Work & Energy
WORK
WORK: It is used by physicists to measure something that
is accomplished. So it results in the equation:
Work = Force x displacement
The symbol for work is the variable = W
W=(F)(d)
You must move something (d) with a force (F) to accomplish work.
Is work being done?
Pushing a car.
Attempting to lift 2,000,000 N.
A snowflake dropping from the sky.
Swimming in a rip current.
WORK
UNITS: Force is measured in Newtons, N, and displacement is
measured in meters, m.
BUT WAIT… we just did torque… and
they were measured in Nm… So scientists named Nm that measure
Work and Energy, and called them Joules (J).
1 Nm of work or energy = 1 Joule
NOW DO IT!
Calculate the work done if a 52.9 Kg brick is dropped 5.0 meters.
SO WHAT IF I PUSH ON A WALL THAT DOES NOT MOVE? HAVE I
DONE WORK?… NO! HAVE I USED ENERGY?... YES! WHAT IS
ENERGY? UH OH… another definition coming…
ENERGY
ENERGY: is the ability to do work… measured in
Joules (J)… AND it comes in two kinds:
POTENTIAL ENERGY: stored up energy
.
KINETIC ENERGY: moving energy
Potential or Kinetic?
Water behind a dam
Wound rubber band
Falling raindrops
The battery while listening to your IPod
POTENTIAL ENERGY
How do you “store up energy?... There are two
kinds we study…
GPE -
GRAVITY POTENTIAL ENERGY: If you put an object up in the
air… gravity will pull it down and make it move some distance…
WORK will be done… an object is moved some displacement… GPE
= mgh… mg=force gravity, h=height… the heavier the object is, the
more force on it, the higher it is, the further down it moves…
GPE = mgh
EPE - ELASTIC POTENTIAL ENERGY:
This is stored in a spring or
rubber band. The stronger the spring, or the further you stretch it, the
more work it can do. So, the EPE=(1/2)(k)(x2) … k = how strong the
spring is, and x= how far you stretch it…
EPE=(1/2)(k)(x2)
KINETIC ENERGY
KINETIC ENERGY = KE: This is moving energy.
Something that is moving will collide and crash into
another object, and move it a distance,… SO, the energy
that it has, will be equal to how BIG it is, m (mass), and
how FAST, v (velocity), it is moving….
KE = (1/2)(m)(v2)
NEW FORMULAS
W=(F)(d)
EPE=(1/2)(k)(x2)
GPE=(m)(g)(h)
KE=(1/2)(m)(v2)
ENERGY is what you “can do”………
WORK is what you “do do”………
and isn’t work doo-doo?
Calculating Work
F
d
m
A mass is being pulled to the right by a force, F , it is moving to
the right so that the displacement is d… BUT is ALL of the
force, F , doing work?.... NO…. because some of the force
is lifting up in the positive +y direction AND the box is not
moving upward in that direction… so (F)(sinϴ) does
NO WORK
This means that the part of F that is pulling to the right in the
+x direction is doing work, because that is the way the box
is moving… so work W = (F)(cos ϴ)(d)
Calculating Work
F
d
m
What about the other forces on the box like weight, mg, pulling
down, or the F-normal, of the ground pushing up…. NO…
they do NO WORK, because the box is not moving up or
down….
Calculating Work
F
d
m
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What about F-friction, is it doing work?... YES… BUT WAIT… THE
FRICTION FORCE IS NOT IN THE SAME DIRECTION THE BOX IS
MOVING!! IT IS TO THE LEFT!!
This force is fighting the work being done by F. It is doing what we call
negative work because it is being done in the OPPOSITE DIRECTION
THAT THE OBJECT IS MOVING… so the work done by friction is
W = -(Wf)(d) = -(µ)(F-normal)(d)
So now go back in your notes to the NET FORCE UNIT, and look up
how to find the force normal on a box with the force F at an angle…
F-normal = [(mg) – (F)(sinϴ)]
Calculating Work
F
d
m
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FINALLY, we can find the TOTAL WORK, Wt, done on
the box…. It is the positive work done by F plus the
negative work done by friction Ff… Wt = W - Wf
QUICK OVERVIEW ON WORK….
1) Work equals Force x displacement… W = (F)(d)
2) Work is measured in Nm called Joules, or J
3) Work is positive if the force, F, is in the same direction as the
displacement
4) If ALL of the force is not in the same direction as the
displacement, find the component of F that is in the same
direction as d… W = (F)(cosϴ)(d)
5) Any force that pushes on the object, but does not move the
object in the direction it is pushing… does NO work
6) Any force that pushes in the opposite direction that the object
is moving, (especially friction), does negative work
POWER
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POWER… What is it? There seems to be political power,
military power, personal power, and automobile engine
power to mention a few….
Physics will concern itself with mechanical power… So
what is that?.... In words it means “how fast you do
work”….. So the formula for power is P = W/t
Let’s see if this makes sense…. W, work is in the
numerator, so if you do more work in the same amount of
time, it takes more power… t, time is in the denominator, so
if you do the same amount of work in less time, it takes
more power… finally, if you do more work in less time, you
need a lot more power!
Units of POWER
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UNITS: If P = W/t… This means that units of power will be
Joules/second or J/s… We call a J/s a Watt or W
Another unit of power is horsepower, hp…. Since a Watt,
W, is in metric units…. To convert to horsepower…. Use
1.00 hp = 755W
We have seen in the previous notes how to find work… W =
(F)(cosϴ)(d)
But how will we find time, t ?... The most common way will
be to use kinematics and the kinematic equations: (vi, vf,
d, a, t)’s…. So keep those equations handy
Finally… If we go back to the other uses of power like
political or military power…. They all seem to mean that if
you can get a lot done, in a short amount of time…. This
takes POWER.
Work Done by a Spring
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Work done by a spring is different… WHY?...
Because the more you displace it, the MORE force it
takes to do it… SO… you can not just multiply F x d,
because F gets bigger as you stretch it…. F IS
CHANGING!... OK… so how do you handle this?
Work Done by a Spring
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Let’s take a spring that is not stretched
or compressed, we will call its length,
Lo.
So Lo is how long the spring is when it
is not stretched or compressed.
Look at the diagram…. x is called the
deformation, it is how far you stretch or
compress the spring from its original
length, measured in meters, m.
Work Done by a Spring
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If we are going to write a formula for the work done
by a spring, or the potential energy stored in a
spring, we have to know HOW STRONG IS THE
SPPRING?
How do you measure this? In words it is “how
much force does it take to stretch or compress the
spring”…. The scientist Hooke gave us a formula:
F = kx
Simply, if we rearrange this formula: k = F/x….
What does this mean in words?....
The strength of a spring, k, is measured by HOW
MANY NEWTONS OF FORCE, F, it takes to
stretch (or compress) a spring, divided by the
AMOUNT YOU STRETCH OR COMPRESS IT, x
(the deformation)…
Work Done by a Spring
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k, the strength of a spring is measured in units of N/m….
An example in words is:
k = 150N/m… It takes 150
Newtons of force to stretch (or compress) this spring a
deformation of 1.00 meter…. THAT is how strong the spring
is….
So if we had a spring, how could we measure how strong it
was… its k?...
Let us hang it down from a post… measure how long it is…
this is Lo, its original length without any force on it…
NOW let’s add a force, F… If it is hanging down the force
will be equal to the weight of the mass we hang on the
spring, mg… What is the deformation, x, or the amount it
stretches?.... This is x, measured in meters… Put a larger
mass on the spring, and again measure, x, the
deformation…. Keep doing this and graph it….
Work Done by a Spring
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You will get a graph like this one at the left…
the force, F, will be equal to the weight, mg,
of the mass hanging on the spring.
The deformation (stretch) can be measured
in meters, m.
The Hooke’s Law constant, k, will be equal
to the slope of the line…
k = ΔF/Δx…. HOW STRONG is the spring in
N/m…
The Elastic Limit is the point when you
stretch a spring, like the one in a ballpoint
pen past the point where it can return to its
original length….
The Breaking Point is where the spring or
rubber band breaks….
Work Done by a Spring
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SO…. How much WORK is done on a
spring?... Or How much ENERGY is
stored in a spring?... They are the
same!... The area under the triangle,
AREA = (1/2)(F)(x)….
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WORK on a spring, OR ENERGY
stored in a spring….
Wspring = EPE (Elastic Potential Energy) = (1/2)(k)(x2)
CONSERVATION OF ENERGY
The Law of Conservation of Energy: ENERGY CAN NOT BE
CREATED OR ELIMINATED…IT CAN ONLY BE CHANGED FROM
ONE FORM TO ANOTHER…
This is one of the most important concepts put forward by scientists of
the last 250 years… It has been used in physics, chemistry, biology,
environmental science, space technology, cosmology, atomic
particle theory and ALL levels of science… BUT what does it really
mean?
The total energy of an object at point a…
+ the work done on it by a (+force)…
- the work done on it by a (-force)…
will equal the total energy of an object at point b…. or
point c…. or point d…. (etc…)
CONSERVATION OF ENERGY FORMULA
ΣEA + WF – Wf = ΣEB = ΣEC = ΣED = etc.
CONSERVATION OF ENERGY FORMULA
ΣEA + WF – Wf = ΣEB
How do you solve this equation?
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ΣEA= This is the TOTAL Energy at Point A….
1) Is there any GPE, gravity potential energy, mgh
2) Is there any EPE, elastic potential energy, (1/2)(k)(x2)
3) Is there any KE, kinetic energy,
(1/2)(m)(v2)
4) ADD UP ALL THE ENERGY AT POINT A….
CONSERVATION OF ENERGY FORMULA
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If an outside force is applied that moves the object in the
same direction that the object moves… ADD WFA
If an outside force like friction acts on an object in the
opposite direction that it is moving… SUBTRACT Wf
**ATTENTION** DO NOT ADD OR SUBTRACT THE
WORK DONE BY GRAVITY OR A SPRING IN THIS
FORMULA!... BUT WHY NOT?... BECAUSE WE
ALREADY DO THAT BY MEASURING THE GPE & EPE
AT THE START AND AT THE END OF THE MOTION
CONSERVATION OF ENERGY FORMULA
ΣEA + WF – Wf = ΣEB
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If an outside force is applied that moves the object in the
same direction that the object moves… ADD WFA
If an outside force like friction acts on an object in the
opposite direction that it is moving… SUBTRACT Wf
**ATTENTION** DO NOT ADD OR SUBTRACT THE
WORK DONE BY GRAVITY OR A SPRING IN THIS
FORMULA!... BUT WHY NOT?... BECAUSE WE
ALREADY DO THAT BY MEASURING THE GPE & EPE
AT THE START AND AT THE END OF THE MOTION
CONSERVATION OF ENERGY FORMULA
ΣEA + WF – Wf = ΣEB
ΣEB= This is the TOTAL Energy at Point B
So let us check:
1) Is there any GPE, gravity potential energy, mgh
2) Is there any EPE, elastic potential energy, (1/2)(k)(x2)
3) Is there any KE, kinetic energy,
(1/2)(m)(v2)
4) ADD UP ALL THE ENERGY AT POINT B….
Now, solve for any of the variables you don’t know……