vectors vector: a physical quantity – magnitude – direction – velocity, force, displacement...

Vectors

• Vector: a physical quantity– Magnitude– Direction– Velocity, force, displacement

• Scalar: a physical quantity– Just magnitude– Age, temperature, speed, distance

• Neither: not a quantity, a quality– Color, shape

Force

• Force = a push or pull• British Units: pounds• SI Units: Newtons• 1 N = 1 kg*m/s²• Can be measured with a spring scale

Vectors: Diagrams

• Vectors = arrows • Length of arrow = magnitude of vector• Direction of arrow = direction of vector

• Example: pushing on a shopping cart

(depending on how tall you are) you don’t push straight ahead, you push on the handle downward and forward…

Shopping and force (forces = a push or pull)The arrow represents the vector: the force you apply to the shopping cart

The length of the arrow indicates how hard you push

The direction of the arrow indicates the direction of your pushImagine that now its not you pushing the cart, but your little brother

The arrow is much shorter, indicating that he isn’t pushing as hard as you

The angle is also not as steep, because he is shorter than you!

Adding Vectors

• Whenever 2 (or more!) vectors are combined, the new vector formed is called the resultant

• If 2 vectors point in the same direction– Add them

• If 2 vectors point in opposite directions– Subtract them – Sign indicates direction

• If 2 vectors point in any other directions– Parallelogram rule

Adding: same directionImagine you are pushing a huge box across the floor

But the box is too heavy, so your friend comes to help

You push with a force of 120 N, and your friend pushes with a force of 150 N

• The vector diagram of the 2 forces looks like this:

The sum of the 2 forces is120 N + 150 N = 270 N

Your push:120 N

Your friend’s push:150 N

Practice with displacement

• Displacement is a vector!– The vector from where you start to where you end

• Displacement 1: walk 20 yards north• Displacement 2: walk 5 yards north• What is the resultant displacement?

• 25 yards north!

Adding: opposite directionImagine you are pushing a huge box across the floor

Even though the box is heavy, your friend doesn’t want to help! In fact, he wants to make it more difficult

You push with a force of 120 N, and your friend pushes with a force of 150 N

• The vector diagram of the 2 forces looks like this:

150 “to the right” + 120 “to the left” makes no mathematical sense.

We use a sign to indicate direction: + “to the left”, - “to the right”

(-)150 N + (+)120 N = -30 N DOES make mathematical sense

(-)30 N = 30 N “to the right”

Your push:120 N

You friend’s push:150 N

Practice with displacement

• Displacement is a vector!– The vector from where you start to where you end

• Displacement 1: walk 20 yards north• Displacement 2: walk 5 yards south• what is the resultant displacement?

• 15 yards north!

Parallelogram Rule

• Tails of the 2 vectors touch• The 2 vectors form adjacent sides of a

parallelogram• The diagonal (from tails to tip) is the resultant

• Example: You are in an airplane flying northbound, but the wind is blowing northeast

Engines push the plane forward; Northbound

Wind pushes the plane forward but also to the East

Parallelogram rule – step 2:Construct the parallel sides

Parallelogram rule – step 1:The tails of the 2 vectors touch, and form adjacent sides of a parallelogram

Parallelogram rule – step 3:Draw the diagonal (from tails to opposite sides)

Again, you can’t add these vectors algebraically – 100 N “north” plus 50 N “northeast” makes no mathematical sense!

Practice with displacement• Displacement 1 = 20 yards north• Displacement 2 = 5 yards northwest• What is the resultant displacement?

20 yds N

5 yds NW

The 2 vectors to be combined are drawn tail-to-tail

Draw/construct parallel sides

Resultant!

Parallelogram Rule: Special Case

• If the 2 vectors are at 90° to each other• The parallelogram they form is a rectangle (or

square)• The diagonal can be found numerically using

the Pythagorean theorem

• Example: you are in a plane flying northbound and the wind is directly east

Engines push the plane forward; Northbound with a force of 300 N

Wind pushes the plane directly Eastbound with a force of 50 N

Parallelogram rule – step 2:Form the parallel sides

Parallelogram rule – step 1:The tails of the 2 vectors touch and form adjacent sides

Parallelogram rule – step 3:Draw the diagonal

300 N

50 N

Now you can actually calculate the magnitude of that resultant vector with the Pythagorean Theorem, because it is the hypotenuse of a right triangle

a² + b² = c²(300 N)² + (50 N)² = c²

Solve for c!

c = 3

04 N 300 N

50 N

Practice with displacement• Displacement 1 = 20 yards north• Displacement 2 = 5 yards west• What is the resultant displacement?

20 yds N

5 yds W

First, diagram the vectors, apply the parallelogram rule

Next, identify the right triangle

20 yds N

Finally, use the Pythagorean Theorem to find the length of the hypotenuse (resultant)!

2 2resultant = 20 5

425

20.6 yds NW

Balanced and Unbalanced Forces

• Balanced Forces:– Tug of war, stalemate– Driving at constant speed on the freeway

All are in Equilibrium: no net force

People on the right pull to the right

People on the left pull to the left

Engine (via torque on axle) pushes car forward

Friction pull car backward

Unbalanced Forces:• Accelerating onto the freeway• Sky diving

Object speeds up or slows down in the direction of net force (the details of this = Newton’s 2nd Law)

Engine pushes car forward

Friction pulls car backward

Weight (the pull of gravity) pulls down

Air resistance (friction) pushes up

Newton’s 1st Law

• All objects remain at rest OR continue in a straight line path at constant speed UNLESS acted on by a net force

• The law of inertia• “inertia” = an object’s tendency to keep doing

what it was already doing

Lab Activity: Treasure Hunt

• Groups of less than 6 students each

• All groups begin at the same starting point: Middle of plaza

• Groups create a “treasure map,” but with notecards: each card gives a single displacement

• 10 displacement vector directions– Must range from small (2 ft)

to large (20 ft)– Must not go in parking lot– Must include all 4 directions

(N, S, E, W)– No intermediate directions

(NW, SE, etc)

– Use lines on plaza patio for direction reference

– Do not number direction cards (they will be mixed up)

– Color code cards and “treasure marker” (“X” on a white paper)

• Your direction cards will be shuffled and given to another team

• Team who finds the “X” of another team wins the “treasure chest”

• BEFORE YOU GOdiscuss: what effect will shuffling the cards have on the success of the treasure hunt?

HOMEWORK/classwork???

8th edition• Pg 52– Review q 6-8

• pg 53 – Exercises 3-4

• Pg 55– Problems 1-2

10th edition• Pg 87– “one step calculations”

1, 2, 4

• Pg. 89– Exercises 31-32

• Pg. 90– Problems 5-6