Lesson: Levers That LiftContributed by: Integrated Teaching and Learning Program, College of Engineering, University of Colorado Boulder
Educational Standards :
Pre-Req Knowledge (Return to Contents)
Students should be familiar with the six simple machines, as discussed in Lesson 1 of this unit, The Advantage of Machines.
Learning Objectives (Return to Contents)
After this lesson, students should be able to:
Introduction/Motivation (Return to Contents)
Today we are ready to learn about three more simple machines. These include the lever, pulley, and wheel-and-axle. These machines may sound unfamiliar initially, but it is very likely you will recognize them when we reveal the many everyday applications, equipment and appliances in which they are found. Although one of the six simple machines is not superior to another, each machine offers its own distinct advantages for various engineering applications. These advantages, along with how engineers use them, will be discussed in today's lesson as we study these next three exceptional machines.
Many engineers today, especially mechanical engineers, are very interested in simple machines and their ability to carry out an immense amount of work with minimal effort. To understand how this is achieved, it is necessary to recall that work is done by applying force to a load and transporting it over some distance. The more force applied and the further the load is moved, the more work is done. This idea is expressed mathematically as
Work = Force x Distance
We know that a specific amount of work needs to be completed for a specific task. However, nature does not specify exactly how this work may be accomplished. This enables engineers to complete the same amount of work with less force by simply moving the load over a greater distance. This tactic of making work easier is performed with simple machines such as the lever, pulley and wheel-and-axle.
The lever is the most familiar of all the simple machines because of its plain construction and extensive use in numerous engineering devices. It merely consists of a rigid beam or rod which freely rotates about a fixed point, also referred to as the fulcrum. By positioning the fulcrum close to a heavy object and applying an effort from far away, levers can be used to lift enormous loads with ease (refer to Figure 1). The object being moved by the lever is often called the load, or output force, while the force applied to the lever is called the effort, or input force. The crowbar is a classic example of how the lever is employed to do work easier. With the crowbar, carpenters can easily extract nails from wood that would be nearly impossible and extremely inefficient without such a handy machine.
Immediately you will see that there is always a fulcrum, load and effort positioned somewhere on the lever, yet it may be difficult to notice how the position of each of these relative to one another can change the characteristics of the lever altogether. For this reason, levers are classified into three different types; called first-, second- and third-class levers (see Figure 2).
The classification of each depends on the position of the fulcrum relative to the effort and load. In a first-class lever, the fulcrum is placed between the effort and load to resemble a seesaw. Examples of this type of lever include a balance scale, crowbar and a pair of scissors.
A second-class lever is when the load is placed between the fulcrum and effort. This lever type has been used in the design of many devices such as a wheelbarrow, nutcracker, bottle opener and conventional door.
Lastly, third-class levers operate with the effort applied between the fulcrum and load. These levers can be found in tweezers, fishing rods, hammers, boat oars, and rakes.
Throughout history, engineers have found the pulley to be the machine of choice when lifting heavy objects over a direct vertical path. The pulley is basically a grooved circular disk which acts to guide a rope or cable pulled around its perimeter, as illustrated in Figure 4. With a single pulley, engineers can change the direction of an applied force; such as pulling a rope down to lift a weight up. However, using a combination of pulleys in a pulley system can change both the amount and direction of the applied effort. To increase the pulley's lifting power, pulley wheels are added to a pulley system so that the effort required to lift objects vertically is largely reduced. This machine is incorporated into the design of various engineering systems such as a crane, whereby huge loads are manipulated with a little force supplied by a relatively small motor. Some cranes can have numerous pulley wheels and a complex array of cables so that the ability to lift heavier objects is even greater. Many other devices employ the pulley in order to benefit from its amazing potential, including an elevator, sailboat and a basic flagpole.
The last simple machine we are going to learn about is the wheel-and-axle, which engineers primarily use to increase a turning or rotational force. This device is composed of a circular wheel directly connected to a circular shaft or axle, and turned to rotate about a common axis (see Figure 5). From this arrangement, you may notice how the wheel and axle operates similar to that of the lever; however, it is different in the sense that it has the ability to increase a rotational force instead of a linear force. Engineers commonly refer to a rotational force as torque. In order to remain consistent with the definition of mechanical advantage, we define the wheel and axle such that the effort or input force is always applied to the wheel and the load or output force is always acting on the axle.
In most cases, the axle is smaller than the wheel and the applied torque is magnified by the machine; however, this configuration is not always the case. In some instances, the axle is larger than the wheel, and the input distance is increased by the machine instead of the input torque.
The mechanical gears found on a bicycle are a great example of the wheel-and-axle at work (see Figure 6). Look at the center of the rear tire and notice the many different sized sprockets that function as gears to drive the rear tire. The gears are directly fastened to the tire which, together, acts as a wheel-and-axle; in this case the gear is the wheel and the tire is the axle. An input force supplied by the moving chain is applied to the wheel (gear), which is then transferred to the axle (tire) so that distance is magnified. The different-sized gears are used to change the mechanical advantage of the machine so that riding a bicycle up a steep slope in a low gear is much easier, while riding over a flat surface in a high gear is much faster.
Other examples of the wheel and axle include a screw driver, steering wheel, jet engine, mechanical gears, and even doorknobs.
Lesson Background & Concepts for Teachers (Return to Contents)
The mechanical advantage of a machine characterizes its ability to do work efficiently and effectively. Therefore, anytime a simple machine is considered for an appropriate engineering system, it is necessary to determine its associated mechanical advantage. In Lesson 1 of this unit, the mechanical advantage of a machine is defined as the number of times a force exerted on a machine is multiplied by the machine. This idea can also be expressed by the general mathematical equation:
For the three simple machines in this lesson, it is important to clarify what the mechanical advantage of a machine reveals about its capabilities. If MA = 1 (Mechanical Advantage = 1), this implies the machine has no effect on alleviating work, since the output force is exactly the same as the input force. Most simple machines provide a mechanical advantage greater than 1 (MA > 1) so that work is made easier; i.e., the input force is increased, and therefore, the mechanical advantage is less than the output force.
There are some cases when it is beneficial to have a mechanical advantage less than 1. On this occasion, work is made harder by the machine since the input force is greater than the output force. This may seem contrary to the purpose of simple machines; however, the force must be reduced by the machine in order for distance to be magnified. This is sometimes very useful in certain engineering applications and surprisingly common among the three machines discussed in this lesson.
To summarize this concept, if MA < 1, distance is multiplied — work is made harder, but faster; if MA > 1, force is multiplied — work is made easier, but slower. With this information, engineers can modify the mechanical advantage of a machine in order to produce an effective, efficient, and very useful appliance.
The advantage of levers, which relates effort and load (or input and output force), depends on how far away each are from the fulcrum. The mechanical advantage of a lever is increased when either the effort is moved further away from the fulcrum or the load is shifted closer to the fulcrum, or both. This idea of leverage can be expressed mathematically by:
In this equation, the distance between the load and fulcrum is called the Load Arm, while the distance from the fulcrum to the effort is called the Effort Arm as shown in Figure 9.
Notice that a single definition for the mechanical advantage of a lever applies to all three classes of levers. However, because of the physical arrangement of each lever type, we note that for second-class levers, MA > 1, yet for third-class levers, MA < 1. First-class levers have the unique option to magnify either the input force or the input distance (MA > 1 or MA < 1). The capabilities of these different lever types provide engineers with an array of choices during the design and selection process of a particular engineering system.
A pulley system operates on the principle that a load can be lifted easier by pulling on a rope or cable that is wrapped between a supporting structure and a rigid fixture attached to the load itself. One extremely common idea — and often a rule of thumb — for a pulley is to calculate its mechanical advantage by counting the number of pulley wheels found in a system. While this practice is generally acceptable, the method does not always provide accurate results. A more precise method for calculating the mechanical advantage of a pulley is by counting the number of ropes or cables which support the load. Then the mathematical relationship is simply expressed as:
This result may seem unrelated to the general definition of mechanical advantage; however, this machine remains in perfect agreement with the definition once the input and output distances are identified. From Figure 11, we can see how the input distance, output distance, and number of support lines are related in a pulley system. In this example, if a pulley has a mechanical advantage of 2 (MA = 2), pulling on the end of the support line with effort over a given input distance will cause the load to be raised a distance equal to half that pulled by the effort. This is true regarding all pulley combinations: the more support lines attached to the load, the more input distance required to raise the load up to a desired height. To conclude, we find that the physical geometry of a pulley system requires its mechanical advantage to always be greater than 1 and only in positive integer values; i.e., MA = 1, 2, 3, etc.
Before stating the mechanical advantage of a wheel-and-axle, it is extremely important to remember that the effort is always applied to the wheel, while the load always acts to resists the turning motion of the axle. Specifically, as effort is applied to the wheel and rotated through an angle θ, geometry dictates the input distance to be the product of θ and the wheel radius. Likewise, since the axle is fastened to the wheel and turned through the same angle θ, the output distance is the product of θ and the axle radius. Then from the general definition, we see the mechanical advantage of the wheel-and-axle depends only on the radius of each, where it can be written as:
This result informs engineers how the mechanical advantage of a wheel-and-axle may be altered to provide the most efficient results in an engineering system. Typically, engineers configure the wheel-and-axle so its mechanical advantage is greater than 1 to benefit from a magnified torque, such as the case with a steering wheel. If engineers prefer the wheel-and-axle to multiply distance and thus do work faster, such as the case with a bicycle, the machine is constructed with its axle larger than the wheel, or MA < 1.
Vocabulary/Definitions (Return to Contents)
Associated Activities (Return to Contents)
Lesson Closure (Return to Contents)
All simple machines are characterized by their ability to provide mechanical advantage, which allows engineers to design devices to make work easier and more efficient. Although one machine is not superior to another, each machine provides its own unique and attractive capabilities which are used by engineers for numerous applications. The lever is capable of quickly increasing either force or distance; the pulley can lift enormous loads over a vertical path; and the wheel-and-axle is used to easily increase an input torque. These three simple machines, combined with the other three (inclined plane, wedge and screw), give engineers a set of extremely valuable tools to effectively carry out work.
Assessment (Return to Contents)
Question/Answer: Ask students the following questions and and discuss as a class:
Voting: Ask a true/false question and have students vote by holding thumbs up for true and thumbs down for false. Tally the votes and write the numbers on the board. Give the right answer.
Lesson Summary Assessment
Discussion Question: Solicit, integrate and summarize student responses.
If MA < 1, then work is harder, but faster
If MA > 1, then work is slower, but easier
Lesson Extension Activities (Return to Contents)
Team Competition: Organize the class into small groups of two or three students each and challenge each group to think of where in engineering systems today the lever, pulley and wheel-and-axle can be found. The group that thinks of the most machines is the winning team. To get full credit, each team must state the engineering device along with the associated simple machine. Examples: Lever: seesaw, balance scales, crowbar, wheelbarrow, nutcracker, bottle opener, tweezers, fishing rod, hammer, boat oar, rake, etc. Pulley: crane, elevator, flagpole, etc. Wheel and Axle: screwdriver, steering wheel, bicycle gears, doorknob, etc.
References (Return to Contents)
Kahan, Peter. Motion, Forces, and Energy: Science Explorer Student Edition. Upper Saddle River, NJ: Prentice Hall, 2002.
Macaulay, David. The Way Things Work. Boston, MA: Houghton Mifflin Company, 1988.
Environment: Hand Tools for Trail Work. Las updated June 16, 2005. Federal Highway Administration, U.S. Department of Transportation. Accessed August 31, 2007. http://www.fhwa.dot.gov/environment/fspubs/05232810/page16.htm
Woods, Michael, and Mary Woods. Ancient Machines: From Wedges to Waterwheels. Minneapolis, MN: Runestone Press, 2000.
ContributorsJake Lewis, Malinda Schaefer Zarske, Janet Yowell
Copyright© 2007 by Regents of the University of Colorado
Supporting Program (Return to Contents)Integrated Teaching and Learning Program, College of Engineering, University of Colorado Boulder
Acknowledgements (Return to Contents)
This digital library content was developed by the Integrated Teaching and Learning Program under National Science Foundation GK-12 grant no. 0338326. However, these contents do not necessarily represent the policies of the National Science Foundation, and you should not assume endorsement by the federal government.