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Mechanical Advantage in Machines
by Ron Kurtus (16 October 2008)
The purpose of a machine is to create a mechanical advantage that will facilitate your ability to move an object against resistive forces. Mechanical advantage (MA) means that the output of the machine is greater than the input. MA is the output divided by the input. There are three types of mechanical advantage: force, distance and speed.
Note: Most science books only consider force mechanical advantage, but we will discuss all three, since they are equally important.
The Law of Conservation of Energy requires that in gaining a mechanical advantage, it will cost you in another factor. For example, increasing output force may cost you by requiring an increase in distance traveled.
Mechanical advantage is most obvious in simple machines, although it can be measured in highly complex machines and even some tools.
Questions you may have include:
- What is force mechanical advantage?
- What is distance mechanical advantage?
- What is speed mechanical advantage?
This lesson will answer those questions. There is a mini-quiz near the end of the lesson.
Useful tools: Metric-English Conversion | Scientific Calculator.
Force mechanical advantage
A machine has force mechanical advantage when it requires less force to do work than would be needed without the machine.
A big problem in moving a heavy object is the force required to do the job. Usually, that force is applied against some other resistive force, such as friction or gravity. A machine provides a force mechanical advantage, where the effort force is less than the load force.
The equation for this is:
MAF = FL/FE
where
- MAF is the force mechanical advantage
- FL is the load force
- FE is the effort force
Note: In this notation, MA is NOT M times A. It simply stands for Mechanical Advantage.
Example of lever
If you apply a 2 Newton force to a lever in order to lift a 14 N weight, the force mechanical advantage of the lever would be:
MAF = 14 N /2 N
MAF = 7
In other words, you could lift 7 times the force that you pushed on the lever.
Cost of moving the load
In order to adhere to the Conservation of Energy Law, the input or effort work must be equal to the output or load work. In other words, the cost of moving the load with a smaller force is that you apply your effort force over a greater distance.
It can be shown that the total work is the same:
W = FEdE = FLdL
where
- W is the work (force times distance)
- FE is the effort force
- dE is the distance the effort force moves
- FL is the load force
- dL is the distance the load force moves
Distance mechanical advantage
A machine has distance mechanical advantage when the output distance is greater than the applied distance.
There are times when you want to apply a force a short distance to increase the distance an object moves. One good example is when you peddle a bicycle. The distance you move the peddles on a bicycle are much less than the distance moved on the circumference of the tires.
The bicycle and other machines may provide a distance mechanical advantage. The equation for this is:
MAd = dL/dE
where
- MAd is the distance mechanical advantage
- dL is the distance the load moves or the output distance
- dE is the distance the effort moves or the input distance
Example of lever
Suppose you wanted to lift a box weighing 2 pounds up to a 6-foot shelf. You could use a form of a lever
Lever configuration provides distance mechanical advantage
Example for bicycle
Consider a bicycle with a 27-inch radius wheel and a 9-inch peddle sprocket radius. (To simplify the example, the chain has a 1-to-1 gear ratio.)
In the case of a bicycle, the circumference of the wheel is C = 2πr where
- π is the Greek letter pi or approximately 3.14
- r is the radius of the wheel
Thus, for a bicycle with a 27-inch radius wheel, the circumference or output distance would be dL = 2*π*27.
and a 9-inch peddle sprocket radius, the distance mechanical advantage would be MAd = 2*π*27/2*π*9 = 3. In other words, the bicycle would travel 3 times as far as the peddles would turn.
Cost for moving further
The cost to move ahead further is that the force applied is greater than if you would simply be pushing the object. This means you must expend more force to move a greater distance. Again, the amount of work is equal.
Relationship between distance and force mechanical advantages
Note that the distance mechanical advantage is the reciprocal of the force mechanical advantage. That is:
MAd = 1/MAF
Since MAd = dL/dE and MAF = FL/FE,
you can see that dL/dE = 1/FL/FE = FE/FL.
Thus,
FEdE = FLdL
This also shows that the work expended is the same.
Speed mechanical advantage
Finally, you may want to increase the speed of an object, such as with a catapult or a bicycle. In this type of machine, you want to have a speed mechanical advantage.
With a catapult, you push on one arm of a level and the end of the other arm moves much faster, throwing the object through the air. With a bicycle, you peddle at a certain speed, but the different sizes between the peddle sprocket and wheels and the gearing results in you going at a faster speed.
The equation for this is:
MAs = sL/sE
where
- MAs is the speed mechanical advantage
- sL is the speed of the load
- sE is the speed of the effort
Since distance equals speed times time, or d = st, there is a distinct relationship between speed mechanical advantage and distance mechanical advantage.
MAd = dL/dE = sLt/sEt = sL/sE = MAs
Thus, Speed MA = Distance MA = 1/Force MA.
Example for catapult
A catapult moves a large rock a distance of 5 meters in 1 second before releasing it at a speed of 5 m/s. The effort end of the catapult moved at a speed of 1 m/s. Thus the speed mechanical advantage is MAs = 5/1 = 5.
Since in the effort end moved 1 meter for 5 meters of the load end, the distance mechanical advantage is MAd= 5.
Also, since MAd = 1/MAF, the force mechanical advantage is MAF = 1/5. This means that the force required to catapult the rock at 5 m/s was 5 times the weight of the rock. Since the force mechanical advantage was less than 1, it would be more appropriate to call it a mechanical disadvantage.
Summary
A machine is used to provide you with an advantage in moving an object or doing work. This is called the mechanical advantage of the machine. There are three types of mechanical advantage: force, distance and speed. Most science books only consider force mechanical advantage, but they are equally important.
The force mechanical advantage is MAF = FL/FE. The distance mechanical advantage is MAd = dL/dE. The speed mechanical advantage is MAs = sL/sE. The relationship between them is Speed MA = Distance MA = 1/Force MA.
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Resources and references
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Mini-quiz to check your understanding
If you got all three correct, you are on your way to becoming a Champion in Physics. If you had problems, you had better look over the material again.
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