MECHANICAL ADVANTAGE OF SIMPLE MACHINES
Mechanical advantage is a term that is used to describe the amount of force that is utilized
internally by some sort of mechanical device. The mechanical advantage allows the device to
perform the task for which it was designed. Many common tools that are used in the home and in
construction make use of this principle.
One of the best ways to understand the idea of mechanical advantage is to consider the simple
action that takes place between a screwdriver and a screw. Force is exerted on the screwdriver,
causing the body of the tool to rotate while at the same time pressing the screw into some sort of
surface, such as a wooden block. The combination of rotational force and forward movement
make it possible for the screwdriver to use mechanical advantage to secure the screw into the
medium.
Lever
The lever is one of the six simple machines. It is made of a board or rigid object that pivots
around a fulcrum. The lever works by transferring an applied force over a distance and exerting
an output force on an object. The lever increases the magnitude of the output force by sacrificing
the distance this force is applied over. The effectiveness of the lever can be shown by calculating
the mechanical advantage (MA) for the lever.
The mechanical advantage of a lever is the ratio of the length of the lever on the applied
force side of the fulcrum to the length of the lever on the resistance force side of the
fulcrum. There are three types of levers - class 1, class 2, and class 3.
How to calculate the Mechanical Advantage in a Lever
1. Identify the fulcrum. This will be the point about which the the board or
rigid object pivots around.
2. Identify the locations of the input and output forces. The input force is the
force that is applied to the lever, oftentimes by a machine or human. The
output force is the force that the lever exerts onto an object.
3. Find the distance between the fulcrum and the input force. This is known
as the resistance arm.
4. Find the distance from the fulcrum to the output force. This is known as
the effort arm.
5. Divide the length of the effort arm by the length of the resistance arm to
calculate mechanical advantage.
MA = L (effort arm) / L(resistance arm)
Only class 1 or class 2 levers can be used to gain a mechanical advantage.
First Class Lever
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Second Class Lever
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Third Class Lever
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Wheel and Axle
The wheel and axle is a simple machine involving two connected circular objects of different
sizes. When the axle turns, the wheel also turns. The mechanical advantage is the relationship
between the force and distance inputted and outputted by the wheel and axle and is directly
related to the radius of the wheel and the axle. The mechanical advantage of a wheel and axle
is the ratio of the radius of the wheel to the radius of the axle. In the wheel and axle
illustrated below, the radius of the wheel is six times larger than the radius of the axle. Therefore,
the mechanical advantage is 6:1 or 6.
Note: The radius is equal to 1/2 the diameter of a circle.
The wheel and axle can also be used to increase speed. This is done by applying the input force
to the axle rather than a wheel.
The increase in the output speed will be directly proportional to the ratio of the diameter of the
wheel and axle. For example, if the diameter of the wheel is 10 inches and the diameter of the
axle is 2 inches, the output speed will be increased 5 times (10:2 or 5:1).
How to calculate the Mechanical Advantage of a Wheel and Axle
1. Find the radius of the larger wheel. In a real-world setting, this value can
be obtained by directly measuring with a tape measure or ruler. In a math
problem, this value is sometimes given.
If it is not, it can be calculated with the formula C=2 (pi) (r) by knowing
the circumference (C) of the wheel and solving for r.
2. Find the radius of the axle. This can be achieved by using the same
methods as finding the larger radius.
3. Divide the radius of the large wheel by the radius of the axle to find the
mechanical advantage.
MA =r (large wheel )/ r (axle)
Wheel and Axle
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Inclined Plane
An inclined plane is a type of simple machine that consists of a slope between two points, for
example a ramp. Since work equals force times distance, changing the distance over which a
force is applied creates a proportional change in the force while the amount of work remains
constant (ignoring friction). The inclined plane allows a smaller force to be applied over a greater
horizontal distance to produce the same change in vertical distance as without the inclined plane.
This change in force and distance is known as the mechanical advantage. In an inclined plane,
the formula for mechanical advantage is the length of the slope divided by the height of the
inclined plane (mechanical advantage = slope length/height).
Inclined Plane
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Pulleys
A pulley, which is derived from the Greek word polos, meaning axis, is a wheel with a groove. A
rope, belt, or cable runs inside the groove. That mechanism can be used alone or connected with
other pulleys in a pulley system. The greater the number of pulleys in the system, the less force it
will take to lift an object. Try to lift a 50-pound boulder with just your arms and then use a pulley
to pick it up. The pulley makes lifting the boulder easier because it reduces the effort required to
lift. But notice that, although lifting becomes easier, you pull a rope that travels a greater
distance than the height to which you lift the boulder. This extra distance decreases your effort
by giving you a mechanical advantage. The mechanical advantage of a moveable pulley is
equal to the number of ropes that support the moveable pulley. (When calculating the
mechanical advantage of a moveable pulley, count each end of the rope as a separate rope).
You can calculate this number for any pulley system using the following methods.
How to calculate the Mechanical Advantage in a Pulley
By Calculation
1. Find the mechanical advantage of a pulley system by noting how many
doubled-up stretches of rope or line must shorten for the load to be lifted.
Denote it with the letter n. For example, if the line passes between two
blocks of pulleys four times then n=4. Proceed to Step 3.
2. Find the mechanical advantage of a leverage system by noting how far the
load's point of contact on the lever is from the fulcrum. Denote it L. Note
how far the force input's point of contact on the lever is from the fulcrum.
Denote it F.
Calculate n = F/L.
3. Write n as a ratio of integers. For example, if n=1.5, then write 3:2,
because 3/2 is equal to 1.5.
This is the mechanical advantage of the system.
Empirically
1. Measure the weight of the load or the force of friction exerted by the load
by lifting or pulling it with a spring scale, but only enough to just make it
move.
2. Attach the pulley or lever system to the load.
3. Attach a measuring device at the point of force input. For example, use a
floor scale if the force is pushing and a spring scale if the force is pulling.
4. Divide the load's force by the input force. This ratio is the mechanical
advantage.
Single Pulley
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Multiple Pulleys
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Wedge
A wedge is characterized by an object that has a defined width at one side that slopes to a point
at the other end. This simple machines allows a force that is applied over a large area to be
concentrated upon an edge or smaller area, such as a knife. This concentration of force is the
mechanical advantage (MA) the wedge provides. Each of the six simple machines offers a
mechanical advantage, and it can be quickly calculated for a wedge. The mechanical advantage
of a wedge can be found by dividing the length of either slope (S) by the thickness (T) of the
big end.
How to calculate the Mechanical Advantage of a Wedge
1. Find the length of the sloped surface of the wedge. For a real-world object,
this can be found in by measuring with a tape measure or ruler. In the case
of a math problem, this value is sometimes given.
If it is not, it can be calculated using the Pythagorean Theorem a2 + b2 = c2
or
the law of cosines cos (a ) / A = cos (b) / B = cos (c) / C
2. Find the width of the large end of the wedge. This too can be found either
by direct measurement or by mathematical calculation.
3. Divide the slope length by the width of the wedge to find the mechanical
advantage.
MA = Slope Length / Width
Wedge
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Screw
A screw is a simple machine that works as a modified incline plane. As the screw is turned, the
screw enters deeper into the substrate. Once inside the substrate, the frictional force of the thread
is intended to prevent the screw from rotating back out of the substrate. The thread of the screw
may be viewed as an inclined plane wrapped around the shaft of the screw. The slope of the
screw is the distance for one complete rotation around the screw while the height of the inclined
plane is the distance between the threads, known as pitch. The relationship between the pitch and
circumference of the screw gives the mechanical advantage. The mechanical advantage of a
screw can be found by dividing the circumference of the screw by the pitch (lead) of the
screw.
How to calculate the Mechanical Advantage of a Screw
1. Measure the pitch of the screw. The pitch of the screw is the distance
between the threads and is determined by measuring the number of threads
per inch (or centimeter) on the screw then dividing one by the number of
threads (pitch = 1 / number of threads per inch or cm). For example, if a
screw has eight threads per inch, the pitch is 1/8.
2. Measure the circumference of the screw. Circumference is calculated by
measuring the diameter of the screw and multiplying by pi (circumference
= diameter of the screw x pi). For example, if a screw has a diameter of
0.25 inches, then the circumference of the screw is 0.79 inches (0.25
inches x 3.14 = 0.79 inches).
3. Calculate the mechanical advantage of the screw by dividing the
circumference of the screw by the pitch of the screw. Using the previous
examples, a screw with a pitch of 1/8 and a circumference of 0.79 inches
would produce a mechanical advantage of 6.3 (0.79 inches/ 0.125 = 6.3).
Screw
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