Tag Archives: vectors

Physics 3 Using Vectors To Show Forces

You are asked to join a game of tug of war by one friend. Each of you grabs an end of the rope and starts pulling. Neither of you can pull the other one.

Another friend comes over and grabs the rope with your friend. What happens?

At first the forces you and your friend exert on the rope are the same and opposite. The result is no force.

Let’s show this using vectors.

using vectors to show tug of war forces

A vector is an arrow pointing in the direction the force is going. In tug of war the forces go away from each other as both sides are pulling on the rope.

When the third person starts pulling, one force stays the same. The other force doubles. Can vectors show this?

using vectors to add forces

There is one vector arrow for each tug of war participant. Each arrow points the way that person is pulling. Two of the arrows are equal and opposite cancelling each other out. That leaves one arrow to show what happens in the game.

When you put the vectors together, they show what happened in your game of tug of war.

 

What Is a Vector?

As you can see, a vector is an arrow. The arrow shaft shows the amount of the force. You can do this with labels or drawing to scale.

The head of the arrow shows the direction the force is acting in.

 

Let’s Draw some Vectors

Materials:

Paper or Graph paper

Pencil

Ruler

Drawing the Vectors:

Draw your block from Project 1

From the bottom center of the block draw a line 2 cm long straight down

using vectors to show gravity

Every object on Earth has this vector arrow pointing down. If that force didn’t exist, everything would float into space.

Put the head of the arrow on the end going down

What does this arrow show? What force is holding the block on the table?

You made the block move by pushing on it. Draw a line to the side of the block. How long should the line be?

using vectors gives information about forces

Using vector arrows for forces makes it easy to see how strong a force is, which direction it is going and how it is acting on an object.

The block moved. Think back to the game of tug of war. As long as the two forces were the same, the forces cancelled each other out.

If the force line showing you pushing the block is shorter than the gravity arrow, will the block move?

Your push vector must be longer than your gravity vector. Let’s make it 3 cm long. Now put the head of the arrow on.

Where does this go? Which way did the force push? It pushed against the block so the head of the arrow points at the block.

Can you draw the pulling force on the block using vectors? Try it. My drawing will be down below.

 

Using Vectors for Changing Forces

Put your block out on the table. This time push on the block from two adjacent sides at the same time.

two forces pushing block

Gravity always pulls on objects on Earth. This time we are interested in two forces pushing on two adjacent sides of the block at the same time. Which way will the block move? Can vectors be used to tell us?

Which way did the block go?

Draw your block on the graph paper again. This time have two pushing forces on adjacent sides on the block.

Now, copy one of the vectors from the tip of the opposite corner. Make sure it points in the same direction and is the same length.

using vectors to show movement

The vector arrows are the same ones as before only moved. they point the same way and are the same length. Moving the vectors lets you draw a resultant vector showing the actual path the block took when acted on by two forces.

Next copy the other vector with the end starting at the point of the other vector. Make sure it points in the same direction and is the same length.

If you draw a final vector from the point of the block to the point of the last vector, you have the direction the block moved when you pushed on it with two forces.

using vectors to show pulling

The vector arrow for gravity still points down. The other vector arrow now points away from the block as you were pulling on it.

Why Use Vectors?

You can see the forces acting on your block, right? You can see some of them but not others.

Using vectors makes what is happening easier to see. As we go on to look at work and simple machines, we will often use vectors to better understand what the forces are doing.

Physics 21 Balancing Forces

Many Projects ago we defined a force as a push or a pull. We found forces could add to or subtract from each other. This Project we will try balancing forces so an object does not move when pulled by three forces at the same time in different directions.

materials needed for project

Question: How do forces balance?

Materials:

Metal or other rigid ring

3 Spring scales [You can use three identical rubber bands but will not be able to measure the forces instead measure the rubber bands]

Protractor

Procedure:

This Project works well with friends to help. If you are working alone like I do, you will need tape to fasten the scales in place.

spring scale

A simple spring scale has a pull strip to zero the scale and two scales. One is in grams for obtaining mass. The other is in Newtons, a unit of force.

Secure one scale to the table

Attach the secured scale and another scale to the ring

Pull on the second scale until it reads the same force as the first scale and  stops moving around the ring balancing forces

Record the measurements on the two scales

Note: I have three spring scales that measure forces in three levels of magnitude. If this is how your scales are, check your measurements carefully as each scale will look different.

Secure the second scale to the table

Attach the third scale to the ring

Pull on the third scale until it has the same force reading as the other two scales again balancing forces

Pull the third scale a little more and record the forces shown on the three scales

Secure the scale to the table

Use the protractor to measure the angles between the scales

Observations:

Record the force you use on your scales (or length of rubber bands):

Draw out where the two scales are on the ring

Draw out where the three scales are on the ring

Measurement of angles between the three scales

Record what happens to the three scales when you pull harder on one

Conclusions:

Do you think you could move one of the two scales pulling on the ring so they were not opposite and still balance the forces? Why do you think this?

balancing forces with two scales

Even if the second scale is next to the first one when hooked to the ring, it will shift to opposite the scale balancing forces with equal and opposite vectors.

If you used vectors to show the forces of the two scales, would they be the same length? Would they point the same direction?

What happened when you started pulling with the third scale?

What happened when you pulled harder on one scale?

What do you think would happen if you pulled on the ring with a fourth scale? Try it and find out.

 

What I Found Out:

All three of my scales used a gram and a Newton scale. All three would register .5 Newtons so I decided to use this amount of force.

I attached one scale to the table, put the ring on the hook and put the hook of another scale on the ring. As soon as I started pulling with the second scale, the hook slid until the two scales were exactly opposite of each other. The scales had to be opposite each other for the forces to balance each other so I couldn’t move one scale.

If I drew vectors to show the forces, the arrows would be the same length because both scales pulled with the same force. The arrows would point in opposite directions.

balancing forces from three scales

Adding a third scale causes the ring to shift. Having the same force with each scale balances the forces. Putting a ruler from the end of one scale to the next will create an equilateral triangle.

The second scale was attached to the table. I put the third scale’s hook on the ring and started to pull it. The hook slid around the ring until the three scales were the same distance apart around the ring.

Whatever force I pulled with on the third scale, the other two scales showed the same force.

Physics 15 Projectile Motion

There’s straight line motion. Our balls and cars have shown us a little about it.

There’s pendulum motion. Nuts on strings showed us about this.

There’s circular motion. We used a nut on a string to find out a little about it.

There’s harmonic motion. A Slinky and springs showed us a little about this.

One last type of motion is projectile motion.

What happens if you throw a ball straight up?

If you don’t dodge, it will come straight down and hit you. Why?

If you throw a ball across a room, it curves down to the floor. Why?

 

Question: How does projectile motion work?

Materials:

Ball

Stopwatch

Paper

Pencil

Procedure:

Toss the ball up from your hand

tossing a ball diagram

The ball leaves the hand, goes up then comes back down into the hand.

Observe how the ball goes up and down

Catch it when it returns to your hand

If you have a friend to help, have your friend gently toss the ball across a space

Observe how the ball moves

thrown ball diagram

Throwing a ball or other projectile gives it an arch shaped path.

Play catch outside with your friend

Start with gentle tosses and gradually throw the ball harder

Observe how the motion of the ball changes as you throw it harder

Get your stopwatch ready. Stand ready to throw your ball straight up.

Start the stopwatch and throw your ball straight up as hard as you can [It helps to have a friend help with this.]

Stop the stopwatch when the ball hits the ground

Repeat this only if you did not get the stopwatch stopped on time

Observations:

Draw your ball going up and down one time

Describe how your ball goes up and down

Draw your ball going across a space

Describe the motion of your ball

Describe how the motion of your ball changes as the throws get harder

Time for your ball to go up and down:

Conclusions:

What makes your ball go up?

Newton’s First Law of Motion says an object in motion will continue that motion unless acted on by another force. What force keeps your ball from going up forever?

Why do you think the projectile motion of your ball changes as you change how you throw it?

vectors for a thrown ball

Throwing a ball is more like the projectile motion people think of because the ball goes over a distance. At the beginning most of the force pushes the ball up, some goes sideways and gravity pulls down. At the top of the arch there is no more force pushing the ball up but it still has force pushing it sideways and gravity pulls it down. When the ball lands, only gravity is still pulling on the ball.

Draw your ball going across a space. Show your ball at the beginning, middle and end of the toss.

Add vectors to show how the forces are acting on your ball at each point to change how it moves.

How does projectile motion work?

Why can’t you use an average time for throwing your ball straight up?

Analysis:

How high did you throw your ball?

Your ball spent half its time going up and half its time coming down. Divide your time in half.

What provided the force to make the ball go up?

vectors for tossing a ball

When the ball first leaves the hand, most of the force is pushing the ball upward with gravity pulling against it. At the top of the loop, gravity and upward force cancel each other out and the ball stops. Then gravity pulls harder than the upward force so the ball falls back down into your hand. This is projectile motion.

What provided the force to make the ball come down?

Remember the formula from the last Project was a = d/t2

This time we know the acceleration is 9.8 m/s2 and the time and want to know the distance. We can rearrange the formula to be d = at2

Calculate how high you threw the ball.

 

What I Found Out

First I found out this Project is much easier with two people and I am only one so pictures were not possible. So I did drawings on my computer.

Tossing a ball up and down in one hand isn’t hard. The ball went up out of my hand then stopped and fell back down into my hand.

Playing catch can be fun. When the ball is tossed easily, it arches up then down into the other person’s hands. As the ball is tossed harder, the arch flattens out until it is almost a straight line.

Throwing or tossing a ball uses force from my hand. Gravity is always pulling down on the ball.

diagram of a ball thrown hard

Throwing a ball harder means it is going faster so gravity doesn’t slow it down as fast flattening the curve the ball makes.

The ball must go fast enough to overcome gravity. The harder I throw the ball, the faster it goes and the longer before gravity slows it down enough to make it fall.

In projectile motion the ball starts off with lots of force pushing it upwards. Gravity pulls a little of that force at a time slowing the ball down. At the top of the arch gravity is equal to the force making the ball go forward. Then gravity is greater making the ball fall down.

When a ball goes straight up, gravity pulls down until the ball stops and starts to fall down. Even if I try very hard, I won’t throw the ball with the same force every time so I must time each throw separately.

When I tossed my ball up, it took 1.94 sec to hit the ground. Half the time is .97 sec. Squaring the time gives .88 s2. Multiplying that by 9.8 m/s2 tells me I threw the ball 8.6 m or about 28 feet up.

Physics 11 Circular Motion

Things move in different paths. So far we’ve looked at straight motion and pendulum motion. What if a pendulum didn’t swing back and forth but went all the way around? This is circular motion. How is circular motion different from pendulum motion?

string, nut, scissors, meter stick, stopwatch

Question: How does circular motion work?

Materials:

String

Nut

Procedure:

Cut a piece of string 1.5 m long

measuring 1.5 m of string

Measure off 1.5 m of string. My string unravels easily so I put a piece of tape over the end to hold it together.

Put a loop in one end big enough to fit on your wrist

loop in end of string

The loop at the end of the string needs to be big enough to slide over a hand but small enough to not slip off the wrist easily.

Tie the nut to the other end of the string [I taped the knot as my string doesn’t hold a knot well.]

nut on string

The nut is tied to the end of the string. Be sure to secure the knot so it will not come loose while you are swinging the nut around. I used tape.

Measure up the string 0.5 m and make a small knot

0.5 m knot

The first knot is tied 0.5 meter from the nut.

Measure up the string 1 m and make a small knot

1 m knot

The second knot is tied at 1 meter from the nut.

!Warning!: Getting hit by the nut can hurt. Hitting something else with the nut can get you into a lot of trouble picking up broken things off the floor.

Put the loop around your wrist

Hold the string at the first knot

Swing the nut back and forth like a pendulum but keep adding force until the nut goes all the way around

Swing the nut around in a circular path several times

Stop the string

Hold the string at the second knot

Swing the nut back and forth like a pendulum but keep adding force until the nut goes all the way around

Swing the nut around in a circular path several times

Stop the string

Observations:

How did you have to move your hand to add force to increase the swing of the nut?

nut moving in revolution

The nut swings at the end of the string. The hand holding the string keeps the nut moving at a fixed distance so it travels in a circular path.

Describe any differences for the longer string

Describe how it felt as the nut moved in a circular path

Describe any differences for the longer string

Conclusions:

Why do you loop the string around your wrist?

If you put a little bit of force into making the string swing, does the nut go all the way around?

Does the nut want to continue in a circular path or does it try to leave that path? Why do you think so?

What will the nut do if you let go of the string? If you decide to test this, be sure you are outside and not swinging the nut toward anything like a window. Take the loop off your wrist, swing the nut so it is going in a circle and let go of the string as it tops the circle. You can get a little idea of what it does by leaving the loop around your wrist, swing the nut by the first know and letting it go at the top of the circle. Be aware the nut could hit you when you do it this way.

Compare the speed of using a short string and using a long string. If you decide to time the swings, have a friend use the stopwatch. It would be easier to get an accurate time if your friend times three to five revolutions instead of one.

Try drawing the vectors to show how the nut travels in a circular motion. Remember one vector will follow the string as it holds the nut in the pathway. Which way will the nut’s forward vector point? Will it be curved or straight? Does gravity have much of an effect on this motion?

 

What I Found Out:

The nut was easy to put on the string. If it hits something breakable like a window, this is bad news. Keeping the string attached to my wrist and taping the knot holding the nut on the string made sure it couldn’t fly off and hit anything or anyone.

My hand swung back and forth to make the nut swing. This hand movement could turn the nut into a pendulum, even one that went very high. It did not make the nut go around in a circular pathway.

I had to move my hand in a circular path to get the nut to go around. With the short string, the nut went around very easily. It was very hard to slow down enough for the nut to not go around.

The longer string took more and bigger movements of my hand to get it started going around. If I slowed down at all, the nut would make only a partial circle and fall down toward the ground.

Once the nut was going around on the long string, I could make the same small movements with my hand to keep it going as long as I kept it going fast enough to go around.

I think gravity pulls on the nut. When the nut is going fast enough, gravity can’t pull hard enough to make it fall. If the nut slow down, gravity takes over and pulls it down.

I could feel the nut pulling on my hand as it went around. There was a bigger pull with the longer string.

When I let go of the string, the nut flew out away from the circular path. I had to keep the loop around my wrist doing this so the nut hit the end of the string and fell to the floor.

The nut was traveling fairly fast around the path making timing challenging. The short string gave me 3.09 sec and 3.06 sec for five times around. The long string times to 3.62 sec and 3.56 sec. The longer string seemed to give a longer time for each revolution. I would wonder how accurate this is because I could not measure the force used to make the nut go around so this may have been very different for the long and short strings.

circular motion vectors

There are three vectors interacting in circular motion. One points in to the center of the circle holding the object in its circular pathway. One is the pull of gravity. One is the straight line motion path the nut would take if the other two forces did not exist.

Drawing the vectors depends a little on where the nut is on the circular pathway. One vector arrow must point down toward my hand. I know this because I had to hold onto the string and felt the nut trying to pull free.

One vector arrow will point down toward the ground. This is gravity. It is a smaller arrow as the nut is going around, not falling straight to the ground.

The last vector arrow goes straight out from wherever the nut is. The nut is trying to go in a straight path. The vector arrow pointing to the hand keeps it from flying off so the straight vector is bigger than the gravity arrow and smaller than the one going to the hand.

Physics 7 Motion and Vectors

For the Projects we’ve done so far we’ve accepted that the paper, the car, the balls and the jar moved. What is motion?

Look up motion in the dictionary. What does it say?

My dictionary says motion isthe act of changing place or moving.

We used vectors earlier to show the direction of a force. Vectors can also help us show where and how far something is moving.

Another concept in physics is displacement. This is how far something moves from its original position. This is not the same as the distance something moves.

Question: How can vectors show how something moves?

Materials:

Sheets of Grid paper [quarter inch is fine]

Pencil

Procedure:

Draw a line across a sheet of grid paper about ten squares from the top

The line is a street

Make a little mark across this line every second square

These marks show blocks

Draw a little house at the middle mark on the line

On a map east goes to the right, west to the left, north up and south down.

Trip 1:

You leave the little house and walk three blocks east to the market

first vector

Above the line you can make a fancy house and market. Below the line is room for the vectors. The first one goes from the house east to the market. It has an arrow on the tip pointing the direction you walked.

To show this draw a line from the mark in front of the house three blocks east or and put a little arrow at the end of the line.

Now you walk three blocks west back to the little house

second vector

You now walk back home so the vector arrow goes from the market to your house. The arrow is now on the end at your house as you walked that way.

Draw another line from the market to the little house and put a little arrow on the end

Conclusions:

How far did you walk?

This is distance. Your total distance is 3 blocks east plus 3 blocks west or 6 blocks.

What is your displacement?

Notice on your graph the arrows are equal and opposite. The vectors say you did not go anywhere.

Displacement is how far something moves away from where it started. In this case the displacement is 0 because you started and ended at the same place.

 

Trip 2:

Draw another line about ten squares below the first line and put a little mark in the middle

street line

The long green line is the street running east and west with a mark showing your house in the middle. You can draw houses above the line if you wish.

This time each square is a block

You go to a friend’s house five blocks west of your house

first vector

The first vector is you going five blocks west so the line goes five squares to the left.

Draw this vector

The two of you decide to go to another friend’s house seven blocks east of where you are

second vector

Going seven blocks east means going past your house plus another two blocks and the vector shows this.

Draw this vector

Later you and your first friend go to your homes for dinner

final vectors going home

You go only a short distance and your friend keeps going so two vectors are needed. I labeled them with a y for you and an f for friend so i would know which was which.

Draw these vectors

Conclusions:

Trip 2:

What distance did you walk?

What distance did your first friend walk?

What distance did your second friend walk?

What is your displacement at your first friend’s house?

What is your displacement at your second friend’s house?

What is your first friend’s displacement at your second friend’s house?

What is your total displacement?

What is your first friend’s total displacement?

Observations:

Your lines and vectors

 

On Your Own

Draw another line about ten squares below your last vector line

Mark your home in the center

Walk the five blocks west to your friend’s house

[Hint: This may be easier if you use more than one color for the vectors such as one to you alone, one for you and your first friend, one for the three of you and one for your two friends.]

The two of you walk seven blocks east to your other friend’s house

The three of you go five blocks east to a park for the afternoon

The three of you go to your house for supper

Your two friends go back to your first friend’s house for the night

Conclusions

What distance did you go?

What was your displacement?

What distance did your first friend go?

What was your first friend’s displacement?

What distance did your second friend go?

What was your second friend’s displacement?

 

What I found Out:

vector graph for the 3rd trip

I used four different colors and labeled the vectors to keep track of them. Another way would be to do three graphs, one for each person. That method would make it easier to see how far and where each person went.

I walked 5 blocks W + 7 blocks E + 5 blocks E + 7 blocks W or 24 blocks. My displacement was 0 because I started and ended at home.

My first friend walked 7 blocks E + 5 blocks E + 7 blocks W + 5 blocks W or 24 blocks.  My first friend’s displacement is 0 because of starting and ending at home.

My second friend walked 5 blocks E + 7 blocks W + 5 blocks W or 17 blocks. My second friend’s displacement is 7 blocks because of starting at home and ending at my first friend’s house.

Physics 4 Vectors

Forces hold things in place and make them move. Some of the forces we can see. Others we know are there but can’t see. We need a way to show all of these forces. That is what vectors do.

Question: How do vectors show forces?

Materials:

Paper

Pencil

Ruler

Procedure:

Open your Journal and write Project 4

Remember Project 1 where the block sits on the table

Draw a table with legs sitting on the floor

drawing of table

Your drawing doesn’t need to be fancy. A simple set of boxes will work like this will work fine.

Draw the block sitting on the table

Gravity pulls down on the block so draw an arrow pointing down from the block

gravity arrow in block

Gravity pulls down on the block so it sits on the table. The vector arrow points down.

Note: Gravity always points toward the center of the Earth which is usually down

If only gravity was pulling on the block, it would fall to the ground so some force is pushing back on the block. The table is pushing back so draw another arrow next to the other arrow but pointing up.

table vector arrow is added

The table pushes back against the block just as hard as gravity pulls it down so a vector arrow the same size pointing up is added to the block.

How long should this arrow be? Vectors show speed and direction. We are not measuring speed but only showing direction in this Project.

Gravity pulls down. If the arrow pointing up is longer showing greater force, there would be more force pointing up than down. The block would float up off the table. It didn’t so the arrow isn’t longer than the gravity arrow.

If the arrow is shorter than the gravity arrow, the force of gravity would be greater than that of the table. The block would pull through the table and fall to the floor. It didn’t so the arrow isn’t shorter.

The arrows must be the same length as the forces are equal and opposite to each other.

gravity vector arrows put in table legs

Since the table is not floating away, gravity is pulling down on the legs so a vector arrow pointing down is put in each leg.

Since the table isn’t floating away, gravity must be pulling down on it too. Draw vector arrows for gravity to hold each table leg on the floor.

floor vector arrows added

The floor pushes up against the table legs just as hard as gravity pulls down on the legs so the arrows are the same length as those vectors but pointing up.

Since the table isn’t sinking into the floor, the floor is pushing back against the table legs. Draw those vector arrows.

push vector arrow added

A vector arrow showing the push on the block is added aimed at the block which was the direction of the force.

Next remember what happened when you pushed on the block. Your finger was a force acting on the block. Draw a vector arrow for that force.

Did the block move? Which way did it move? Since the block moved, there was no force pushing back against your finger so there will be no arrow.

all vector arrows drawn

Every pair of force vectors have the arrows equal and opposite except for the last pair. The pushing force arrow must be longer than the block resistance force arrow for the block to move.

Now wait a minute! When I pushed against my block of wood, the end of my finger flattened so the block did push back. But the block moved so the force the block pushed against my finger was much less than the push my finger gave the block. I will draw a little arrow from the block toward my finger.

paper airplane

Paper airplanes are fun to fly. They fly and fall because of forces pushing and pulling on them. Those forces can be drawn as vectors.

Now let’s draw vectors for a paper airplane:

Draw the airplane flying

What force made the airplane fly?

You threw it so you exerted a force on it. Draw that arrow pushing the back of the airplane.

thrust vector on airplane

Throwing a paper airplane pushes it forward so the vector arrow pushes against the tail end.

Is gravity acting on the airplane? Gravity acts on everything on Earth so draw an arrow pointing down for gravity.

air vector on airplane

Like the sheet of paper, air pushes up on the wings of the paper airplane so the vector arrow points up toward the wing.

What is pushing up on the wings to keep the airplane up? Air pushes up.

Does the air keep the airplane up all the time? It didn’t keep mine up. So there is an arrow for the air pushing the wings up but it is less than the gravity.

gravity vector arrow on airplane

The paper airplane doesn’t fly forever so gravity pulls down on it which the vector arrow shows.

Where would the arrow for the fan pushing the airplane go? Draw it in.

Notice that this arrow is with the one from you throwing the airplane so the two add up.

air pushing airplane forward

When the air from the fan pushes the paper airplane, the vector arrow pushes against the airplane’s tail adding to the thrust vector from you throwing it.

Where would the arrow go for when the fan pushed against the airplane? For my airplane it would be the same length as the one for throwing the airplane because I did see it stop my airplane once.

air vector against airplane

When the air from the fan blows against the paper airplane, the vector arrow must point toward the airplane.

This last set of drawings shows one way vectors help a physicist understand the forces acting on an object. When forces act together, they add up. When forces act against each other, they cancel each other out.

 

Another way vectors show how forces work is shown with the car going down the ramp.

Draw the ramp with a car on it.

The car is racing down the ramp so the vector arrow goes down the ramp. Or does it?

car on ramp

The car is moving down the ramp so the vector arrow points down the ramp but is this correct?

Gravity is pulling the car down but gravity pulls straight down. So there should be an arrow pointing down from the car.

gravity arrow on car

Gravity pulls down on the car so a vector arrow points down from the car.

But the car moved down the ramp. So there is another arrow from the tip of the gravity arrow to the ramp spot where the car will be after a certain amount of time.

all force vector arrows on car

The gravity arrow and the forward arrow meet at the vector arrow on the ramp’s point as the two add up to that vector.

In this case vectors show the movement of the car is made up of two different vectors, one pulling down and one pulling across.

 

About Vectors

Vectors usually show both direction and acceleration. They are a way to see how forces add and subtract from each other so you can tell where an object will go when several forces push or pull on it at the same time.

For now the accelerating force we will work with will be gravity. The next Project will look at some ways gravity pulls objects down.