“‘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.)