Mechanical Advantage
Introduction
This module is derived from activities that I created and used with students between their
freshman and sophomore years in high school. The concept was to expose students to the
engineering concept of Mechanical Advantage, coupled with the math concepts necessary
to analyze it. I believe this is easily adaptable to lower grades because the mechanical
principles are ones that even elementary school students have used, and the basic math
can be reduced to a very simple level.
My experience from applying this material in the classroom is that students exhibited a
huge gain in comprehension of the mechanical concepts (which was to be expected, since
they were new). But they also showed significant gains in their ability to apply ratios and
solve simple algebraic relationships. I believe the math gain was the result of tying the
math concepts to relevant real world applications for everyday life.
Addresses the following Indiana Standards
Number Sense
Computation
Algebra
Problem Solving
Sixth Grade
6.1.6, 6.1.7
6.2.1, 6.2.2, 6.2.5, 6.2.6, 6.2.7
6.3.1, 6.3.5, 6.3.8
6.7.1, 6.7.3, 6.7.4, 6.7.11
Eighth Grade
8.2.1, 8.2.4
8.3.1, 8.3.2
8.7.1, 8.7.3, 8.7.4, 8.7.12
Lesson Materials
Mechanical Advantage occurs any time we use a tool or mechanism to increase the effect
of the effort that we apply. We encounter examples of this every day.
For effect, I usually begin this discussion by bringing in a mechanical floor jack. First I
ask the strongest student in the class if they think they can lift a heavy desk. Then I roll
the jack under the desk and ask the smallest girl in the class to raise the desk. Using the
mechanical advantage supplied by the jack mechanism, she can do this easily.
© 2005
P. Hylton & W. Otoupal
I follow this with a discussion of simple tools that students have all used, which amplify
their input force by mechanical advantage. A good first example is a pair of pliers.
When we squeeze the handles of the pliers, the force that the pliers apply to whatever
they grip is greater than the force we apply. In simplest terms, the mechanical advantage
is the force applied by the pliers divided by the force we apply to the pliers.
Mechanical Advantage =
Force Applied by the Mechanism_
Force We Apply to the Mechanism
In other words, if we apply 5 pounds of force to the handles, and the pliers applied 25
pounds of force to the item that they are gripping, then the mechanical advantage is the
ratio 25/5 which is equal to a mechanical advantage of 5, which is sometimes expressed
as “5 to 1” and written 5:1.
The ratio of the forces is inversely proportional to the distances from the hinge, or pivot
point. In the case of our pliers, a simplified diagram would look like this:
5 inches
1 inch
Item being gripped
(force applied by
pliers is 25 pounds)
Pivot Point
Applied Force on each
Handle = 5 pounds
Mechanical Advantage = Gripping Force = Length of Handle
Handle Force
Length of Grip
= 25 pounds = 5 inches
5 pounds
1 inch
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= 5
P. Hylton & W. Otoupal
or
5:1
This example can be applied to numerous simple tools such as wire cutters, hedge
clippers, scissors, etc. Let’s consider a different type of pliers, commonly referred to as
channel lock pliers.
How would we determine the force applied to the item we grip with this tool?
10 inches
5 inches
4 inches
Applied Force on each
Handle = 8 pounds
3 inches
The ratio of the forces is still inversely proportional to the distances from the pivot point.
So
Clamping force =
Handle force
Handle length__
Clamping length
But which length do we use for the clamping length, 3 inches, 4 inches, or 5 inches?
Since the length we want is the distance from the pivot point, it is the 5 inch length that
we want. (This is also a good opportunity to use the geometry concept of a 3-4-5 right
triangle.)
Therefore the proportional ratios that define the mechanical advantage of this tool (using
x as the unknown force applied to the item being gripped) are
_x_ = _10_
8
5
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P. Hylton & W. Otoupal
Depending on your preferred approach,
You could either multiply
both sides by 8
You could cross multiply first
OR
x (5) = 8 (10) = 80
_x_ (8) = _10_ (8)
8
5
which reduces to
x = 16 pounds
and then divide by 5
x = _80_ = 16 pounds
5
Either way, we discover that the mechanical advantage provided by this tool is 16:1.
More complicated mechanisms can provide greater mechanical advantages. Vice grip
pliers like the ones shown below can provide advantages of greater that 100:1, but are
harder to analyze.
The next mechanical advantage concept that we want to talk about is leverage. Every
grade school student who ever sat astride a teeter-totter at the playground has
experimented with leverage. If a small child sits on one side and a larger child sits on the
other side, the teeter-totter doesn’t want to rock. The larger child must slide closer to the
pivot point in order to balance the ride, similar to what is shown here.
We can use a lever to lift something heavy, without having to apply as much force as the
weight of the object. The lever has a point of rotation, similar to the pivot in the
previous examples, only this time the pivot point is often called the fulcrum. The inverse
proportionality relationship still exists for the forces and the lengths. So by adjusting the
lever, we can reduce the load required to lift an object, as shown in the following figures,
both of which use a 10 foot lever, but with the fulcrum moved by 1 foot..
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P. Hylton & W. Otoupal
X = Load
Required to Lift
the 100 lb
100 lb
4 ft
Y = Load
Required to Lift
the 100 lb
100 lb
6ft
3 ft
ft
__X__ = 4 ft
100 lb
6 ft
7 ft
__Y__ = 3 ft
100 lb
7 ft
X (6) = 4 (100)
Y (7) = 3 (100)
X = 66.6 lb required to lift 100 lb
Y = 42.8 lb required to lift 100 lb
So by moving the fulcrum only 1 foot out of 10, we can reduce the required load from
66.6 lb to 42.8 lb (a 35% decrease in applied effort!)
Here is a similar, but slightly more difficult problem. My cat has lost his favorite toy
under the heavy dresser at home. I can’t reach it and the dresser is loaded and weighs
400 lb, so it is too heavy for me to move by myself. If I place a lever under the dresser as
shown, how much force must I apply in order to lift the dresser and rescue the missing
catnip mouse?
X = Force
Required to
Lift Dresser
400 lb Dresser
2 ft
5 ft
First, we must think of how much upward force is required to lift the right side of the
dresser. If it is evenly loaded, then it shouldn’t surprise anyone that the left legs are
holding 200 lb and the right legs are holding 200 lb. So to lift the right side of the
dresser, we must lift upward with 200 lb. This is actually a leverage problem unto itself
if we think of the 400 lb as being a downward force at the center of the dresser, as shown
in this simplified sketch of the bottom of the dresser:
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P. Hylton & W. Otoupal
400 lb Weight
Y = Upward Force
Required to Lift the Right
Side
Pivot Point
2 ft
2 ft
Set up the inverse proportionality ratios using the distances to the pivot point again.
__Y__
400 lb
=
2 ft
4 ft
Y = 200 lb, just as we thought it should.
So now we can set up the ratios to determine the load I have to place on the lever in order
to rescue the catnip mouse.
__X__
200 lb
= 2 ft
5 ft
X = (200 lb) (2 ft) = 80 lb to lift the dresser and rescue the toy
5 ft
Let’s move on to another device that gives us mechanical advantage,……the pulley.
Consider the two pulley systems shown below. Assume that the black blob on the right
side of each system weighs 100 lb. How hard would we have to pull on the rope on the
left side of each system in order to raise the 100 lb blob?
The answer is NOT 100 lb for both cases. Consider the pulley system on the left. If the
blob is 100 lb., then it is clearly pulling on the rope with 100 lb, so to raise the blob by
pulling on the opposite end of the rope, we would have to pull with 100 lb. Now look at
the pulley system on the right. How many ropes are pulling up on the smaller red pulley
that is attached to the blob? There are two ropes. Ropes have two interesting
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characteristics. First they can only exist in tension, not compression. (You can’t push a
rope! You can only pull it.) Secondly, if you do pull on a rope, it will have the same
tension anywhere in that rope. In other words, if you tie a rope to the wall and pull with
25 lb of force, there will be exactly 25 lb of force anywhere in the rope.
So if you return to our last figure, there are two ropes that affect the smaller red pulley
and its attached 100 lb. blob. Since ropes can only be in tension, the two pieces of rope
must be pulling up on that small red pulley. And since the tension in a rope must be the
same anywhere, it makes sense that each of them must be carrying 50 lb of weight due to
the 100 lb. blob. That means to lift the 100 lb. blob, we would only have to pull on our
end with 50 lb. of applied force. That’s mechanical advantage, for sure.
This advantage can be increased by adding more pulleys. When you combine pulleys in
this fashion, it is called a block and tackle system
Block and tackle systems have been used for centuries. They were used to raise the
stones that were used to build the Egyptian Pyramids. They were used on old time
sailing ships to raise the sails. Today’s students would generally not have much of a feel
for this. However, several recent popular movies such as Master and Commander and
Pirates of the Caribbean involved old time sailing ships. Ask the students how they
think those huge heavy canvas sails were raised on those old ships with just one sailor on
a rope on each side of the ship. They used old wooden block and tackle systems like this.
And by running more and more loops through the block and tackle, the mechanical
advantage was increased until the weight of the sails became manageable by one man.
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P. Hylton & W. Otoupal
Mechanical
Advantage
1:1
2:1
2:1
3:1
4:1
We have now discussed a significant number of simple systems which utilize mechanical
advantage, and we have shown how to analyze them using simple ratios and basic
algebra. My recommendation is that after these principles have been demonstrated, the
students be presented with the opportunity to examine and analyze such systems. Bring
in selected hand tools that use mechanical advantage.
Acquire some small pulleys from a hardware store, and using string, allow students to
design, build and analyze their own pulley systems (they can be hung from door frames,
chalkboard trays, desks, or any number of handy objects). First have them diagram their
design in a journal or lab book, explaining their thought process. Have the students
predict the mechanical advantage that they believe they should get from their pulley
system. Then they should construct their design. Using simple fish scales (another
hardware store acquisition) the actual applied loads can be measured. It is important to
follow this activity by writing their observations in their journal and trying to explain
anything that turned out differently from their expectations. (Note that the most common
mistake is to forget that the pulleys themselves have weight, and in many designs, that is
a factor that the students ignore.)
Encourage some students to look at more complicated systems, like the floor jack. Have
students keep a journal of their observations as they examine and work with the tools and
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systems. Encourage investigative participation. Hands on activity with the tools and
systems should increase student interest in the activity.
Do not be afraid to have the students analyze systems that are more complicated, even if
you are not sure exactly how to analyze them yourself. Using the simple descriptions that
you have already seen, you and the students can work your way through just about any
mechanical advantage evaluation.
As a final example, here is the logic that would be used on the floor jack
W = Weight to
be Lifted
These are
hinged joints
located at
four places on
the jack
F = Force you can apply to
the jack handle
60 inch long
handle
3 inches
1 inch
First, figure out the advantage gained at the handle end
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P. Hylton & W. Otoupal
X = 60
F
3
X
3
F
So X = F (20)
Pivot Point
Realizing that the force X is the same at the lifting end as at the handle end, figure
out the advantage gained at the other end.
W
W = 3
X
1
X
So W = X (3)
3
Pivot Point
1
Combining these two results, W = X (3) = F (20) (3) = F (60). Which means that the
weight that can be lifted is 60 times the force that you apply. THAT is Mechanical
Advantage. And there is no reason that pre-high school teachers and students cannot
work their way through these simple engineering and math concepts.
For more information, contact Professor Pete Hylton at Indiana University / Purdue
University at Indianapolis at phylton@iupui.edu.
© 2005
P. Hylton & W. Otoupal