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Mechanical Advantage

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Mechanical Advantage

Did you ever use a hammer to break open a piggy bank? When you use a hammer to hit something, you apply force to the end of the handle. The head of the hammer, in turn, applies force to the piggy bank or whatever object you are striking. The head of the hammer moves farther than you move the handle with your hand. For all machines—not just hammers—the user applies force (input force) to the machine over a certain distance (input distance). The machine, in turn, applies force (output force) to an object, also over a certain distance (output distance). A machine makes work easier by changing the force or the distance over which the force is applied.  

What Is Mechanical Advantage?

How much a machine changes the input force is its mechanical advantage . Mechanical advantage is the ratio of the output force to the input force, so it can be represented by the equation:

\mathrm{Actual\;Mechanical\;Advantage=\frac{Output\;force}{Input\;force}}

Note that this equation represents the actual mechanical advantage of a machine. The actual mechanical advantage takes into account the amount of the input force that is used to overcome friction. The equation yields the factor by which the machine changes the input force when the machine is actually used in the real world.

Ideal Mechanical Advantage

It can be difficult to measure the input and output forces needed to calculate the actual mechanical advantage of a machine. Generally, an unknown amount of the input force is used to overcome friction. It’s usually easier to measure the input and output distances than the input and output forces. The distance measurements can then be used to calculate the ideal mechanical advantage. The ideal mechanical advantage represents the change in input force that would be achieved by the machine if there were no friction to overcome. The ideal mechanical advantage is always greater than the actual mechanical advantage because all machines have to overcome friction. Ideal mechanical advantage can be calculated with the equation:

\mathrm{Ideal\;Mechanical\;Advantage=\frac{Input\;Distance}{Output\;Distance}}

You can watch a video about actual and ideal mechanical advantage at this URL:

http://video.google.com/videoplay?docid=8517358537561483069#

A Simple Example

Look at the ramp in the Figure below . A ramp is a type of simple machine called an inclined plane. It can be used to raise an object off the ground. The input distance is the length of the sloped surface of the ramp. This is the distance over which the input force is applied. The output distance is the height of the ramp, or the vertical distance the object is raised. For this ramp, the input distance is 6 m and the output distance is 2 meters. Therefore, the ideal mechanical advantage of this ramp is:

\text{Ideal Mechanical Advantage}=\frac{\text{Input distance}}{\text{Output distance}}=\frac{6 \ \text{m}}{2 \ \text{m}}=3

An ideal mechanical advantage of 3 means that the ramp ideally (in the absence of friction) multiplies the input force by a factor of 3. The trade-off is that the input force must be applied over a greater distance than the object is lifted.

Mechanical advantage for a ramp

Q : Assume that another ramp has a sloping surface of 8 m and a vertical height of 4 m. What is the ideal mechanical advantage of this ramp?

A : The ramp has an ideal mechanical advantage of:

\text{Ideal Mechanical Advantage}=\frac{8 \ \text{m}}{4 \ \text{m}}=2

Mechanical Advantage of Different Types of Machines

Many machines—including inclined planes such as ramps—increase the strength of the force put into the machine but decrease the distance over which the force is applied. Other machines increase the distance over which the force is applied but decrease the strength of the force. Still other machines change the direction of the force, with or without also increasing its strength or distance. Which way a machine works determines its mechanical advantage, as shown in the Table below .

Strength of Force Distance Over which Force is Applied Mechanical Advantage Example
increases decreases >1 ramp
decreases increases <1 hammer
stays the same (changes direction only) stays the same =1 flagpole pulley

Summary

  • The actual mechanical advantage of a machine reflects the increase or decrease in force achieved by the machine. It takes into account the force needed to overcome friction.
  • The actual mechanical advantage can be calculated with the equation: \mathrm{Actual\;Mechanical\;Advantage=\frac{Output\;force}{Input\;force}}
  • The ideal mechanical advantage of a machine reflects the increase or decrease in force there would be without friction. It is always greater than the actual mechanical advantage because all machines must overcome friction.
  • The ideal mechanical advantage can be calculated with the equation: \mathrm{Ideal\;Mechanical\;Advantage=\frac{Input\;Distance}{Output\;Distance}}
  • The mechanical advantage of a machine may be greater than, less than, or equal to 1, depending on the type of machine.

Vocabulary

  • mechanical advantage : Number of times a machine multiplies the input force; calculated as the output force divided by the input force.

Practice

Watch the video about mechanical advantage at the following URL, and then solve the problems below.

http://www.youtube.com/watch?v=K9RloG1rbNY

  1. Juan uses a lever to lift a rock. He places the rock on the short arm of the lever and pushes down on the other arm of the lever. He applies 30 N of force over a distance of 1 m. The other end of the lever moves in the opposite direction and raises the rock a distance of 0.2 m. What is the weight (force) of the rock?
  2. What is the mechanical advantage of the lever in question 1?

Review

  1. What is the mechanical advantage of a machine?
  2. How is the actual mechanical advantage of a machine calculated?
  3. How does ideal mechanical advantage differ from actual mechanical advantage? Why is the ideal mechanical advantage of a machine always greater than the actual mechanical advantage of the machine?
  4. A hammer has an input distance of 3 cm and an output distance of 9 cm. What is its ideal mechanical advantage?

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