally .
KEi + PEi = KEf
+ PEf
(1/2)mvi2 + mghi = (1/2)mvf2 + mghf
(note that this is equivalent to saying DKE = DPE, or ½mvf2 – ½mvi2 = mg(hf-hi)
(1/2)*m*(25 m/s)2 + m*(9.8 m/s2)*(0 m) =
(1/2)*m*(0 m/s)2 + m*(9.8 m/s2)*hf .
Here we see that the mass cancels out, and we
have one equation in one unknown (hf):
hf = (1/2)*(25 m/s)2 / (9.8 m/s2) = 31.89 m.
Observations
We should note two things from this example:
Conservation of Energy is a scalar equation,
and so has no information about directions.
This makes it easier to solve, but gives less
information in the answer.
Conservation of Energy makes no mention of
time (only initial and final). This removes t
from the problem - making it easier but also
giving us less information in the answer.
Escape Speed
In the previous example, we threw something up that
went about 32 meters high. How fast would we
have to throw something to make it escape
from the earth altogether?
We can use Conservation of Energy again, but
we need the more general form for potential
energy due to gravity.
To escape the earth, rf = infinity!
We start with ri = Rearth = 6.4 x 106m.
Escape Speed
S Energiesinitially = S Energiesfinally .
KEi + PEi = KEf
+ PEf
(1/2)mvi2 + -Gmearthm/ri = (1/2)mvf2 + -Gmearthm/rf
(note that this is equivalent to saying DKE = DPE, or ½mvf2 – ½mvi2 = -Gmem/rf - -Gmem/ri)
We see that m is in each term, so we cancel it.
(1/2)*(vi)2 - (6.67x10-11 Nt*m2/kg2) *(6.0 x 1024 kg)/(6.4x106m) =
(1/2)*(0 m/s)2 - (6.67x10-11 Nt*m2/kg2) *(6.0 x 1024 kg)/(infinity)
We again have one equation in one unknown (vi).
Escape Speed
We have used vf = 0 m/s since this is the
minimum speed we need at the end. We
could have more speed when we escape, but
we’re looking for the lowest speed for the
object to still escape; this means that both
the terms on the right side = 0.
Solving for vi = vescape = [2*G*Mearth /Rearth ]1/2
= 11,180 m/s = 25,000 mph.
Friction and Energy Loss
Can we use Conservation of Energy if we
have friction? What happens with friction?
We convert kinetic energy into heat!
What “formula” do we use for how much
energy is “lost” to friction, that is, how
much energy is converted from kinetic to
heat?
Friction
We start from the basic definition of energy:
the capacity to do work, where work =
Force thru a distance:
Elost = Ffriction * s. We still have Ffriction = mFc.
Where does this Elost go in the equation for
Conservation of Energy: on the initial or
final side? Is it a positive or negative
amount of energy?
Friction
Since some of the initial kinetic energy will go
(transform) into some heat, the Elost should be
a positive term if it is on the final side, or a
negative term if it is on the initial side.
S Energiesinitially = S Energiesfinally .
KEi + PEi = KEf + PEf + Elost
where Elost = + Ffriction*s = + mFcs .
Friction - example
Problem: If the coefficient of friction
between a block of wood and the concrete
floor is 0.50, how far will a block of wood
slide on the floor before coming to rest if it
starts with a speed of 10 m/s ?
We recognize this as a Conservation of
Energy problem with kinetic energy and
with friction (Elost).
Friction - an example
We are given: vi = 10 m/s; vf = 0 m/s; m = .65
We are looking for s (the distance of slide).
S Energiesinitially = S Energiesfinally .
KEi = KEf + Elost
where Elost = + Ffriction*s = + mFcs .
From S Fy = 0, we have Fc = mg. Therefore:
(1/2)*m*vi2 = (1/2)*m*vf2 + m*m*g*s .
Friction - an example
½*m*vi2 = ½*m*vf2 + m*m*g*s
We notice that there is an m in each term so it
cancels out!
½*(10 m/s)2 = 0 + (.65)*(9.8 m/s2)*s
This is one equation in one unknown (s):
s = ½*(10 m/s)2 / (.65)*(9.8 m/s2) = 7.85 m.
Power
We now know what Force and Energy are, but
what is Power?
Power
We now know what Force and Energy are, but
what is Power?
The definition of Power is that it is the rate of
change of Energy from one form into
another:
Power = DEnergy / Dt .
The units of power are: Joule/sec = Watt.
Another common unit is the horsepower, hp.
The conversion factor is: 1 hp = 746 Watts.
Example: Power
What is your power output when you climb
stairs?
In this case, you are changing your potential
energy (mgh) in time, so … P = Dmgh / Dt
if your mass = 70 kg, gravity is 9.8 m/s2, and
you climb steps of height 10 meters in a
time of 20 seconds:
P = 70 kg * 9.8 m/s2 * 10 m / 20 sec = 343 W
or 343 W * (1 hp / 746 W) = .46 hp .
Example #2: Power
What is your average power output per day?
If you eat 2,000 Calories per day, (and
assuming you do not gain or lose weight),
that energy must be converted into energy
you use throughout the day.
P = 2,000 Calories / day = (2,000 Cal)*(4,186
joule/Cal) / [(24 hours)*(60 min/hr)*(60
sec/min)] = 97 Watts.
Example #3: Power
How powerful must a car engine be (on
average) if it is to accelerate a 2,000 kg car
from zero to 65 mph in 20 seconds?
This is a power question. The change in
energy is in the form of kinetic energy. We
should convert 65 mph into metric form:
vf = 65 mph * (1 m/s / 2.24 mph) = 29 m/s.
vi = 0 (starts at rest); Dt = 20 sec.
Example #3: Power
Power = DEnergy / Dt =
[(1/2)*m*vf2 - (1/2)*m*vi2] / t =
[(1/2)*(2,000 kg)*(29 m/s)2 - 0 ] / 20 sec =
42,000 Watts * (1 hp / 746 Watts) = 56 hp.
Note that this is the average power.
Force and Power
We know how force is related to energy, and
how energy is related to power. Can we
relate power to force?
Work = F s , Power = DWork /Dtime =
F Ds /Dt (but Ds/Dt = v), so Power = F v .
Power, like work, is a scalar.
Note that if F is constant, Power must go up
as speed (v) goes up! If Power is constant,
F must go down as v goes up.
Force and Power
At very low speeds, even a small power will
give a rather large force!
On cars with manual transmissions, you
normally don’t rev up the engine (high
power), and then pop the clutch! This
causes tremendous forces that can break the
car!
What do you pay for:
Force, Energy, or Power?
What do you pay MLG&W (for example) for:
force, energy or power?
What do you pay the gas station for:
force, energy or power?
What do you pay for:
Force, Energy, or Power?
What do you pay MLG&W (for example) for:
force, energy or power?
What do you pay the gas station for: force,
energy or power?
In both cases you pay for ENERGY!
Cost of Energy
What is the cost of energy?
We saw before that we could do work at the
rate of a couple 100 Watts, but that was
hard work!. If we worked for 40 hours a
week, how much useful work would we
perform?
Work = Energy = Power * time.
The MKS unit of energy is the Joule, but this
is a very small unit. Another common unit
of energy is the Kilowatt*hour.
Cost of Energy
In terms of kilowatt-hours, if you worked at
the rate of 200 Watts for 40 hours, you
would do 8 KW-Hrs of work.
How much does the power company charge
for a KW-hr of energy?
How much would you make for your 8 KWHrs of work (assuming MLG&W paid you
exactly what it charges us for that same amount of
energy)?
Cost of Energy
In Memphis, MLG&W charges about 9 cents
per KW-hr. Thus, if you were to work for
the power company providing power, you
would earn about 8 KW-hr/week * $.09 =
$.72/week (72 cents per week)!
As we see, energy is quite cheap! The reason
our utility bills are so high is that we use so
much energy - especially when we heat (or
cool) things (like air and water)!
Computer Homework
The computer homework program, Energy
and Power, is the first program on
Volume 2. This program has questions
about the concepts we have covered in this
set of slides.
The labs on Atwood Machine and Hooke’s
Law also deal with these concepts.