Unit 1: Normal Forces

1. Solid objects do not exert forces.

2. When two stationary objects push on each other, the “stronger” one exerts a larger force.

3. When two bodies interact, the “stronger” body pushes with a greater force than the weaker body.

4. When two bodies push on each other, the harder one pushes with a greater force than the softer one.

This unit contains two lessons and one laboratory activity with springs, designed to introduce students to some fundamental considerations about the nature of force. The introduction in the previous chapter gave an overview of the instructional strategies to be used in this unit. The laboratory activity stands alone quite well and could be done early in the year before or after the two lessons in the unit. We strongly recommend using the lab before teaching any other units about force in this document, as analogies to springs are used extensively in later units.

The two lessons contain a series of bridging situations analogous to the target problems and to the anchors (see Figure 1.1 and the concept diagram in Figure 1.2). The foam pad and flexible board cases are examples that have some features in common with the target problem and some with the anchor—but also have some features that are qualitatively different. For example, the foam pad deforms perceptibly whereas the table does not. The basic strategy of this unit is to use these intermediate examples to bridge the conceptual gap between target and anchor, and to help students transfer the correct intuition about an upward force in the anchor situation to the target problem.

You are urged to draw the concept diagrams on the board as the lessons proceed so these examples will be available for easy reference, and to encourage students to make comparisons between situations. Draw the bridging situations between the target and anchor situations as shown on the concept diagram.¹

In accordance with our general strategy of appealing to mechanisms and causal agents and not just to rules, the lessons then introduce a microscopic model—one that is visualizable—consisting of an imaginary network of spring-like bonds between molecules that pushes back when it is deformed. We do not introduce the molecular model for its own sake, but to provide an alternative to the concept of a perfectly rigid body. (Rigid bodies do have some use in physics, but in this context they get in the way of learning.)

A number of interesting questions and examples are raised by students in the discussions during this unit. The teacher may wish to anticipate some of the difficulties below.

• Students have occasionally raised the question of an “ideal table” which would not bend at all. Here we recommend allowing discussion if there is a lot of interest in this issue. If students press for teacher response in this area, it would seem reasonable to respond that physicists believe that tables, even the most solid ones, are made of atoms with spaces in between them. Therefore, whatever conclusion we come to about the ideal, perfectly rigid table would be difficult to apply to the real world.

• Others ask: “How does the table know how to push on an object?”, “Is it intelligent?” This is another important discussion question. It may be fruitful to direct the discussion back to the spring at this point. The physicist’s view is that the larger weight deforms the table more so that the table provides a greater normal force.

• A harder question is: “If a feather is sitting next to a football player on the table, how does the table know how hard to push on each?” Here one needs to refer to both compression and bending of the imaginary “network of springs” in the table, the compression under the football player being larger.

While the term “normal forces” is used as a title for this unit, these words have not been used within the lessons. Should the teacher choose to introduce the term at this stage it is important to clarify the meaning of the word “normal” in this context. Many high school math curricula do not use the word “normal” to mean perpendicular so some students may interpret “normal” as meaning something like “ordinary.” In any case the “normal force” idea will be introduced in Unit 3, which includes friction forces.

Some students may view the car pushing on the hydrant problem in Day #2 as different from the book on the table problem in Day #1 because (a) the force of gravity is more predominant in the table problem, and is an “action at a distance,” and (b) the touching surfaces are vertical in one case and horizontal in the other. To physicists and teachers of physics, the similarity of the two situations may seem obvious, but to students struggling with these new ideas they may seem quite different.

An objective of this unit is to help students understand that the magnitude of forces between objects depends on the distance between the centers of the objects. An appeal to the molecular model introduced in Day #1 should help make this point clear to students. A stiff and a soft object can exert equal forces if the soft object is compressed more. Understanding this point will also set the stage for later work in a number of lessons.

The target problem for this lesson is the “book on the table” problem shown in the concept diagram. The key question we are asking is: “Does the table exert a force on the book?”

We recommend that you avoid the question of the equality of the opposing forces in this lesson, but concentrate on the existence of the upward force. Once that is well established, you can tackle the equality issue in Day #2.

The anchor in this lesson refers to the “hand on the spring” situation shown in the concept diagram. Here we ask the question: “Does the spring exert an upward force on your hand?” We have found in diagnostic tests that the great majority of high school students (even before they have taken any physics) have a correct intuition about this question.

1. a book or a large feather

2. a large spring (mattress spring is a good choice)

3. a block of soft foam

4. a pair of metersticks to use to simulate a flexible table (or perhaps thin balsa wood sticks if you are using a feather instead of a book)

5. a laser and a small mirror (an alternative light source with a narrow pencil beam will work)

6. voting sheets (with introduction on the back)

7. homework sheets – Normal Forces – Day #1

8. Concept Diagram sheets – Normal Forces – Day #1 (attached to homework)

1. The lesson should help the student construct the conceptual model that solid objects are elastic and that they exert a force when a force is exerted on them.

2. The lesson should help the student construct the microscopic model that solids are elastic because of spring-like bonds between their molecules.

1. Introduce voting sheets

   • Pass out voting sheets with the sheet that explains their use. If this is your first use of the sheets, have students read the introduction about the boomerang and then answer any questions. Emphasize the fact that these questions do not count on their grade, but that they are important.

2. Introductory remarks about force

   • We are beginning to investigate the nature of the concept called “force.”

   • Suggest the following definition (on the board): “A FORCE IS A PUSH OR A PULL OF ONE OBJECT ON ANOTHER OBJECT.”²

   • Notice this definition is only a starting place and will be refined and expanded during this course. Consider the fact that this definition does not tell you what a force is not, so we will be spending time clarifying and building on this notion as used by physicists. For example, we will not include social forces or economic forces.

3. Book on the table – target problem

   • Draw the diagram.

   • Vote #1 – question:

     “Does the table exert an upward force on the book?”³ or “Does the table push up on the book?”

   • Choices:

     • Yes

     • No

   • Please circle the makes sense score on your voting sheet and write in a comment if you wish.

4. Discussion of the target problem

   The discussion can extend 15-20 minutes and should be as lively as possible with a variety of inputs from students. Students may introduce some of the ideas below on their own. The teacher is urged to remain neutral during this discussion.

5. Demonstration and discussion – introduce the anchor

   • Demonstrate pushing down on a spring (like a bed spring).

   • Vote #2 – question:

     “Does the spring exert an upward force on the hand?”

     Choices:

     • Yes

     • No

     Remember the makes sense score.

   • Discussion Questions:

     • “How is this situation like the book on the table?”

     • “How are the situations different?”

     • “Could someone who felt the table and the spring are different please tell us why?”

6. Book on a soft foam pad (bridging example)

   • a)

     Introduce the idea of the book on a soft foam pad.

   • b)

     Vote #3 – question:

     • Does the foam exert an upward force on the book?

   • c)

     Discussion question:

     • Is this case the same or different from the table case? Why?

7. Book on a flexible board (bridging example)

   • After discussion of the foam, introduce the idea of the book on a flexible board (demonstrate with metersticks as table top). Balsawood sticks may be needed to show a deflection with a feather.

   • Vote #4 – question:

     Does the flexible board exert an upward force on the book?

   • Discussion questions:

     • Is this case the same or different from the table case? Why?

     • Does the situation change as you think about the board gradually becoming thicker? Why?

     • At what thickness does it become rigid?

8. Later discussion⁴

   As the discussion/conflict evolves, the following questions might be used to challenge those people who seem to believe the table pushes up.

   • Challenge: Where does the force of the table come from?

9. Vote #5 – Target problem again

   Repeat Vote #1: “Does the table exert an upward force on the book?” Comment that you are really interested in the makes sense score.

10. Springy atomic model for solids

    • Introduce a springy atomic model for solids (draw Figure 1.7).

    • Consider the solid table to be made of atoms connected by bonds that are somewhat like springs.

    • Ask the class how this view of a very solid table could explain the mechanism with which the table can exert an upward force. (Refer to anchor picture if necessary).

11. Demonstration (teacher stands on the table)

    Laser beam reflecting from a small mirror near the center of the lecture table deflects when the teacher steps near the mirror.⁵

12. Vote #6 – Final vote on the target problem

    Repeat Vote #1: “Does the table exert an upward force on the book?” (again)

13. Defining the Normal Force

    If the teacher is satisfied that most students have grasped the concept, then the upward perpendicular force of the table on the book can be identified as a Normal Force. See the Physicists’ View preceding this lesson.

14. Summary question

    Which of the diagrams on the board most helped you imagine that the table exerts an upward force on the book?⁶

15. Homework

    Pass out the homework for Day #1 with the attached Concept Diagram, Normal Forces Day #1.

1. It is suggested that this lesson is most exciting and effective if the teacher withholds his/ her position as long as possible (until the atomic model). Students are almost certain to have strong feelings on both sides of the struggle and these should be drawn out to make the class more exciting. Should some students complain that they really need to know the correct answer, just assure them that you plan to make your opinion very clear in the near future. If only one side seems to be represented, a bit of “devil’s advocacy” may be a useful strategy.

2. Should students complain that they don’t know what you really mean by “force,” assure them that they are asking an important question and the struggles in this lesson will help them become clearer about what physicists mean by “force.”

3. The definition of force given in this lesson (a push or a pull) suggests muscular force, and more generally contact forces. Some physicists may object that this is too limited a definition. But the most rigorous definition is not always the most useful for a beginner. We want to start by building on the learner’s intuitive ideas. There will be time for the concept of force to be refined and generalized. It would be useful to point out to students that this definition is an initial one, to be elaborated as their physics understanding grows.

4. If the equality of the forces arises during discussion on the first day, avoid it by saying you want to concentrate on whether the force exists and how to explain it.

5. Introducing the molecular model as a mental image also makes it available later for the study of collisions, friction, and tension. It is important to point out that this is only a model, and that all models have limitations.

6. The demonstration (see Figure 1.8) with the teacher standing on the table is fun and very helpful to some students. The teacher should practice this a bit, but the general setup in Figure 1.8 worked well for our classes. The downward deflection of the light will be greater if the mirror is placed between the side and the center of the table closer to the light source. Even a small deflection of the spot on the wall seems to impress students.

7. When the term “normal” is introduced, the teacher should emphasize the fact that this means perpendicular to the surface whether the surface is horizontal, vertical, or inclined. This should also be emphasized during the discussion of the homework problems.

Unlike the previous lesson, where we were mainly concerned with the existence of forces from static objects, the main objective of this lesson is to help students build the notion of equality of the forces exerted on one another by a pair of static objects in contact. Although these sources are also equal in accelerated situations, we deal with the static case here for simplicity and reserve the dynamic case for treatment in Unit 9. The target problem in this lesson is the “car pushing on the hydrant” problem (see concept map Figure 1.9): in which we ask students to compare the magnitudes of the force exerted by the car on the hydrant and the force exerted by the hydrant on the car. There is no motion. The anchor situation is a person pushing with the hand against a spring attached to a vertical wall with the same question of relative magnitude of the forces exerted by hand on spring and spring on hand.

An additional anchor (to be used if the first one doesn’t seem to convince some students) is the case of a cart with opposing forces pushing on each side. The goal is to have students understand and apply the notion that the object will move (accelerate) if the forces acting on it are not balanced.

As in Lesson 1, the strategy is to determine initially that most students have the correct concept for the case of the anchor(s), and then to bridge the conceptual gap between target and anchor(s) by suggesting several intermediate examples (the intermediate diagrams in Figure 1.9). Students are asked to make comparisons between these intermediate examples and the target and anchor(s), so that the students are able to extend the correct intuition in the anchor case to the target problem.

The closing portion of the lesson allows for completion of some in-class experiments with rubber bands of unequal strengths. Another optional extension uses magnets of unequal strength.

1. pairs of connected, unequal rubber bands for the entire class—include a metal ring on each end.

2. a bed spring and an automobile spring

3. pairs of magnets of equal mass but different strengths (optional section)

4. voting sheets

5. homework sheets – Normal Forces – Day #2

1. The student should understand that when any two static objects interact (push or pull on each other), the forces are equal in magnitude and opposite in direction.

2. The student should understand that the size of the force of interaction depends on the degree of deformation in the objects.

3. The student should understand that when two static objects interact the forces are equal and opposite even if the two objects deform by very different amounts.

1. Voting sheets

   Pass out voting sheets (perhaps a brief reminder here about the meaning of the makes sense score).

2. Homework discussion

   Discuss any matters of concern that the students have related to the homework problems from Day #1. A show of hands on questions 4 a) and 4 b) can be very informative for the teacher.

3. Generalization presented

   Try to establish the generalization below after reviewing the homework items.

   • “Matter is compressible (“squishy”) and touching objects (even passive ones) push on each other.”

4. Introduce the target problem

   • Draw the target problem of the car pushing on the hydrant.

   • Explain that the car slowly touches the hydrant and then pushes hard. Carefully explain the three choices below before the students vote. Point out the fact that the car is pushing hard, but there is no motion at the time we are comparing the forces.

   • Vote #1 – choices:

     Explain the choices:

     • F_{c on h} > F_{h on c}

     • F_{c on h} < F_{h on c}

     • F_{c on h} = F_{h on c}

5. Discussion of Vote #1

   • Share responses to the fire hydrant question.

   • Encourage input from students with differing views.

6. Present the anchor problem (pushing a spring)

   • Draw the picture of the anchor situation.

   • Demonstrate pushing on a board on a spring.

   • Think of the forces acting on the thin board between the hand and the bed spring. Compare the force of the hand on the board to that of the spring on the board. (Demonstrate if possible with a bed spring and a thin board.)

   • Discussion question:

     What would happen to the board if one force were larger?

7. Vote on a bridging example – Hand on spring without board

   • Demonstrate the hand pushing on the spring without a board. Explain that we are considering the force of the hand on the spring and the force of the spring on the hand.

   • Vote #2 – choices:

     Explain the choices

     • F_{h on s} > F_{s on h}

     • F_{h on s} < F_{s on h}

     • F_{h on s} = F_{s on h}

   • Discuss responses to Vote #2.

   • Challenge people voting for equality with the question:

     How does the spring “know” how hard to push back?

8. Introduce the automatic force equalizer

   • Introduce the idea of a spring as “an automatic adjustable force equalizer.”

   • Demonstrate pushing a stiff spring and a soft spring into each other. A bed spring and a truck spring will get the students’ attention.

   • Ask students to explain how the springs both adjust to exert equal and opposite forces.

9. Consider this optional example if some students are still having conceptual difficulties:

   • Question:

     What happens to a cart if you push on two sides with different forces?

   • Draw Figure 1.12.

10. Introduce the bridging examples

    • Consider a car pushing on a very strong rubber hydrant.

    • Draw Figure 1.13.

    • Vote #3 – choices:

      • F_{c on h} > F_{h on c}

      • F_{c on h} < F_{h on c}

      • F_{c on h} = F_{h on c}

    • Discussion questions:

      • How does this compare to the original problem?

      • If the car is stiffer than the hydrant, should it exert more force?

11. Another bridging example to offer (optional)

    • Consider a person pushing on a wall.

      Ask, “How does the wall ‘know’ how hard to push back?”

    • Offer the comparison with the hand pushing on the spring.

    • Review the “automatic adjustable force equalizer” concept.

12. Introduce an experiment to verify the “automatic adjustable force equalizer” concept ⁷

    • Demonstrate the rubber band apparatus. (No motion is allowed for testing the rubber bands). Explain that one student will close his/her eyes and the other student will attach the stretched apparatus to the outstretched fingers of the first.

    • Vote #4 (precedes the experiment) – question:

      Which of my hands will feel the greater force, the strong band or the weak band?

    • Choices:

      • strong > weak

      • strong < weak

      • strong = weak

13. Class experiment activity (rubber bands)

    • Pass out one rubber band setup to each pair of students. Have them test the static case with a partner’s eyes closed.

    • Take turns and continue until most students get the same result (only a few minutes).

    • Collect the equipment.

    • Question: How is it possible for the “strong” band and the “weak” band to produce equal forces? How does each work?

14. Generalization

    • Ask for a generalization about how stretching things work when connected.

    • Consider the situation below if another concrete example is needed.

    • Thought experiment question about Figure 1.16:

      Which is larger? The force of the hand on the stiff spring or the force of the wall on the soft spring?

    • Point out that springs work the same way for compression.

15. Formal presentation of a generalization ⁸

    Try to draw out a generalization of the static third law. List the previous situations covered. We’ve been looking at a number of situations. What do these situations have in common? Can someone give a general statement concerning the forces between objects that interact with each other?

    “F_{A on B} = – F_{B on A}” (explain notation as needed).

16. Vote #5 – The target problem again

    • Repeat Vote #1 – The car pushing the metal hydrant

17. Homework

    Assign Homework – Normal Forces – Day #2.

18. Force at a distance example (optional)

    Two magnets of unequal strength but similar size and weight are needed.

    • Now we consider a force that works at a distance.

    • Demonstrate to the class that one magnet is much stronger than the other. (One will pick up many paper clips).

    • Vote #6 – (comparing strong and weak magnets) question:

      Which hand will feel the greater force?

      • S > W

      • W > S

      • S = W

    • Experiment (optional): Pass out pairs of magnets to groups of students. Have each student try to say which is stronger while keeping her/his eyes closed and moving the magnets near each other. Magnets should have similar size and weight.

      -or-

      Demonstrate with a volunteer student, if you have only one set of magnets.

    • If time permits, introduce the baby magnet model. Students should be willing to accept the notion that one can identify a smallest unit of magnetism. Note equal number of lines each way leads us to expect equal forces (we assume baby magnets are equal in strength.)

The first part of this lesson should be a brief review of the homework and summary of last class drawn into a generalization. Try not to get bogged down here, for there are many good ideas suggested in this lesson.

The major thrust of the lesson is concerned with the equality of the contact forces. Some students will find a line of argument that “forces must be equal because unequal forces would produce motion” to be rather unconvincing. Perhaps a careful discussion of the forces exerted by two rubber bands of different stiffness cemented together or two different springs pushing together will be more helpful for these students. The teacher’s style can be quite flexible in the main body of the lesson, encouraging students to introduce bridges or offering bridges from the lesson plan that seem the most helpful. We would, however, still urge the teacher to withhold his/her professional position on the answer until very near the end of the lesson.

The teacher should carefully avoid being drawn into a discussion of the interactions between bodies in the dynamic case. If students are interested, assure them that the case for interacting forces during motion will be carefully considered later in the course. Try to stay focused on the static case for the purposes of this lesson. It is clearly not the intent of these lessons to deal with the third law in the dynamic case. The third law in constant velocity and acceleration cases should follow the study of inertia. That issue is the main focus of Unit 9.

The notion of a spring as an “automatic adjustable force equalizer” seems to be a helpful way of getting students to confront one of our major objectives. It is probably easiest to see a spring under tension or compression adjust its length so that the force of the spring balances the force of the outside agent acting on the spring.

If students are expected to write answers to the homework questions with emphasis on the mechanisms involved, this assignment will probably be more helpful to them. They should be urged to think in terms of springs while remembering all matter is made of atoms with spring-like bonds.

The homework problems include some questions with unequal forces—partly to keep students worried that the correct answers about forces are not always “equal.” In other words we want them to keep thinking and analyze each situation. Problem 9 in the homework is placed at the end intentionally as it raises issues that the teacher may wish to avoid at this time. It may be included if the optional section about magnets was included as part of class discussion.

The rubber band apparatus in Figure 1.15 is a bit simplified. A small metal key ring or a ring made from a paper clip must be added to each end so the student cannot feel which band is thicker.

This experiment provides an important opportunity for students to handle springs and take measurements of spring elongation (or compression) and applied force. An extension of the simple spring experiment involves connecting two springs, of differing spring constant, in series to find the behavior of the combined spring system.

In order to extend students’ reflection in this area, the lab may be extended to examine rubber bands, which are non-linear in behavior. We also suggest the section on spring compression, although the apparatus needed for this part is more difficult to arrange.

The teacher may elect to have students measure force with a spring scale (force measurer) calibrated in newtons or use masses and tell students to assume 100 g masses weigh 1 Newton each. In any case students need to become familiar with the Newton as a unit of force.

This experiment calls for students to plot elongation vs. force (as in Y vs. X) rather than force vs. elongation as is normally done. At this stage in the physics course many students think of force as the cause of the elongation and thus we suggest force as the independent variable for this early lab. Some teachers may feel they would rather introduce the graph in the F = –kx form to be used later in the course.

If the class has not been exposed to the term “elongation,” it will be necessary to be sure there is a clear understanding about the difference between elongation and length.

1. ring stands (one per group)

2. rings (from which to suspend springs)

3. two springs per group (different K values)

4. spring scale (one per group) (range depends on spring constant)

5. string

6. rubber bands (two different per group)

7. spring compression apparatus (if available)

1. Introduction to Springs Laboratory Experiment

1. Normal Forces – Day #1

2. Normal Forces –Day #2

Name: __________

Period: __________ Date: __________

1. Apply different amounts of force to your spring (using the Newton scale or masses) and measure the resulting lengths. Record your data in a neat and properly labeled data table and include a column for elongation. The elongation of the spring is the change in length as compared to the original or natural length with no force applied.

2. Plot the elongation (change in length) of the spring vs. the applied force in newtons. Collect data for the three cases below. Plot the three lines on one graph.

   • weak spring

   • strong spring

   • combination of both springs (in series)

3. Questions

   • Using your graph, compare the elongations of the weak spring, the strong spring, and the combined springs when the same force is applied.

   • If the springs in case a, b, and c are each elongated by the same amount, which spring would exert the greatest force on your hand? Which spring would exert the least force on your hand?

   • Do your graphed points produce lines or curves? Do they go through the origin? Explain why or why not.

   • If someone were to plot the length of the entire spring instead of the change in length of the spring (elongation)—how would this graph be different from graphs that were plotted correctly?

1. Collect and neatly tabulate the same data for the rubber band cases.

   • weak rubber band

   • strong rubber band

   • combination of both rubber bands (in series)

2. Plot the elongations vs. force for the rubber bands again including three cases on one graph.

3. Questions:

   • Do the rubber bands show the same pattern as the spring graphs? Explain.

   • Would a strong rubber band be a fair substitute for the spring in your spring scale? Why or why not?^*

1. Apply different amounts of force to your compression spring, and tabulate your data.

2. Plot the compression, (i.e. change in length) of the spring in millimeters vs. the applied force in newtons producing compression.

3. Questions:

   • If the compression spring were the same as one of your elongation springs, how would you expect the slopes of the two graphs to compare?

   • Predict and sketch what you might expect to find if you obtained and plotted the data in a compression experiment for a:

     • weak spring

     • strong spring

     • combination of two springs

   • What do you think there is in the internal structure of a spring (or rubber band) that allows it to behave in the manner you discovered in lab?

Name: __________

Period: __________ Date: __________

Name: __________

Period: __________ Date: __________