Building the Ancient Pyramids

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Building the Ancient Pyramids
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• Located on the Giza Plateau, the Great Pyramid
  (or Pyramid of Khufu) is truly marvelous
• How did the ancient Egyptians build it?

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Some Stats

• Constructed from approximately 2580-2560 BCE
• The worlds tallest structure (480 ft, now 460ft)
  for over 3000 years
• The pyramid contains 2 million blocks, each
  weighing 1.5 tons
• Some materials were transported over 500 miles to
  the construction site
• A truly massive undertaking (no pun intended)

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• Lets focus on how the blocks were transported to
  their actual locations on the pyramid
• The pyramid has a height of 480 ft
• How did they transport massive stones to such
  heights?

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Masters of Simple Machines

• The Egyptians were smart enough to realize the
  practical value of simple machines
• A machine is a device which can multiply or
  change the directions of forces
• The primary machine used by Egyptians was the
  ramp, or inclined plane

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• What does the ramp do for us (think back to the
  lab)
• It reduces the amount of necessary force to
  transport an object from one location to the other

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Egyptian Ramps
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Some Numbers

• The force necessary to lift a 1.5 ton block
  straight up in the air is about 14,000N
• The force required to push it up a ramp inclined
  at 15 is 3500N
• The ramp reduces the amount of force necessary to
  move the block

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Mechanical Advantage

• Without the machine, wed need 14000N to move the
  block
• With the machine, we need 3500N
• The ratio of these two forces is called
  Mechanical Advantage (MA)
• In this case, our MA is 14000N/3500N, which
  equals 4

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Which Force Goes Where?

• MA Output Force/Input Force
• Input Force is the force we apply, using our
  machine
• Output Force is the force wed need without the
  machine (or the effective force we get with the
  machine)

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Other Simple Machines

• Pulleys
• Levers
• Screws

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Pulley Demo

• What is the Mechanical Advantage of this pulley
  system?

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A Choice of MA

• You have the choice of using four simple machines
  to achieve a task. Machine 1 has a MA of 1.2
  Machine 2, 0.5 Machine 3, 4.5 Machine 4, 1
• Which would you choose?
• Why?

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• Which machine would do more harm than good?

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The Catch

• Using the ramps, the Egyptians needed less force
  to push the blocks up the ramp
• What was the tradeoff?
• They had to move the stones further than they
  would have by lifting them straight up

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Think Back to the Lab

• What did you notice about the values in the far
  right column in your data table?
• They should have been constant
• What does Force x distance represent?
• Work

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Work

• Were all familiar with work
• Love it, hate it, we all have to do it
• Scientific work however, is a bit different
• Work Force x Displacement

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Units?

• Theres a better way to talk about work than
  Newtons x meters
• To better deal with work, we define a new unit,
  called a Joule (J)

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• Find the work done by a 10N force that moves an
  object 30m
• 300J

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Direction Matters

• Imagine two situations
• In situation 1, you pick a book up off the ground
  and lift it up 1m
• In situation 2, you walk for 5 miles with the
  book already in your hand
• In which situation do you do more work on the
  book?

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• One can only do work on an object if the force is
  in the direction of motion, or against it
• The entire force does not have to be in that
  direction a piece of a force can also do work on
  an object

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A Better Definition

• Work Force x Distance x cos q
• W Fd cos q
• Note F and d represent magnitudes, not
  vectors
• q is the angle between the force vector and your
  direction of motion

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• In an airport, a traveler pulls her luggage
  across the ground with a 20N force, which is at
  an angle of 35 with the horizontal
• If she moves the suitcase 10m, calculate the work
  done by her pull
• What is the work done by friction?

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Machine Efficiency

• In an ideal machine, the work you do with the
  machine the work you would do without the
  machine
• In reality, these values are never the same

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Why?

• Think about an Egyptian worker pushing a stone up
  the ramp
• Which other forces does the worker experience?
• Friction
• Hes got to push a little harder than he would
  without friction, which results in more work

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• The ratio of the work we get out of the machine
  to the work we put into the machine is called
  efficiency
• In an ideal machine, it is always 1
• In reality, it is always less than 1

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• Efficiency Output work/Input work
• If we multiply this number by 100, we can
  represent efficiency by a

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Input vs. Output

• Using 40N of force, I push a 80kg object 5m up a
  ramp. Assuming the object starts on the ground
  and ends 2m high

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Give the input and out work, respectively

• 1. 80J, 40J
• 2. 160J, 200J
• 3. 200J, 160J
• 4. 40J, 80J

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• In a rope/pulley system, a 4kg mass is raised
  vertically 0.2m. To accomplish this, a 25N
  horizontal force is applied over a distance of
  0.4m
• What is this pulleys efficiency?
• 78-80

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and - Work

• So what happens when your angle 180, and work
  is negative?
• Work is a scalar, so the and do not represent
  direction
• They give us some indication of the energy gain
  or loss of a system

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What is Energy?

• Scientifically, energy gives us the ability to do
  work
• Like work, it has units of Joules

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• Suppose I do 40J of work on an object what is
  this objects change in energy?
• (use a sign if it loses energy)
• Bonus What if it does 40J of work on me?

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Work/Energy Theorem

• A fancy way of saying what we already know
  doing work on an object gives it energy and
  doing work on an object takes energy away

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Energy Types

• Kinetic
• Potential
• Radiant
• Like work, energy is a scalar quantity

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Kinetic Energy

• Kinetic Energy (KE) is energy associated with
  motion

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How Much KE?

• We calculate an objects kinetic energy by the
  equation
• KE ½ (mass) x (speed)2
• The 2 may not seem important to you, but it
  explains everything from why baseball players
  cork bats to why comets are more dangerous than
  asteroids

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An elephant (1) and mouse (2) both have the same
amount of Kinetic Energy. Which is moving faster?
Do they have the same amount of momentum?
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Assume the elephant from the previous problem
(m1500kg) is running at 10m/s. How much kinetic
energy does this elephant have while running?
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Two objects (A and B) have the same mass, but
object A is moving 3 times as fast as object B.
What is the ratio KEA/ KEB?
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Corking Bats and Dangerous Asteroids

• The 2 may not seem important to you, but it
  explains everything from why baseball players
  cork bats to why comets are more dangerous than
  asteroids

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Potential Energy

• Potential is stored energy that has the
  potential to do work
• A stretched rubber band, a compressed spring,
  TNT, a nucleus, and a rock on the edge of a cliff
  all have potential energy

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Gravitational Potential Energy

• Gravitational Potential energy is the potential
  energy of an object due to its position
• GPE weight x height mgh

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• So
• The higher an object relative to the ground, the
  more gravitational potential energy it has

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Imagine 3 objects, A, B, and C. Object A is
located 2m above the ground object two has ½ the
mass of A and is also 2m off the ground Object C
has ¼ the mass of A and is located 10m off the
ground. Rank the GPE of these objects in
increasing order

• 1. A, B, C 2. B, C, A
• 3. B, A, C 4. C, A, B
• 5. A, C, B

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An astronaut in full space gear climbs a vertical
ladder on Earth (1). Later, she makes the same
climb on the moon (2). In which location does
her GPE change less?
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A 10kg object is lifted from the floor to a
shelf, 2m off the ground. What is the change in
this objects GPE? Bonus How much work did it
take to get the object up to the ledge?
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Elastic Potential Energy

• The energy stored in a stretched spring
• Depends upon the amount the spring is
  stretched/compressed, the type of spring

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Chemical Potential Energy

• A combustion reaction releases a large amount of
  energy
• What is this energys origin?
• From the chemical bonds between atoms

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Nuclear Potential Energy

• Using a nuclear device, we can take an object the
  size of a soccer ball and destroy a city
• The energy released comes from the nuclei of the
  constituent atoms

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Electrical Potential Energy

• Energy dependent upon the location of charged
  particles
• We can use this energy to move charges
  (electricity)

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Energy Conservation

• One of the pillars of science is the law of
  energy conservation
• You can neither make nor destroy energy, you can
  only change it from one form to another

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• Basically, in any situation you always start and
  end with the same amount of energy
• That energy can be rearranged in different ways

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Pendulums
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• Energy Skate Park
• http//phet.colorado.edu/new/simulations/sims.php?
  simEnergy_Skate_Park
• Mass/Spring Applet
• http//phet.colorado.edu/new/simulations/sims.php?
  simMasses_and_Springs

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Energy Conservation Clips

• http//www.youtube.com/watch?vtbhT6KbHvZ8
• http//www.youtube.com/watch?vyVE81BHBD0E

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Suppose I throw a 5kg ball up in the air.
Assuming it has 200J of kinetic energy when it
leaves my hand, how high will it rise?
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