Authors: Peter Anderson, Wade Goodridge, Sarah Lopez, Natalie Shaheen, Ann Cunningham

## Class Size

- This lesson was originally written for 15 students.

## Lesson Structure

90 minutes

- Introduction to Force: 10 minutes
- Vectors Add, Subtract, Components: 65 minutes
- Newton’s 3rd Law: 5 minutes
- Tension and Compression Introduction: 10 minutes

## Objectives

Students will be able to:

- Identify force as a vector with magnitude and direction.
- Classify forces as balanced or unbalanced.
- Identify a diagonal vector as composed of x- and y-components.
- Calculate the x- and y-components of a vector given the magnitude and angle from horizontal using sine and cosine functions.
- Add and subtract vector components in the x-direction and y-direction.

## Prerequisite Knowledge

- Familiarity with the XY coordinate plane
- Familiarity with using a protractor, ruler, and T-square

## Accessibility

- Spring scale with tactile labels
- Tactile protractor
- Accessible calculator with sine and cosine functions
- Sensational BlackBoard

## Materials

- Geoboards, 5” x 5” - 1 per student
- Rubber bands, size variety - 3 per student
- Spring scale (fish scale), 1,000 N (10 lbs.) with tactile labels - 1 per 3 students
- Could also use a spring loaded kitchen scale

- Sensational BlackBoard - 1 per student
- Pen, ball point 0.5 mm - 1 per student
- Ruler, Braille - 1 per student
- Protractor, tactile - 1 per student
- Rope, thick, for tug-of-war, ~40 feet long - 1 (2nd rope optional)
- Calculator, accessible with sine and cosine functions - 1 per student
- Handout: Vector Practice (PDF for hardcopy production only) showing a 6” vector at 45°, on a coordinate plane
- T-square notched at 1” intervals - 1 per student
- Construction foam, blue, cut to 4” x 4” (or scrap) - 1 per student

*Note: Refer to Accessible Lab Equipment & Instructional Materials for additional information regarding specialized tools/materials. *

## Preparation

- Cut construction foam into ~4” x ~4” pieces or collect scraps of construction foam.

## Formulas

$$$\mathrm{Vector\_X}=\mathrm{Vector\_Length}\cdot cos(\mathrm{Vector\_Angle})$

$\mathrm{Vector\_Y}=\mathrm{Vector\_Length}\cdot {\mathrm{sin}\left(\mathrm{Vector\_Angle}\right)}^{\mathrm{}}$

## Procedure

### Introduction to Force: 10 Minutes

**Welcome students and introduce topic.**- Tell. “Welcome everyone, in this lesson we will be calculating vectors inside different pieces of your structure to describe the effect of an applied force. In other words, we’ll be figuring out how a structure can stand up and stay standing when there is weight on it. This will help us understand how forces balance throughout our structure which allows us to design the structure so that it won’t collapse.”

**Introduce force.**- Tell. “Force is a push or pull acting on an object. You can push hard or soft and in any number of directions. Sometimes this push or pull can cause movement, such as if you were to push your chair back from the table, but other times there may be no movement, such as if you pushed with all your might against the wall.”

**Units.**- Tell. “We measure force in units of pounds (lbs.) or newtons (N). This describes how hard something is being pushed or pulled. Note that we also measure weight in pounds, because weight is actually just a force describing how much your body is pushing down on the earth due to gravity. Likewise, the weight of a building is a force it exerts on the ground. For reference, a pound is about the weight of a package of spaghetti or a can of beans.”

**Pull on scale.**- Do. Pass out the spring scales. Direct students to pull on the hook of the spring scale softly, medium, and hard.

**Explain.**- Tell. “This scale is designed to measure a pulling force; in particular, how much the weight of a fish would pull the scale down if the fish was hanging on the hook. As you pull on the hook, you can feel the indicator move to show how much pulling force you are exerting on the hook. Can you pull on the hook with exactly 2 pounds? What about 5 pounds? What is the maximum amount these scales can measure?”
*Note: if using spring-driven kitchen scales instead of fish scales, modify this description appropriately. (Pushing instead of pulling, etc.)*

**Forces that balance.**- Tell. “As we noted earlier, sometimes forces cause movement, and sometimes they don’t. When a force doesn’t cause movement, it is because there is another force working to cancel it out. For example, when your weight pushes against the floor, you don’t fall through the floor because the floor is pushing back up against you. Or, when you pulled on the scale with exactly 2 pounds, the scale wasn’t moving, because the spring inside it was pulling back against you with an opposite 2 pounds of force. We’ll talk more about forces that cancel in just a bit.”

**Review results.**- Tell. “So, a force can be a push or a pull and it can go in different directions. Force can also be of different magnitude or amount. A light push has less force than a hard push, so the magnitude is smaller. Because a force has both a direction and a magnitude, it is called a vector”

### Vectors: Add, Subtract, Components: 65 Minutes

**Vectors.**- Tell. “A vector is an idea in math that is used to describe any quantity that has both a magnitude and a direction. When we simplify a force down to just direction and magnitude, we can use vectors to describe, draw, add, and subtract forces.”

**Vector arrow.**- Tell. “We use arrows to draw force vectors. The arrowhead tells you the direction of the force. The length of the arrow tail shows you the size. We’re going to use rubber bands to draw these forces on a geoboard. You can ‘draw’ a vector by looping the rubber band around the peg where you want the vector to start, and then stretching and hooking it on a peg where you want the vector to end.”

**Geoboard how to.**- Tell. “A geoboard is a 5” x 5” grid of short pegs that you can stretch rubber bands on to.”

**Vector exercises.**- Do. Pass out the geoboards, rubber bands, rulers, and protractors. Lead the students through these exercises, checking their work after each one.
- Teach. Starting anywhere, draw a vector going to the right of size 2. The 2 can represent 2 pounds or 2 newtons.
- Teach. Make a vector going up of size 4. You will need to start at the bottom to fit it.
- Teach. Make a vector that goes 3 units right and 3 units up on a diagonal angle. How long is this vector? (Have students measure with a ruler. Should be approximately 4.2 units.) What angle is this vector pointing at, relative to horizontal? (Help students use protractors, and measure up from horizontal. Should be 45°.)
- Teach. Make a vector going down 3 units and left 3 units, starting in the upper right. How long is this vector? (Still 4.2.) What angle is this vector? (225°. Stop and explain how to get to 225. 180 to get from the positive x-axis around to the negative x-axis, and then another 45 to get down to the vector.)

**Introduce force addition and subtraction.**- Tell. “When 2 forces go in the same direction and operate on the same object, they add together. Or, when they go in the opposite direction, they can cancel out. To figure out how much total force will be left, we add the vectors up, using negative numbers for opposite directions.”

**Introduce the tug-of-war.**- Tell. “We are going to play tug-of-war a few different ways to feel how vector addition and subtraction works.”

**Conduct the activity.**- Do. Direct 2 students to grab onto each end of the tug-of-war rope. Have a third student (or yourself, as the instructor) stand with their hand passively on the middle of the rope to narrate what is happening to the rope.
- Do. Direct only one person to pull. The center narrator should tell that the rope moves towards that person. Have seated students draw a vector to the right of size 1 on their geoboards.
- Do. Have the second “puller” (from the other side of the rope) stand with the first puller and pull in the same direction. The rope moves towards them more.
- Make a vector to the right of size 1.
- Starting from the end (right side) of the first vector, make a second vector to the right of size 1. This is called arranging the vectors “head to tail.” Arranging vectors this way will show the sum of the vectors. The total force can be seen as a vector going from the beginning of the first vector to the end of the last vector.

- Do. Have the second puller return to their side, and direct the pullers to try to pull in opposite directions with equal force. Count to 3 before pulling. Center of rope stays (approximately) still.
- Draw a vector to the right of size 1.
- Starting from the arrow end of the first vector, draw a vector to the left of size 1.
- Ask. “Where did the end of the second vector end up?”
*Back where the first started.*Explain. “This is called cancelling out or being balanced. Our 2 volunteers’ forces cancel out.”

- Do. Add the rest of the students to the 2 ends of the rope, and have them pull. This can be a methodical, controlled process of adding 1 student at a time, and observing what happens to the middle each time, or you can use it as a fun break to get students moving and have them go all in on a traditional round of tug-of-war. Either way, conclude with a discussion of how the rope moving means that there is more force on 1 side than the other, but when the rope is still, the forces are balanced.

- Do. Direct 2 students to grab onto each end of the tug-of-war rope. Have a third student (or yourself, as the instructor) stand with their hand passively on the middle of the rope to narrate what is happening to the rope.
**Optional extension: 3-way tug-of-war.**- Do. If you want to explore the tug-of-war analogy further with the class, you can tie a second rope to the middle of the first one to create 3 branches pulling away from each other. Start with 1 student pulling on each branch, and someone at the central knot to narrate what is happening there. Demonstrate how the angle of the branches affects how the forces add together, by having 2 students move towards each other (but not exactly together) and pull against the third. Then have 1 of them change their angle to move towards the third student, and pull against the first. Have students spread out equally, and try to pull with equal force. Point out that although no 2 forces are directly canceling each other, because they are not going in exact opposite directions, the addition of all 3 forces is canceling out.

**Cleanup the tug-of-war.**- Do. Coil and stow the rope out of the way.

**Vertical forces.**- Tell. “Forces don’t always go just to the right or left, sometimes they go up and down too, like the force of you on the chair you are sitting in and the chair pushing back on you. They add together the same way.”

**Combine vertical and horizontal forces.**- Tell. “Forces don’t always go one direction. Sometimes a vertical force can combine with a horizontal force. To combine the vectors, draw the vectors head to tail, then make a new vector that connects the first tail to the last head. This new vector is the total force and is called the resultant.”

**Resultant diagram.**- Teach. This first diagram students create should show 2 vectors and a resultant.
- Make a vector of size 3 to the right.
- Starting from the right end of the first rubber band, make a vector of size 4 going up.
- Ask students the size of the resultant. Have students make the resultant by stretching a rubber band from the left end of the first vector to the top of the second one. Have students measure the resultant with a ruler.

- Teach. This first diagram students create should show 2 vectors and a resultant.
**Review section.**- Tell. “So we know what forces are and how they can combine to make motion or not motion. We also know that the same force can be created by 1 rope pulling an object in 1 direction, or multiple ropes pulling at different angles so that the forces add up to the same magnitude and direction. In either situation, the effect on the object will be exactly the same.”

**Reality check.**- Ask. “Do any of you want your structure to move or flex in the wind? Would you want to be in a building that was moving?” No.
- Tell. “Most buildings are built so as not to move when forces are applied. To make this work, the building needs to be strong enough to cancel out any forces that push or pull on it.”

**Diagonal vectors.**- Tell. “We have talked about adding vectors to get a resultant but now we will consider the reverse situation, where we will start with a resultant, and split it into separate pieces. First, make a vector starting in the lower left going up 1 and to the right by 2.”

**Components defined.**- Tell. “We know that this vector can be made of 2 vectors, one of size 2 to the right and one of size 1 going up. These pieces of the vector that only go sideways or only go up-and-down are called components. The horizontal one is called the x-component and the vertical direction is called the y-component.”

**Calculating components.**- Tell. “For a diagonal vector at a known angle, measured from the horizontal, we can figure out its x-component with a function (button on calculator) called cosine (cos) and its y-component with a function called sine (sin).”

**Using the calculator.**- Tell. “While this is an easy calculation for a calculator, we have to be careful to type it in correctly. We have to find the sin or cos of the angle, and then multiply that number by the vector’s length.”

**Calculate components of a vector.**- Tell. “For example, suppose we want to calculate the x- and y-components of a vector that is 5 units long at a 30° angle. We’ll calculate the y-component first, so we will be using the sine function.”
- Teach.
- Y component. Direct students to calculate sin(30°)*5 with the following instructions. If students are getting odd results, make sure calculators are in ‘degree’ mode, not ‘radian’ mode.
- If students are using standard single-function calculators, tell, “Type in ‘30,’ ‘sin’, ‘x,’ and ‘5.’” Say the answer. 2.5.
- If students are using algebraic calculators, tell, “Type ‘sin,’ ‘30,’ ‘),’ ‘x,’ and ‘5.’” Say the answer. 2.5.

- X component. Direct students to calculate cos(30)*5 as above. Use instructions specific to the type of calculators students are using. Say the answer. 4.3 (roughly).

- Y component. Direct students to calculate sin(30°)*5 with the following instructions. If students are getting odd results, make sure calculators are in ‘degree’ mode, not ‘radian’ mode.

**Why we would do this.**- Tell. “This is an important tool because adding diagonal vectors together requires either very precise drawing and measuring or complicated geometry. If we can use only x- or y-vectors instead, it becomes very easy to add them.”

**Sample problem: Measure example.**- Do. Pass out <vector handout> with 6” vector.
- Ask. “What is the size and angle of this vector?”
- Students Do. Have students measure the length and angle of the vector on their handout. Should be 6” long at a 45° angle.

**Sample problem: Determining components with math.**- Ask. “If a vector goes up 6 units at 45°, what are its x- and y-components?”
- Students Do. Have students calculate sin(45)*6 and cos(45)*6. (Both are about 4.24.)

**Sample problem: Determining components with the graph.**- Tell. “Measure a straight line down from the vector’s head to the x-axis using the T-square ruler. Measure how much X it covers.” About 4¼.
- Tell. “Draw a straight line left from the vector’s head to the y-axis with the T-square ruler. Measure how much Y it covers.” About 4¼.

**Comparing.**- Ask. “How do these two results compare?” About the same. “Which method was easier for you? Which method do you think was more accurate?”

**Review results.**- Tell. “We can always break a diagonal force down to 2 components that operate with the same net effect, and this is sometimes very convenient when we want to add 2 forces together and they are not on the same line.”

### Newton’s 3rd Law: 5 Minutes

**Action/Reaction forces.**- Tell. “So, to complicate matters, forces always occur in pairs. We call one the action force and one the reaction force. When you push on something, it also pushes back on you. For example, if you are pushing a heavy box across the floor, the box is pushing back on you, and squishing your fingers a bit.”

**Find reaction forces.**- Tell. “Push down on your table. You are exerting an action force on the table. What do you think is the reaction force?” The table is pushing back on you. “Assuming you didn’t break the table, the action force and the reaction force are canceling out, and the table doesn’t move. You can usually find the reaction force by thinking about where the action force is coming from and going to (e.g., FROM you TO the table) and then switching the directions (e.g., FROM the table TO you).”

**Brief review.**- Tell. “If you push down on the table, the table must push up on you. If you stand on the floor, the floor must push up against you. Typically, you only notice these forces when either they can’t hold you, like stepping in mud, or the floor is flexible, like a trampoline. In design, an engineer will determine what forces are at play, so that they can design the structure to counter them. Statics is the field of study for situations where all the forces balance each other out.”

**Equilibrium.**- Tell. “When all of the forces on an object cancel each other out, we say that the object is in equilibrium. This happened when there was a stalemate in tug-of-war, or when you pushed on the table but it didn’t move. We can describe this physically by saying that if the forces balance out, the object will not move (or, technically, if it is already moving, it will not change velocity). We can also describe this with math by writing an equation that shows that the sum of all the forces is equal to 0. This kind of equation is called an equilibrium equation, and is used very frequently in structural engineering. You will see this again in a few more lessons.”

### Tensions and Compression Introduction: 10 Minutes

**Tension and compression.**- Tell. “There is 1 more way we categorize force. When a force is applied to an object, it will resist that force in 1 of 2 ways, tension or compression. Tension is when an object is resisting being pulled apart. For example, the rope we just used was being pulled in 2 directions away from its center, so it was in tension. If an object is resisting being squished or crushed, it is in compression.”

**Compression foam.**- Do. Pass out the squares of construction foam. Direct students to push on either side of it with a fist lightly at first.
- Tell. “When you push on it, the foam is under compression. It feels 2 forces squeezing it towards its center, 1 from your hand and 1 from the table.”

- Do. Pass out the squares of construction foam. Direct students to push on either side of it with a fist lightly at first.
**Compression failure.**- Do. Direct students to push harder until the foam crushes.
- Tell. “Every material has a point at which it can’t resist the force anymore. If it is a compression force, it will deflect or bend, then crush. If it is a tension force, it will deflect or get longer and narrower, then snap. These deflections can be useful to figure out which pieces are under which type of forces. Different materials react differently to compression and tension, so engineers have to consider this carefully when choosing materials.”

**Review structural failure.**- Ask. “Does anyone want their house to break?”
*No*. “Does anyone want their house to move?”*No*. - Tell. “Typically, we don’t want the parts of our house to move very much. When beams move, they can break, or can cause other parts of the house to break. In the next lesson, we will start to calculate what forces your structure will need to resist in order not to break.”

- Ask. “Does anyone want their house to break?”

## Standards Alignment

*NGSS Standards Alignment:*

- SEP 5 - Using Mathematics and Computational Thinking
- CCC 2 - Cause and Effect: Mechanism and Explanation
- HS-ETS1-2

*Standards Alignment:*

- CC.9-10.R.ST.3, CC.9-10.R.ST.4, CC.9-10.R.ST.5, CC.9-10.R.ST.7, CC.11-12.R.ST.3, CC.11-12.R.ST.4, CC.11-12.R.ST.5, CC.11-12.R.ST.7
- CC.9-12.N.VM.1, CC.9-12.N.VM.2, CC.9-12.N.VM.3, CC.9-12.N.VM.4, CC.9-12.SR.T.8