• During the lesson, use the following checklist to evaluate students’ understanding and ensure that they master targeted learning goals and understand the deeper meaning behind mechanical advantage:

• Student demonstrates with simple machines that unbalanced force can be used to produce mechanical advantage.

• Student understands that mechanical advantage means that less force is required to perform the same task.

• Student understands that mechanical advantage is realized by exerting less force over a greater distance.

• Student understands that mechanical advantage does not reduce the amount of overall work produced.

1. Open the lesson with the following narrative:

“In Egypt, the Great Pyramid of Khufu stands 481 feet tall. It is made from limestone blocks that each weigh, on average, 5,500 pounds, and there are 2.3 million of them. Since powered cranes were not invented until 4,400 years after the pyramid was built, there are people who seriously believe that it must have been built with the help of an advanced civilization from another planet.

“A less dramatic explanation is that the ancient Egyptians had the one machine they needed: the inclined plane. We still rely on it today. When you are in a car driving to the top of a mountain, the driver does not park at the bottom of a cliff, wrap a rope around the axle, and have the car winch itself to the top. Instead, the driver follows the road as it angles up the mountain, usually going back and forth in a series of switchbacks. Each switchback amounts to an inclined plane. The inclined plane of the pyramid builders was probably a dirt ramp.

“In previous lessons in this unit we spoke of unbalanced force as something that just happens to generate movement. But you can set up an unbalanced force that generates movement in a desired manner, usually so that less force is involved. With an inclined plane, that unbalance is built in.

“The energy (or work) needed to move something is the amount of force applied over the distance that the object is moved, or:

Work = Force × Distance

(W=F×D)

Let’s say that W is the amount of force needed to lift an upright piano four feet into the back of a truck. Let’s say four people were needed to lift it. But if you slide the piano up a ramp that’s 16 feet long, the force required to move the piano is a quarter as much. One person could do it. The work is spread over a longer space or time, and the ramp supports a certain portion of the piano’s weight during the process.

“But keep in mind that, by the time you get the piano into the back of the truck, the total amount of work, or energy expended, is the same. In that sense the result is the same whether you lifted it straight up or slid it up the ramp. The one person sliding the piano up the ramp probably took at least four times longer than the four people who lifted it directly. And that one person was probably four times more fatigued than any of the four other people who lifted it directly.

“This reduction in force that the ramp gives you is called mechanical advantage. The work required is still the same, but the mechanical advantage lets you reduce the force you need to apply while you’re engaged in the task, or accomplish a larger task than would otherwise be possible. (There are also opposite situations, where you want to apply greater force to accomplish the task in less time or space, such as taking an elevator instead of the stairs.)

“Again, the inclined plane involves an unbalanced force, one that is unbalanced on purpose. The net result is that the object gets lifted, but less force needs to be available since the force is applied over a greater distance.

“In this lesson we’ll explore the mechanics of the inclined plane, using tools available to the pyramid builders.”

2. Conduct the following experiment with help from students:

• Stack up a couple of books and create an inclined plane leading to the top of the books with, for instance, and upside-down clipboard.

• Measure and record the length and height of the inclined plane.

• Measure and record the length of a rubber band at its fullest extent without stretching.

• Possibly using a bent paper clip as a hook, use the rubber band to lift a modest-sized object, such as a stapler, off the table and onto the books. Measure and record the length of the rubber band while the object is suspended.

• Place some pens or pencils with round cross-sections under the object on the inclined plane, to act as rollers.

• Drag the object, attached to the same rubber band, up the inclined plane.

• Measure and record the length of the rubber band as the object moves.

• Raise one end of the inclined plane to make it steeper, and drag the object up again, and again measure and record the length of the rubber band as the object moves.

• Raise the inclined plane to different heights and measure each trial.

The measurements gathered so far would look something like this:

Length of rubber band	Use
5.7 cm	slack
6.9 cm	dragging up inclined plane 3.8 cm high, 33.02 cm long
8.9 cm	dragging up inclined plane 10 cm high, 33.02 cm long
10.2 cm	dragging up inclined plane 15.2 high, 33.02 cm long
15.2 cm	vertical lift

(The example is in inches but centimeters will work the same.)

3. Lead the class in a discussion of the basic findings: the rubber band was stretched considerably less when it was dragging the object up the inclined plane than when it was lifting the object straight up. Meanwhile, as the inclined plane was made steeper, more stretching was needed to pull the object up the slope.

The stretching of the rubber band will serve as a rough measure of the force exerted to move the object, with more stretching indicating greater force.

On the inclined plane, the force exerted was unbalanced, with part of the force directed upward and part directed sideways. The part of the force that was directed upwards, against gravity, was largely exerted by the ramp itself. As the slope became steeper, the rubber band had to support more and more of the weight of the object as well as move it.

Meanwhile, with W=F×D, increasing D (distance traveled) ought to decrease F (force exerted), as in the case of pushing something up a long ramp as opposed to lifting it straight up. The experiment demonstrated that this was the case.

4. Mention to students that all other basic machines (lever, pulley, wheel-and-axle, screw, and wedge) all create mechanical advantage by trading distance for force.

The lever is a rigid bar that is pivoted on a fulcrum (like a see-saw, but usually off-center) to exert an unbalanced force. The end that is closer to the fulcrum moves a smaller distance but exerts more force when the opposite end is moved. Therefore, W=F*D produces a dramatically different result at opposite ends of the rod, with a different D resulting in a larger or smaller F.

A pulley is a rope supporting a weight that hangs over a grooved wheeled. The force is unbalanced because part of the weight of the object is supported by the fixed end of the rope, and the rest is supported by whoever is holding the other end of the rope. Using several pulleys in parallel to support a weight is called a block-and-tackle, and the mechanical advantage equals the number of pulleys used in parallel. With two pulleys, for instance, the advantage is a factor of two, but that also means that the amount of rope you have to pull to lift an object is twice the height you are lifting it.

A wheel-and-axle is basically a lever that rotates on its fulcrum. A common example is a doorknob, or a fishing reel, or a bicycle pedal with gears.

(Caution students against confusing the wheel-and-axle and the wheel. A wheel that rolls passively is not a machine, but a device that minimizes friction.)

The uses of simple machines are endless. For instance, a wedge is an inclined plane that is inserted into an object, to split it. A screw is a wedge that is wrapped around a cylinder. It can be used to move objects (as with a screw jack) or clamp objects together (as with construction screws.)

5. The group will further analyze the results of the experiment by calculating the force needed to move the object, based on the amount that the rubber band stretched. The result will be figured in rubber band units, henceforth called RBUs. RBU’s will be used later in the lesson to extract work, expressed in Newtons.

Review that force is expressed in Newtons. A Newton (N) is a unit of force equal to the amount of force required to accelerate a mass of 1kg at a rate of 1m per second per second.

Remind the class that they will not be calculating Newtons during this lesson but they need to know the background of force for future lessons and applications.

A member of the class should use a calculator to find the ratio of the rubber band during each trial to its original length. Divide the difference between the stretched rubber band’s length and the slack length by the slack length to give you the force.

If the slack rubber band had a length of 2 cm and stretched length of 4 cm, the force would equal 1 RBU.

If the RBUs were added as another column in the previous example, the results would look like this:

Length of rubber band	Use	RBUs
5.7 cm	slack	0
6.9 cm	dragging up inclined plane 3.8 cm high, 33.02 cm long	0.2105
8.9 cm	dragging up inclined plane 10 cm high, 33.02 cm long	0.5614
10.2 cm	dragging up inclined plane 15.2 cm high, 33.02 cm long	0.7894
15.2 cm	vertical lift	1.6666

6. The force exerted in each trial has now been quantified. The group will next compare the work performed by dragging the object up the inclined plane, versus lifting it vertically to the same height.

For each time the object is dragged up the incline, the group should multiply the RBUs measured while dragging the object times the length of the inclined plane. (In our example, the plane was 13 inches long.) Then students should multiply the RBUs recorded when the object was lifted vertically, times the height of the inclined plane for that trial.

Added to our example, the results would look like this:

Length of rubber band	Use	RBUs	W=F*D in RBU inches
5.7 cm	slack	0	Inclined plane	Vertical lift
6.9 cm	dragging up inclined plane 3.8 cm high, 33.02 cm long	0.2105	6.9507	6.333
8.9 cm	dragging up inclined plane 10 cm high, 33.02 cm long	0.5614	18.5374	16.666
10.2 cm	dragging up inclined plane 15.2 cm high, 33.02 cm long	0.7894	26.0695	25.3323
15.2 cm	vertical lift	1.6666	---	---

7. Lead students in a discussion of the results: the total work performed was almost (but not quite) the same (in RBU inches) whether the object was lifted vertically or dragged up an inclined plane to the same height. However, less force had to be applied to accomplish the task when using the inclined plane. The ratio of the difference is the mechanical advantage of that plane.

In the example, slightly more work was required for the inclined plane, presumably because of friction. Any discrepancies could result from measurement errors and deformation of the rubber band.

8. Separate students into at least two groups, one to drag, one to measure the rubber band, and possibly a third to handle the calculations. After lifting an object with a rubber band, each group should set up its own inclined plane, with an object to drag, and rollers. Students should record three separate trials at different heights, with their own rubber band, as in the example, and figure the work performed. The use of a calculator, or access to a computer spreadsheet, is advisable.

The class should then reconvene to compare results, and discuss the source of any differences.

Extension:

• Students can additionally calculate the ideal mechanical advantage (MA) offered by the inclined plane in each trial, and the actual MA that was realized. They can then calculate the mechanical efficiency of the system.

Ideal MA is the distance over which the effort is applied divided by the distance over which the object is moved. In the first trial of our example, the effort was applied over a 13-inch inclined plane, while the object was moved upwards 1.5 inches. The ideal mechanical advantage (MA) therefore is a factor of 8.67.

The actual MA, meanwhile, is the output force divided by the input force. In this case the output force is the weight (in RBUs) of the object lifted, reflected by the RBUs recorded when the objected was lifted vertically. The input force is the RBUs recorded when the object was dragged up the inclined plane. In our example, the trial using a height of 1.5 inches involved an output force of 1.6667 RBUs and an input fore of 0.2222 RBUs. This resulted in an actual MA of 7.5.

Mechanical efficiency, meanwhile, is the actual MA divided by the ideal MA, expressed as a percentage. In our example, it would be 7.5 divided by 8.67, giving 0.867 or about 86 percent.

Added to our example, the results might look like this:

Because of friction, efficiency can never be 100 percent. Steeper planes caused less friction and therefore produced higher mechanical efficiency.