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The Mechanical Advantages of Simple Machines

Lesson Plan

The Mechanical Advantages of Simple Machines

Objectives

Students will:
  • examine how simple machines are used.

  • measure the mechanical advantages of a Class 1 lever.

Essential Questions

  • How does the mechanical advantage offered by simple machines help us do work?

Vocabulary

  • Machine: A device that allows you to do work by transmitting or modifying force or motion.

  • Mechanical Advantage: The ratio of the output force produced by a machine to the applied input force or the factor by which a mechanism multiplies the force applied to it.

  • Work: Force times distance moved. W = F x D.

  • Inclined Plane: A simple machine involving a flat surface with one end higher than the other.

  • Lever: A simple machine consisting of a bar that rotates on a fixed point (its fulcrum, or pivot point) so that force applied to one point on the bar will exert force against a load on another point on the bar.

  • Wheel-and-Axle: A simple machine consisting of a modified lever that rotates in a circle around a center point or fulcrum.

  • Pulley: A simple machine utilizing a wheel or set of wheels with grooved edges over which a rope or chain can be drawn in order to change the direction of a pulling force and increase the capacity for lifting weight.

Duration

45 minutes/2 class periods

Prerequisite Skills

Prerequisite Skills haven't been entered into the lesson plan.

Materials

  • meter sticks, either metal or wood

  • small postage scales/balances (you can also use rubber bands with double the number of rulers)

  • various objects weighing less than 1 kg

  • calculators (optional)

  • copies of Mechanical Advantage Worksheet (S-8-7-3_Mechanical Advantage Worksheet and KEY.doc)

Related Unit and Lesson Plans

Related Materials & Resources

The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.

  • Lives by Plutarch, written about 96 A.D.; available in various translations and titles. The section on the Roman general Marcus Claudius Marcellus covers the 214–212 B.C. siege of Syracuse and the impression made by Archimedes’ machines.

  • The Histories by Polybius. Harder to find than Plutarch, book eight of this Greek manuscript, written in 130 B.C., describes Archimedes’ machines in greater detail.

  • The tale of Archimedes using “death rays” in the form of parabolic solar mirrors to destroy Roman ships originates with later classical writers and may be apocryphal.

Formative Assessment

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    During the lesson, monitor students’ understanding of the following points, and adjust their understanding as necessary, to help ensure that they can master the targeted learning goals within the time frame:

    • Work equals force times distance, so we can perform the same work using less force over a greater distance.

    • Mechanical advantage is the difference between the force we need to perform a task with the help of a machine, and doing the same task without the machine.

    • The amount of work involved in a task remains the same, with or without mechanical advantage.

    • By machine we mean (rather than modern engines and motors) the basic, classical mechanical devices of the lever, inclined plane, pulley, and wheel-and-axle.

    • The screw and wedge are basically inclined planes.

    • A wheel-and-axle is not the same as a wheel, which is not a simple machine but a device that reduces friction.

    • There are three classes of levers and this lesson concerns Class 1, where, like a see-saw, the fulcrum or pivot point is between the force and the weight.

    • The mechanical advantage of a Class 1 lever is the same as the ratio between the length of the lever on the side of the fulcrum with the weight, to the length of the lever on the other side of the fulcrum where the force is applied.

Suggested Instructional Supports

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    Active Engagement, Explicit Instruction
    W:

    Students will get hands-on experience on the topic of mechanical advantage by gauging the effect of using a Class 1 lever.

    H:

    The students will be presented with a genuine historical anecdote involving a man (Archimedes) changing history purely through his understanding of mechanical advantage.

    E:

    Students will be provided with the experience of experimenting with a Class 1 lever and establishing its mechanical advantage.

    R:

    Students will be asked to reflect on the many classical machines they use every day, since they are built into common tools, such as scissors and can openers.

    E:

    Students will be able to express their understanding by gathering actual mechanical advantage data on a Class 1 lever, and through class discussion.

    T:

    The lesson plan is flexible and relates to different learning styles, as it involves hands-on material, observation, listening, manual activity, and math practice. It also involves a dramatic historical anecdote.

    O:

    This lesson is designed to demonstrate the mechanical advantage of classical machines and confront students with the widespread nature of their use in daily life. Their learning will be reinforced by doing a hands-on but simple experiment where they will establish the mechanical advantage of a Class 1 lever.

Instructional Procedures

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    “‘Give me a place to stand, and I will move the Earth.’ Those were the words of ancient scientist Archimedes (pronounced Ark-a-me-dees) after studying the properties of levers. What he meant is that he could calculate the force needed to move any weight, given a lever long enough and a suitable fulcrum.

    Coming from him, those words have extra meaning. He lived from about 287 to 212 B.C. in Syracuse, a port city in Sicily, an island just off the toe of Italy. Syracuse was then a rich city-state that was part of the Roman Empire, but after Hannibal invaded Italy by marching over the Alps with elephants, Syracuse rebelled and joined Hannibal’s side. After surviving Hannibal, the Romans sent an expedition to retake Syracuse in 214 B.C. Their navy charged into the port—and huge machines reached out over the walls, picked up the ships off-center and capsized them. Other Roman ships were hit by an accurate stream of heavy rocks. Attacking on land, the Roman infantry was routed by an devastatingly accurate stream of projectiles. Demoralized, they would panic at the site of a rope or board being held out over the wall.

    For the moment, the machines had won.

    Let’s define our terms. In conversation we use the word machine for any device that performs a task using mechanical or electrical power. It may include a component that converts fuel or electricity into mechanical force, such as wheels being turned by an engine or motor.

    That’s not what Archimedes had. In this lesson we are going to use the word machine in the classical sense that Archimedes would have used: a device that lets you change the magnitude (or, in some cases the direction) of an applied force. The devices that Archimedes built probably resembled modern construction cranes, but the input power, magnified by simple machines, was all supplied by muscles.

    But you don’t have to be defending a city against the Romans to make full use of simple machines. Any time you do anything more complicated than picking up something with your bare hands, you are probably relying on one or more (or an assembly of many more) simple machines.

    And the word simple is used on purpose. By some analyses there are only four simple machines:

    • Lever: A simple example is a claw hammer used to loosen nails. Pulling the handle several inches with moderate force moves the claw about an inch to pull a nail out of a piece of wood.

    • Inclined plane: A good example is the loading ramp that drops from the back of a furniture truck. Sliding things up the ramp into the back of the truck involves less force than lifting them straight up into the back of the truck. (A wedge is two inclined planes stuck together, and a screw is an inclined plane wrapped around a shaft.)

    • Wheel-and-axle: A good example is a doorknob or a water faucet handle. By turning the large outer wheel, you magnify the turning force being applied to the smaller central axle or hub shaft. (Some insist that this is really just another example of the lever.)

    • Pulley: A good example is the hoisting tackle used by construction cranes to lift objects. Each set of grooved disks that the rope is threaded through increases the distance the rope has to be pulled to lift the object, and decreases the necessary force by the same amount.”

    Lead the class in a discussion of simple machines found in everyday life, or even around the classroom. Scissors are two levers bolted together. Their cutting edges are a wedge, which is two inclined planes. A mechanical can opener includes a lever, a wedge, and a wheel-and-axle. Bike pedals amount to levers attached to a wheel and axle. A water faucet is a wheel-and-axle that turns (inside the valve) a screw, which is an inclined plane wrapped around a shaft.

    The reason we use simple machines is important because they let you perform the same amount of work with less force.

    Remember, work is force times distance, as in:

    W = F × D.”

    Write the equation on the blackboard and point out that it implies that you can decrease the force involved in a task by increasing the distance over which it takes place.

    This is especially important in situations where you just don’t have the force to do the basic task. You may not be strong enough to lift a refrigerator directly into a truck. But you are probably strong enough to slide it up a ramp into that truck.

    But keep in mind that W does not change. Getting the refrigerator into the truck takes the same amount of work whether you lift it directly or slide it up a ramp. (In fact, thanks to friction, using the ramp may slightly increase the total work involved.)

    The difference in the value of F between using a machine and not using one is the mechanical advantage of the machine.

    All simple machines offer mechanical advantage, which is why we use them. With levers, the mechanical advantage comes from the ratio between the length of the lever on either side of the pivot. The longer the side on which the force is being exerted over the side doing the work of lifting or moving, the greater the mechanical advantage.

    There are actually three kinds of levers:

    • Class 1, with the fulcrum between the Force Side and the Weight Side. Examples are a crowbar and a see-saw, although the latter is not designed for mechanical advantage.

    • Class 2, with the weight between the fulcrum and the force. An example is a wheelbarrow when the user lifts the handles. The wheel is the fulcrum, the contents are the weight, and the operator is the force.

    • Class 3, with the force between the fulcrum and the weight. Your forearm works this way, with the elbow as the fulcrum, the muscles in the middle of your arm as the force, and whatever your hand is holding as the weight.

    Today we are going to experiment with a Class 1 lever.”

    Divide up the class into teams. Each team should fill out the Mechanical Advantage Worksheet (S-8-7-3_Mechanical Advantage Worksheet and KEY.doc), following its instructions, gauging the force needed to lift the weight four different times with the pivot point at a different position each time.

    Caution students on these points:

    • The object should be balanced on the ruler, clear of the table.

    • The weight side measurement of the lever should be from the pivot point (fulcrum) to the nearest edge of the object, not to the opposite end of the ruler.

    • The force should be applied to the end of the ruler away from the table, straight down.

    • The force should be applied gradually until the object is lifted just clear of the table.

    • The worksheet specifies what numbers to divide by what other numbers. The calculator should be used for this purpose.

    Check each team to make sure students are getting useable results.

    After each team is finished, reconvene the class and go over the results, making these points:

    • The ratio between the two sides of the lever should be approximately equal to the mechanical advantage that was actually achieved for each trial.

    • Possible sources of error include the difficulty of reading the postage meters upside down, the crude nature of the scales, inconsistent positioning between trials, and the weight of the ruler.

    • Getting a mechanical advantage of 8 should have been possible, but should be more than 1 as long as the weight side was shorter than the force side.

    • The amount for work needed to lift the object is the same each time. Work is force times distance, and the force side of the lever was moved proportionally more than the weight side.

    Using insights like these, Archimedes knew exactly how much force he needed for the machines he used to defend Syracuse. He held the Romans off for more than two years, until the Romans got through with a surprise night attack on a weak point in the walls. A Roman soldier arrived at Archimedes’ house to summon him to the Roman commander. ‘Do not disturb my circles,’ Archimedes supposedly replied—presumably he was involved in a calculation, as he would have used geometry to solve problems that we now solve with trigonometry. Annoyed, the solder killed him instead.”

    Extension:

    • Taking Archimedes at his word, we will provide him with a lever that could move the Earth. Putting his weight into it, he might briefly exert a force of 60 kilograms. The Earth weighs about 6×1024 kilograms. If the weight side of the lever (the side between the fulcrum and the Earth) is 1 meter long, how long would the force side have to be? (Since he is exerting 6×101 kilograms of force, the lever needs to give him a mechanical advantage of 1023, so the force side would have to be 1023 times longer than the weight side. That would make it 1023 meters long. Since a light year is about 1016 meters, if would be about 10 million light years long.)

    • Identify some of the practical problems involved with using a lever to move the Earth. (The fulcrum would have to be immobile, or many times heavier than the Earth, and the lever would be impossibly long and massive and strong, and the force side would have to travel 1023 times farther than the weight side. But Archimedes’ point is that the leverage could be calculated, as we just did.)

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DRAFT 11/19/2010
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