Vectors and Forces

Introduction

There are two kinds of numbers we use in everyday life (although you may not realize it): scalars and vectors. A scalar is just a plain number – 6 apples, $4.50, 100 miles. A vector is a number that has a direction, and probably is easiest to think of in terms of driving – 100 miles north, or 60mph east on 64/40 (If you just said “100 miles” or “60mph”, those would be scalars.).

Scalar addition is comfortable and familiar, but vector addition is probably kind of new. An easy way to see how vectors add is to look at forces.

Think of a ball resting on the floor, with a string attached to it. If you pull on the string (or, in physics terms, put a tension force, T, on the string), the ball comes toward you (physics: accelerates, a).

But if you have a friend who grabs a second string on the same ball, as long as your friend pulls in the opposite direction with the same strength (tension, T), the ball won’t move. The forces are equal in strength (magnitude) and opposite in direction, so they balance and the acceleration is zero. We can use equations to describe this:

(Tyou – Tfriend) = Ftotal = m × a = 0

This is the basic idea behind what you’ll be doing today. Since there’s only so much you can do with one dimension, we’re going to expand things and work with two. The basic equation is exactly the same, but there are always going to be two separate calculations, one for each dimension.

Materials

Table with paper on it

3 spring scales

4 weights attached to strings that can be hooked onto the scales

Colored pencils (3 or 4 colors)

Protractors

Calculator

Methods

A. The simplest case:

1. Hook two scales together, and attach a weight right to the middle of the scales.

2. With a partner, hold the scales so that the weight is right over the star in the middle of the paper.

3. Pull on the scales, always keeping the weight right over the star, and take turns reading the weight that your scale registers. Then, both of you pull a little harder (but keep the weight still) and read out the weights on your scales.

4. What do you notice about the weights?

B. What if you add acceleration?

1. Hook two scales together.

2. With a partner, hold the scales steady so they both read (***).

3. While still pulling on the scales, one of you should pull harder so the scales accelerate (go from holding still to moving towards the person who’s pulling harder). Right at the beginning of the acceleration, both of you should pay attention to what number your scale reads.

4. Whose scale has the bigger number?

5. Diagram the forces at the middle of the two scales here. Draw the arrows with different sizes to represent the different forces from each of the two scales. Make sure the arrows point in the direction of the force.

C. Ready for two dimensions!

1. Find a third partner with a scale, and a fourth partner with a pencil.

2. Hook all three scales together, and add hanging weights from the middle of the three scales (where they all hook together), as well as one from each scale near where you’re holding it.

3. One person should stand at the head of the table, another should stand at the side of the table (that is, 90 degrees from the first person), and the third should adjust their location and strength of pull so that the center weight holds still over the star on the table.

4. The person with the pencil should now mark exactly where the center weight hands, as well as where each of the 3 individual scale weights are hanging. The people holding scales should notice what their scale registers.

5. The table will look something like this: Draw lines and write down weights so it looks like this:

6. Draw axes on the table. They should be perpendicular to each other (90º) and should run parallel to the two forces that are 90 degrees apart. Name your axes (X and Y, perhaps?).

7. Measure the angle from the axes of the third force, and split the forces into X and Y components using trigonometry. Get someone to help you if it’s confusing.

8. Add the forces together. Do they add to zero? If not, look back and check your work.

D. Predicting an unknown.

1. Again, hook all three scales together with weights like you did in C. Stand around the table, again with two people 90 degrees away from each other and the third wherever they have to be to balance the forces, and hold the center weight still over the star.

2. Everyone look at your scale weight, but DO NOT share it with your partners! (Also, don’t forget it…)

3. The fourth partner should mark down the positions of the weights with dots, as in C.

4. Choose your axes.

5. Split your forces into components. The person who marked down the positions of the weights should team up with the third person (whose force doesn’t run parallel to one of the axes) to figure out that force’s X and Y components. The two people whose forces run along the axes should team up together.

6. Can each team predict the other team’s forces? Think about what you would need in each direction to balance out the force and make everything add to zero.

7. The two teams should compare predicted and actual forces. Did it work?

Discussion Question

In D, you all balanced your forces so they added to zero. If the person whose forces ended up right along an axis had suddenly pulled hard, startling everyone, and made the center weight accelerate, would the forces along that axis have added to a positive number, a negative number, or zero? What about the axis perpendicular to that person’s force (+, - or 0?)?

Assessment Question:

Break forces A, B and C into components based on the axes drawn. Calculate and draw the X and Y components of the force that’s required to balance out forces A, B and C. Figure out the magnitude of the balancing force.