# Scissors_ Salad Tongs and Tin Sn

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```					Scissors, Salad Tongs and Tin Snips:
Uncovering Algebra in Everyday Mechanisms

Materials What is a Mechanism? Levers have a lot of class How to use a pair of scissors Salad tongs geometry Finding a function Hooking up Hooke’s Law Visualizing May the force vectors be with you May the Extreme force be with scissors you Extreme scissors

The purpose of these activities is to develop some basic concepts of algebra and geometry through a study of common utensils and tools. Students begin by examining some of these devices, and trying to find their common features. Through this activity, they develop ideas about input, output, motion and force. Next, they sort some common mechanisms according to lever classes, and then do a qualitative exploration of mechanical advantage. Students then collect data showing the relationship between input and output position of a pair of salad tongs. The result is expressed by a linear function. To investigate force, as well as distance, students use spring scales to find the force exerted by the return spring of a pair of salad tongs. An investigation of angular dependence leads to a discussion of vectors. This is followed by a study of the relationship between force and distance, which provides an example of a nonlinear function, as well as an introduction to the Law of the Lever, which is a special case of Conservation of Energy. The concluding activity looks at the tin snips, as an example of a compound lever. How can the mechanical advantage be calculated for this device? The answer involves an introduction to composite functions.

1

Materials

Brass paper fasteners, 1” or 11⁄2” length; markers or felt-tip pens; duct tape; rulers; graph paper.

Scissors, nutcrackers, tin snips, salad tongs, hole puncher

At least six additional mechanisms per group. These might include can openers, eyelash curlers, glue stick or lipstick, tweezers, staple removers, egg beaters, ice cream scoops (with movable blades), folding chairs, ironing boards, pop-up books, etc. Some should be devices that incorporate more than one lever, such as nail clippers, can opener with rotating handle, corkscrew with two arms, vise grips, bicycle handbrake assembly, umbrella, tin snips, garden shears, pencil sharpener with clamp, pizza tray holder, pedal-operated wastebasket, ice cream scoop with thumb lever, etc.

Thin cardboard sheets (about 10”x10”) and strips (about 2”x10”) for making mechanisms; these can be obtained from discarded cereal boxes, express mail envelopes, or thin cardboard cartons

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Materials (continued)
Small pieces of cardboard of assorted weights; scrap sheet metal if available

Three sample “Mystery Mechanisms” with inside concealed. These can be made from file folders or express mail envelopes.

Spring scale, in the 5 N. (500 g.) or 10 N. (1000g.) range; each of these should be equipped with about a one-foot loop of string through the hole, so that it can attach to a salad tong arm more easily.

2

What is a Mechanism?

Procedure: Each group is provided with a few mechanisms. They also look for additional examples of mechanisms, both from around the room, and also from among their personal effects. They are asked to determine the characteristics these devices have in common. They then share these characteristics with the whole class, which then develops a brainstorming list of what makes something a “mechanism.” The job of a mechanism is to transform the force and motion supplied by the user to the force and motion needed to do a task elsewhere. Some things these mechanisms have in common are that each one needs a human being to operate it, and each is designed to do a job. Some tasks involve cutting, holding, crunching, mixing, or squeezing; but there are also mechanisms whose job is simply to fold or unfold, retract or extend, etc. The point on the device where the user applies a force is called the input, while the place where the job takes place is the output. As they look around the room, students should be encouraged to extend their view of mechanisms to include such items as, light switches, door locks, doors, door knobs, cabinet handles, window latches, crank-operated pencil sharpeners, folding chairs, etc. They may also identify mechanisms they have brought with them, These might include retractable ball-point pens, book bag buckles, key chain latches, and so forth. Interesting issues may come up about whether books, cell phones, calculators, watches, etc., should be considered mechanisms. Raising and thinking about these questions are far more useful than trying to answer them definitively.

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Levers have a lot of Class
Sorting Mechanisms Procedure: Students are introduced to the terms “fulcrum”, “effort,” and “load,” and asked to identify these points on several of their devices. Using the Sorting Mechanisms Worksheet, they then sort the mechanisms from “What is a Mechanism?” according to the arrangement of the effort, fulcrum and load: effort-fulcrum-load (1st class lever); effort-load-fulcrum (2nd class lever); or load-effortfulcrum (3rd class lever). “Input” and “output” are general terms that could be used with any system. A special language is used in describing the most basic type of mechanism, the lever. The input is often called the effort, and the output is called the load. As with the terms “input” and “output,” both the “effort” and “load” could describe either a force or a location. In addition to the effort and the load points, every lever has a pivot that allows the rest of the device to rotate. In the language of levers, the pivot is called the fulcrum. Several important characteristics of lever operation depend on the arrangement of the three elements: fulcrum, effort and load. There are three different possibilities, labeled as first-, second- and third-class levers below.

5

Sorting Mechanisms (continued)
To see what class a particular lever falls into, identify the fulcrum, effort and load, and compare their arrangement with the diagrams above. The diagrams below show how this can be done for some examples from each class.

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Compound Levers
Procedure: A compound lever consists of one lever operating another. Students should have encountered some compound levers during the sorting activity, such as the nail clipper.Using the Compound Levers Worksheet they should identify each of the nail clipper’s levers and find the effort, fulcrum and load for each one. Then ask each group to select one compound lever; make a sketch of it showing each of the simple levers; and label the effort, fulcrum and load on each one. Based on this drawing, they should be able to tell the class of each lever. Then ask them to represent the compound lever in their sorting categories. The nail clipper is an example of a compound lever. The image to the right shows the two levers that make it up: the handle and the upper jaw. Notice that the load of the handle is also the effort of the jaw; in other words, the handle is in series with the jaw. From the drawing, it should also be clear that the handle is second-class, while the jaw is third-class. Some other compound levers are shown to the left. Clockwise starting at the top, these include: • a pair of vise grips (1st class lever as input to 1st class lever) • a pizza-tray holder (same as vise grips) • a grapefruit sectioner (1st class lever as input to 3rd class lever) • a tea-bag strainer (reverse of grapefruit sectioner) • a pair of garden shears (2nd class lever as input to 1st class lever) • a pair of tin snips (see Extreme Scissors for an extended discussion)

An interesting issue is how to represent these compound levers in sorting according to first, second- or third-class. One solution is to use a Venn diagram.

7

Design Challenge 1
Procedure: Demonstrate sample mystery mechanisms A & B shown below. Challenge students to create these two mechanisms, using paper fasteners and cardboard.

To help students solve these challenges, ask them to examine the directions of the input and output in each case. Do they travel in the same or opposite directions? Then suggest that they look at the first-, second- and third-class levers they have sorted. In which of the classes do the input and output go in the same direction? Which class makes them go opposite ways? If the input and output are in the same direction, which class has the output at the end? Based on this analysis, students should conclude that the mystery mechanism A must contain a third-class lever, while B is based on a first-class lever. Complete solutions are shown below.

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How to use a Pair of Scissors

Procedure: Introduce this topic by asking students about how they might place a piece of cardboard so a pair of scissors could cut it easily. Then ask each group to try their idea on a small piece of heavy cardboard using the How to use a Pair of Scissors Worksheet. For each position of the cardboard, have them identify the effort, fulcrum, load, effort arm and load arm; and explain how the two arms change as the cardboard is moved towards the center. When a pair of scissors doesn’t cut a piece of cardboard very well, what do you do? Most people would suggest moving the cardboard from A to B puttting it closer to the pivot. Why does this work? Some would say that the blades are sharper nearer the pivot, but that might not always be true. A more reliable explanation has to do with levers. Let’s look again at the two configurations, using the language of levers. Here are some new terms:

Effort arm = Distance from fulcrum to input, or effort point Load arm = Distance from fulcrum to output, or load point effort arm Mechanical advantage = Ratio of load arm Later, we’ll see why this ratio is called “mechanical advantage.” For now, let’s think about what happens when we move the cardboard from A to B.

9

How to use a Pair of Scissors (continued)

The image shows how the effort arm is the same, whether we cut at point A or point B, but the load arms are different. At point B, the load arm is shorter than at point A. Also notice how far the blades move at A and B. Not only is the load arm shorter at B, but the blades don’t have to move as far. Now consider what happens to the mechanical advantage as we move from A to B. Because the load arm is in the denominator, reducing it will make the ratio larger, increasing the mechanical advantage. As the mechanical advantage increases, it requires less effort force to overcome the same load, making the job easier.

10

Input and output ranges, effort arm and load arm Procedure: Using the Salad Tongs Geometry Worksheet each group measures the input and output ranges of motion of a pair of salad tongs, and calculates the ratio of (input range)/(output range). Does it matter whether the measurements are made using arc lengths or chord lengths? This issue can be explored either by taking measurements, and/or by plane geometry. Students then measure the effort arm and load arm and take a second ratio: (effort arm) /(load arm). How do the two ratios compare? The image below shows the four quantities that need to be measured. The input and output ranges show how far the input (effort) and output (load) actually move when the mechanism is used. As we have seen in the previous activity, the effort arm and load arm are the distances of the input (effort) and output (load), respectively, from the fulcrum.

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To measure the input and output ranges, it’s easiest to do some tracing on a large sheet of paper. First, trace the outline of the tongs in both open and closed positions.

The next task is to measure the lengths of the two ranges, as well as the lengths of the effort arm and load arm. (Recall the definitions of effort and load arms from p. 9.) Then take the two ratios, and compare them. In other words, input range output range effort arm load arm

Does

=

?

Students may have some difficulty in measuring the ranges, because both are arcs, not straight lines. To get around this problem, they may wonder whether they could measure the straight-line distance from end-to-end of the range, instead of having to bend the ruler around an arc. The straight line connecting both ends of an arc is called a chord.

12

The point of the measurement was to find the ratio (input range) / (output range). Is the ratio of arc lengths the same as the ratio of chord lengths? One way to answer this question is to make both sets of measurements and compare the two ratios. Another way is to use plane geometry. The graphic on the left below uses similar triangles to show that: input chord length (BC) = output chord length (DE) effort arm (AB) load arm (AD)

Meanwhile, on the right side, you can see from the definition of arc length, and the fact that both arcs subtend the same angle θ, that the ratio of arc lengths also comes out to be effort arm over load arm.

ABC & ADE are similar triangles, therefore: BC / AB = DE / AD rearranging: BC / DE = AB / AD i.e., ratio of chord lengths = ratio of arm lengths

By formula for arc lengths, arc1 = r1θ; ARC2 = r2θ; (θ in radians) then: arc1 / ARC2 = R2 / r1 = AB / AD i.e. ratio of arc lengths = ratio of arm lengths

So, both methods do produce the same ratio. Notice how plane geometry helps us come up with a useful result! 13

Design Challenge 2
Procedure: Using the Design Challenge 2 Worksheet, each group is asked to create a cardboard mechanism in which: 1. 2. There are two outputs controlled by a single input, and both outputs move in opposite directions, and The two outputs have the same range of motion.

Mystery mechanism “C” can be achieved by combining the third- and first-class mystery mechanisms “A” & “B” from Design Challenge 1. The first- and third-class levers need to be placed in a parallel combination, so the same input operates both outputs simultaneously.

+ =
14

Design Challenge 2
The second part of the problem requires students to redesign this mechanism so that both outputs move the same distances. The lever on the left side is third-class, while that on the right side is first-class. Furthermore, we now know that: input range output range = effort arm load arm

Also, the same input operates both levers, so the input range of motion must be the same for both. Therefore, the output ranges will match so long as the ratio (effort arm)/ (load arm) is identical for both the first- and the third-class levers. The graphic below identifies these arms on the two levers of the mystery mechanism.

Looking closely at the graphic, it seems that the lengths may have to be adjusted slightly to make the ratios come out the same. Then the two output ranges can be measured again to see how well it really works!

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Finding a Function
Collecting input and output data Procedure: On the input and output ranges already drawn, use a flexible ruler to mark off distances, as in the image below. Then use these calibrated input and output ranges to collect and tabulate data showing the relationship between the input and output variables. Show your results on the Finding a Function Worksheet

We have used the words “input” and “output” to describe various features of the salad tongs. The same terms show up in algebra, and are crucial to the concept of a function. This activity explores the relationship between input and output variables, and develops methods of showing the function that relates them. The first step is to calibrate the input and output ranges traced in the previous activity. This is done by marking distances along each range using a ruler.

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Finding a Function
The next task is to collect data showing the relationship between input and output locations along the calibrated ranges. One arm of the salad tongs should be held stationary, as in tracing the input and output ranges. Then move the other arm through its range in steps, recording both the input and output locations at each step. The graphic to the right shows the tongs in the position where the input is at 2.0 and the output is at 4.0, i.e., (2,4). It is important to include (0,0) as a data point.

These measurements should be taken and tabulated for at least 5 or 6 locations. A typical data table (not for the diagram above) is shown below:

Input (x) (inches) 0 1 2 3 4
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Output (y = f(x)) (inches) 0 1.75 3.6 5.3 7.0

Finding a Function
What can we learn from this data? In order to draw conclusions, it is helpful to represent it in more than one way. Ask students how they could present the data to make the patterns clear. Some ideas might be: • find the ratio of y:x or x:y at each point • making a bar graph • plotting points on a Cartesian coordinate grid Each of these methods has something to recommend it. For example, taking the ratio x:y for each data point (except for x = 0, y = 0) will give approximately the same result. A bar graph shows how the input and output vary together. A plot on an x-y coordinate grid, as shown below, provides another way of looking at the data. By drawing a straight line as close as possible to the points, you can see that the relationship between input and output is linear. In other words, one set amount of movement at the input will result in another set amount of change at the output. In this case, approximately 1 inch difference at the input always causes the output to move by about 1.75 inches. This ratio, (1 / 1.75), is the mechanical advantage. If we take the reciprocal, 1.75, we have the slope of the graph.

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Hooking up with Hooke’s Law
Procedure: Students find the return spring that restores the salad tongs to its open position, when the arm is released. Using a spring scale, they then learn to measure the force exerted by the return spring against an arm of the tongs. Then, they use the Hooking up with Hooke’s Law Worksheet to investigate how this force varies as the arm moves through its range. The relation ship is described by something called Hooke’s Law. First students should look for and sketch the return spring inside the salad tongs. Then they will develop a method for finding the force exerted by this spring. The simplest device for measuring force is the spring scale. One minor modification makes it much easier to use a spring scale. Pass about a foot of string through the hole that holds the hook, and make a knot in the string. This modification will make it possible to grab the tongs arm with the measuring end of the scale. The next task in using a spring scale is to calibrate it. When a scale is calibrated, the small metal indicator will point to zero, without any force pulling on the hook or string. To achieve this, push or pull the metal tab on top, to slide the pointer until it reaches zero.

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Hooking up with Hooke’s Law
Force is quantity that tells you how hard it is to push or pull something. To measure the force generated by the return spring, loop the string around one arm, hold the other arm stationary, and pull on the body of the scale. If the string slips off, a little piece of tape can help to secure it. The reading in Newtons (N.) will be where the pointer stops on the dial. The Newton is the only standard metric unit for force. Unfortunately, many spring scales are calibrated only in grams (g.), which is a unit of mass, not force. The conversion from mass to force uses the gravitational constant, which is 9.8 N./kg. (on earth), or equivalently, 102 g./N, so the reading in N. is about 100 times the reading in g. Students should try to make these measurements several times, pulling on the scale by different amounts, but keeping the scale at right angles to the arm. It is likely that their measurements will not all agree. If they do not pull hard enough, the arm of the tongs will remain in the fully open position, and they will not have overcome the force of the return spring. If they pull too hard, the arm will move to the fully closed position, and some of the force will be used up in just pushing the two arms against one another. Within the middle of the range, the amount of force should be proportional to how much the spring is stretched, illustrating a physical principle called Hooke’s Law.

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Visualizing Vectors
Procedure: Using the Visualizing Vectors Worksheet, students explore how the force varies with the angle at which the spring scale is pulled. These observations lead to an exploration of vectors. Besides the variation of force within the range, there is another issue illustrated in the picture. The angle at which the spring scale is placed, with respect to the tong arm, will also affect the result. If students try this measurement several times, with various orientations of the scale, they are likely to make two important discoveries: 1. The measurements depend on the angle of the scale relative to the arm. 2. The measured force is smallest when the scale is perpendicular to the arm. In the graphic, we see how vectors can be used to represent and explain these two observations. A vector has a magnitude, just like an ordinary number, and also a direction, showing which way it leads. The magnitude of a force vector is just the amount of force; in other words, the reading on the spring scale. The direction of each force vector is the direction of the spring scale away from the arm. Vector A represents the force measured on a spring scale that is perpendicular to the arm, while B is the force measured at an oblique angle. Now we can express observations 1 and 2 in the language of vectors: 1. The magnitude of the force vector depends on its direction. 2. The magnitude is least when the force vector is perpendicular to the arm. 21

Visualizing Vectors
First let’s look at Vector A. It shows the force actually needed to balance the return spring. It is all of the force recorded on a spring scale that is held at right angles to the arm. None of this force is “wasted” in doing anything else. Next, consider Vector B. Its magnitude represents the force measured by a spring scale at an oblique angle to the arm. Some of this force is in the same direction as Vector A, but some is not. If we construct another vector, which we’ll call X, we can use it to show the part of B that pulls along the arm, rather than at right angles to it. Notice that X is perpendicular to A, and parallel to the arm itself. This force cannot have anything to do with opposing the return spring, which is completely taken care of by A. All X does is pull the arm lengthwise, tending to make the whole pair of tongs move downwards. So X is an “extra” piece of B that can only make it bigger than A. When we add them together, A + X = B. Now we can explain observations 1 and 2. When the spring scale is at a right angle to the arm, the “extra” vector X disappears. As the angle of the force vector (measured on the spring scale) increases from 90 degrees, the X vector increases, thereby making the magnitude larger. The same thing happens when the angle decreases, making an acute angle. Only the perpendicular vector will have no extra piece, and therefore the minimum magnitude. To see how the vectors relate to the actual measurements, it is useful to sketch and measure the lengths of a vector corresponding to each measurement. Each vector should be drawn at the same angle as the string leading to the spring scale, and all vectors should terminate in the same construction line, parallel to the arm. The actual length of each vector should then be proportional to the force measurement on the spring scale. To test how well this works, compare the ratios of force measurements to the ratios of the lengths of the corresponding vectors.

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May the Force be with You
Force changes with distance Procedure: Using the May the Force be with You Worksheet each group uses a spring scale to measure the force needed to balance the return spring at various points along one arm of a salad tongs. They collect data showing the relationship between force and distance, and graph the data to show the functional relationship. These two variables are then multiplied to form a constant quantity called energy, illustrating the Law of Conservation of Energy. We have introduced the term mechanical advantage, but up to now have calculated it only from distance measurements. To see what the “advantage” is, students now make force (F) and distance (x) measurements at different locations along the arm. x is always the distance from the fulcrum to the effort, which is supplied by the spring scale. To make the distance measurements easier, it is convenient to tape a ruler along the arm, making sure that the zero mark on the ruler is next to the fulcrum. Based on what we have learned in “Visualizing Vectors,” the force measurements should always be taken at a right angle to the arm, and approximately in mid-range between open and closed positions. Using the template makes it easier to use the same position each time.

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May the Force be with You Distance (d) inches 2 4 6 8 10 12 Force (F) Newtons 6.2 3.2 2.3 1.9 1.5 1.2

For each location, record both force and distance, and then make a table showing all of these measurements. The table shows some typical data.

It might look like this.

Notice how different this function is from the one in Finding a Function. Not only does the output decrease as the input is increasing, but the curve connecting them is not straight. The slope is constantly changing; in other words, a change in input from 2 to 4 inches has a much bigger effect than a change from 10 to 12 inches. This type of function is called non-linear. What do you notice about the force vs. distance? The force seems to go down as the distance goes up! In other words, both force and distance are changing in opposite directions. How can you get a number out of this data that does not change – i.e., that stays constant. Someone may come up with the idea of multiplying force times distance. Trying that out with the data gives the results shown below. 24

May the Force be with You Distance (d) inches 2 4 6 8 10 12 Force (F) Newtons 6.2 3.2 2.3 1.9 1.5 1.2 Force times distance (F x d) 12.4 12.8 13.8 15.2 15 14.4

Notice that when force and distance are multiplied, the result comes out nearly the same each time! This happens because the product of force and distance gives a new quantity called work or energy. Because energy can be neither created nor destroyed, the energy input to the device is always the same as the energy output, and the energy output is based on the return spring, which doesn’t change. In other words, the nearly constant values in the third column are an example of the Law of Conservation of Energy. Some students may ask why the results in the third column aren’t exactly constant. This is an interesting question, which can lead to a discussion of the sources of error in this experiment. Since the distance we have measured is the effort arm, and the force is effort force, the relationship we have discovered is: (EFFORT FORCE) x (EFFORT ARM) = ENERGY INPUT = CONSTANT The load arm and load force have a similar relationship: (LOAD FORCE) x (LOAD ARM) = ENERGY OUTPUT = CONSTANT Why is this quantity constant? The answer is that the load arm and load force are designed into the salad tongs by the placement and stiffness of the return spring. Moving the spring scale along the arm hasn’t done anything to change that. In addition, the Law of Conservation of Energy tells us: 25

May the Force be with You
ENERGY INPUT = ENERGY OUTPUT In other words, the energy input has to remain constant because the energy output is, and both have to be equal to conserve energy. Combining all three relationships gives us: (EFFORT FORCE) x (EFFORT ARM) = (LOAD FORCE) x (LOAD ARM) The last equation, which was discovered by Archimedes, is called the Law of the Lever. By rearranging it, we reach the conclusion that: EFFORT ARM LOAD ARM = LOAD FORCE EFFORT FORCE

This is the ratio we have called Mechanical Advantage. Now we can see why. The purpose of many levers is to amplify the load force, for a given amount of effort force. Their ratio is the “bang for the buck” that expresses the “advantage” the lever provides. The equation shows that this ratio can be calculated easily, just by measuring the lengths of the two arms, and taking their ratios. Because force goes up as distance goes down – a consequence of energy conservation – the ratios are inverted when you alternate between force and distance. In other words, to maximize the load force for a given effort force, increase the effort arm relative to the load arm. This is the principle behind the long handle on a jar opener, nail clippers or bolt cutter. Alternatively, as we have seen in How to Use a Pair of Scissors, you can accomplish the same thing by reducing the load arm, while keeping the effort arm constant.

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Extreme Scissors
Which has more cutting power: a scissors or a tin snips? Procedure: Using the Extreme Scissors Worksheet students try cutting a heavy piece of cardboard or a thin piece of sheet metal with a pair of scissors. Then they try doing the same thing with a pair of tin snips, and decide which one cuts more easily. To confirm their observations, they measure the mechanical advantage of each device. A pair of scissors and a pair of tin snips have the same basic function: to cut things. However, a tin snips will cut through sheet metal, or a thick piece of cardboard, while a scissors will not. How come?

The fact that a tin snips can cut through thicker materials than a scissors suggests that it can cut with more force. The increased force is at the load end, because the user is supplying about the same the effort force with his or her hand. The ratio of these two is what we have been calling mechanical advantage or M.A. (see p. 25): MECHANICAL ADVANTAGE (M.A.) = EFFORT ARM LOAD ARM = LOAD FORCE EFFORT FORCE

In Salad Tongs Geometry, we found that the same ratio can be gotten by measuring the input and output ranges of motion. Combining all of these, we have: M.A. = EFFORT ARM LOAD ARM = INPUT RANGE OUTPUT RANGE = LOAD FORCE EFFORT FORCE

So, we have three different ways of measuring the same thing! Let’s find out how the scissors and tin snips compare in mechanical advantage. 27

Extreme Scissors
The hardest way to measure M. A. is by force ratio, because it requires the use of two spring scales at the same time. The M.A. of the scissors can be measured either by (effort arm)/ (load arm) or (input range)/ (output range). The tin snips have more than one lever on each side (see below), so it’s not clear how to measure the effort and load arms. That leaves (input range)/ (output range). We’ll measure the mechanical advantage of each device using input and output ranges, and compare the results. Table 4 shows some typical data:

Device Scissors Tin Snips

Input Range (cm) 6.0 9.5

Output Range (cm) 2.5 2.0

According to this data, the tin snips has nearly twice the M. A. of the scissors, which would explain its superior cutting power. Where does the extra M. A. come from? Deconstructing a pair of tin snips Procedure: Students examine a pair of tin snips closely, identify all of the levers, and find the effort, fulcrum and load of each one. Look closely at the tin snips. It resembles a pair of scissors on top, and a pair of scissors on top operated by a pair of nutcrackers on the bottom.

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Breaking it down and building it up
Notice how the scissors and the nutcrackers are arranged. In the language of Sorting Mechanisms, each side of the tin snips is a compound lever. Students should be able to make a drawing of the device, showing the effort, fulcrum and load of each lever, and then identify the class of each one. Since the right side is the mirror image of the left, we only need to worry about one side. The drawing might look like this ->. From this drawing, it should be easy to determine that each side of the tin snips consists of a second-class lever (nutcracker) whose output is the input to a first-class lever (scissors). But what’s the purpose of the nutcracker? What does it add to the scissors? Breaking it down and building it up Procedure: Students calculate the mechanical advantage of the tin snips by measuring the mechanical advantage of each lever separately, and then deciding how to combine them to get the total. Then they create and test their own “tin snips” using a nutcracker, a pair of scissors and some duct tape. We know that the difference between the scissors and the tin snips is that only the tin snips has a second-class lever (nutcracker) operating the blades. Also, we have seen that the tin snips has nearly twice the mechanical advantage of the scissors. Where does this extra M.A. come from? The obvious answer is: the second-class lever supplies the extra oomph! Following this idea through, let’s measure the M.A. of each of the levers of the tin snips, and see how we could combine them to get the total. 29

Breaking it down and building it up
To measure the mechanical advantage of each lever, it is easiest to use the ratio of (effort arm)/ (load arm). Recall the definitions of effort arm and load arm from Salad Tongs Geometry: Effort arm = Distance from fulcrum to input, or effort point Load arm = Distance from fulcrum to output, or load point This shows how to use these definitions to find the effort arm and load arm of each lever:

The table shows some typical data:

Lever (type) “nutcracker” (2nd class) “scissors” (1st class) Tin snips (compound)

Effort arm (cm) 13 5

Input range (cm)

Output range (cm)

9.5
30

2.0

4.75

Composite functions
Procedure: Ask each group to come up with a method for predicting the mechanical advantage (M.A.) of a compound lever from the M.A.’s of the simple levers that make it up. Each group should present their idea to the class, and explain why they think it will work. To address this problem, it is helpful to represent the tin snips by another function, which relates the effort force (x) to the load force (y). If we call this function y = f(x), the function is simply the mechanical advantage times the input, i.e., effort force. It works because: LOAD FORCE x EFFORT FORCE = M.A. x EFFORT FORCE LOAD FORCE = EFFORT FORCE Effort force (x)

Tin Snips
y = f(x) y = (tin snips MA) x

We can do the same thing for the nutcracker and scissors, each taken separately. We’ll call our new functions g (for the nutcracker) and h (for the scissors): Effort force (x)

Nutcracker
z = g(x)

z = (nutcracker MA) x

Effort force (z)

Scissors
y = h(z)

y = (tin snips MA) z

Noticing that the output of the nutcracker is the input to the scissors, we can connect the two functions together: Effort force (x)

Nutcracker
z = g(x) z = (nutcracker MA) x z

Scissors
y = h(z)

y = (tin snips MA) z

Finally, we can put everything in the same box, and eliminate the arrow connecting them:

Composite
y = h(g(x))

Effort force (x)

y = (nutcracker MA) (scissors MA) x 31

Composite functions
Now look at the first and last diagrams on the previous page. The inputs and outputs are the same, but the equations inside the boxes are different. However, they have to mean the same thing, because a tin snips is made by using a nutcracker as the input to a scissors. So, we can conclude that the boxes are equivalent, which means that: f(x) = h(z) = h(g(x)) (tin snips MA) = (nutcracker MA) · (scissors MA) The second relationship is the one we were looking for. Here is a way to verify it. From the last two diagrams on the previous page, it should be clear that: TIN SNIPS effort force = NUTCRACKER effort force, and TIN SNIPS load force = SCISSORS effort force; simply because the input to the tin snips is the nutcracker and the output of the tin snips is the scissors. Also, NUTCRACKER load force = SCISSORS effort force, because the output of the nutcracker is the input to the scissors. Using the definition of mechanical advantage, plus these three relationships, the M.A.’s of the scissors and nutcracker become: SCISSORS M.A. = NUTCRACKER M.A. = SCISSORS load force = SCISSORS effort force TIN SNIPS load force ; NUTCRACKER load force NUTCRACKER load force . TIN SNIPS effort force

NUTCRACKER load force = NUTCRACKER effort force

SCISSORS M.A. x NUTCRACKERM.A. = = = TIN SNIPS load force NUTCRACKER load force x NUTCRACKER load force TIN SNIPS effort force TIN SNIPS load force = TIN SNIPS M.A. TIN SNIPS effort force

where the last line cancels cross-terms and uses the definition of M. A.. 32

Composite functions
Everything we have said could apply to any compound lever. There are two important conclusions from this: 1.A compound lever is represented by a composite function h(g(x)), whose input z is itself the output of a function of x, g(x). It’s no coincidence that the words “compound” and “composite” are almost the same! 2.The Mechanical Advantage (MA) of a compound lever is the product of the MA’s of the two simple levers. Here is another experiment you can do. Construct a model of a tin snips using an actual scissors and nutcrackers, and some duct tape. Figure 34 shows how. The duct tape will need to be fairly tight, or the nutcracker won’t transmit most of the force to the scissors. Try out this model “tin snips” on sheet metal or heavy cardboard, and compare the results with those of the scissors alone, as well as the real tin snips. Then repeat all the measurements of effort and load arms and mechanical advantage.

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