VECTORS

VECTORSVECTORS

Scalars are quantities which are fully described by a magnitude alone.

Vectors are quantities which are fully described by both a magnitude and a direction.

Do you remember the difference between a scalar and a vector?

1. The length of the line represents the magnitude and the arrow indicates the direction.2. The magnitude and direction of the vector is

clearly labeled.

The magnitude of a vector in a scaled vector diagram is depicted by the length of the arrow. The arrow is drawn precisely to length in accordance with a chosen scale.

Scaling!!!Scaling!!!

Direction!!!Direction!!!

Sometimes vectors will be directed due East or due North. However we will encounter vectors in all sorts of directions and be forced to find the angle!

Compass Coordinate System

Compass Coordinate System

• Δx = 30 m @ 20º E of N

• V = 20 m/s @ 30º W of N

• a = 10 m/s2@ 40º W of S

• F = 50 N @ 10º S of E

Navigational System?

Use the same scale for all vector magnitudes

E

S

N

W

To Draw direction:

Ex. 20º E of N: Start w/ North and go 20° East

All these planes have the same reading on their speedometer. (plane speed not speed with respect to the ground (actual speed)

What factor is affecting their velocity?

A B C

Easy Adding…

The resultant is the vector sum of two or more vectors.

1. Select an appropriate scale (e.g., 1 cm = 5 km)

2. Draw and label 1st vector to scale.

*The tail of each consecutive vector begins at the head of the most recent vector*

3. Draw and label 2nd vector to scale starting at the head of the 1st vector.

4. Draw the resultant vector (the summative result of the addition of the given vectors) by connecting the tail of the 1st vector to the head of the 2nd vector. (initial to final pt.)

5. Determine the magnitude and direction of the resultant vector by using a protractor, ruler, and the indicated scale; then label the resultant vector.

A. Tailwind

(with the wind)

120 km/h20 km/h

80 km/h

100 km/h

100 km/h

20km/h

=

B. Headwind

(against the wind)

=

80 km/h

100 km/h

60 km/h

Resultant

C. 90º crosswindUsing a ruler and your scale, you can determine the magnitude of the resultant vector. Or you could use the Pythagorean Theorem.

Then using a protractor, you can measure the direction of the resultant vector. Or you could use trigonometry to solve for the angle.

1. Find the resultant force vector of the two forces below.

25 N due East, 45 N due South

25 N, East

45 N, South

Decide ona scale!!!

51 N59º S of E

51 N31º E of S

0o

N

180o

270o

An airplane is flying 200mph at 50o N of E. Wind velocityis 50 mph due S. What is the velocity of the plane?

Scale: 50 mph = 1 inch

E

N

W

S

An airplane is flying 200mph at 50o N of E. Wind velocityis 50 mph due S. What is the velocity of the plane?

Scale: 50 mph = 1 inch

E

N

W

S

An airplane is flying 200mph at 50o N of E. Wind velocityis 50 mph due S. What is the velocity of the plane?

Scale: 50 mph = 1 inch

200 mph

50 mph

VR = 165 mph @ 40° N of E

2. Find the resultant velocity vector of the two velocity vectors below.

700 m/s @35 degrees E of N; 1000 m/s @ 30 degrees N of W

V1

Vr

V2

A bear walks one mile south, then one

mile west, and finally walks one

mile north. After his brisk walk, the bear ends back where he

started.

What color is the bear???

Intro to Vectors Warm-up

In what direction is the leash pulling on the dog?

What would happen to the upward and rightward Forces if the Force on the chain were smaller?

1) Find the resultant Magnitude:__________

of the two vectors Direction:___________

Vector #1 = 20.5 N West

Vector #2 = 14.3 N North

Vector Diagram

24.99 N

34.90º N of W

V2

V1

VR=?

2) Find the component of the

resultant = 255m 27º South of East

Vector # 1 _______ Direction__________

Vector # 2________ Direction__________

Vector Diagram:

115.8 m

227.2 m

South (-)

East (+)

Vr

V2

V1

Conventions:

+

+

-

-

1. Draw vector diagram. (Draw axis)

2. Resolve vectors into components using trig:

Vadj = V cos θ Vopp = V sin θ

3. Sum x and y components:

ΣVxi ΣVyi

4. Redraw!! Determine resultant vector using Pythagorean’s Theorem and trig:

Magnitude= √(Σ Vxi)² + (Σ Vyi)²

Direction Action: θ = tan-1(opp/adj)

Practice: Find FR = Fnet =?

200 N due South, 100 N at 40º N of W

Answer: Fnet = N @ ˚ W of S

Skip

An airplane flies at an engine speed of 100 m/s at 50º W of S into a wind of 30 m/s at 200 E of N. What is the airplane’s resultant velocity?

Solve using the components method!!

How far has the plane traveled after 1 hr??

Answer: 75.52 m/s @ 28.54˚ S of W

168.89 miles per 1 hour

You Try!!!You Try!!!

A motor boat traveling 4.0 m/s, East encounters a current traveling 3.0 m/s, North.

a. What is the resultant velocity of the motor boat?

b. If the width of the river is 80 meters wide, then how much time does it take the boat to travel shore to shore?

c. What distance downstream does the boat reach the opposite shore?

Equilibrium

Balanced? or Unbalanced?

Forces are in Equilibrium

Are these Objects are in Equilibrium?

50 N

151 N

25 N

15 N10 N

102 N

5 kgCeiling

The First Condition of Equilibrium: The upward forces must balance the downward forces, and the leftward forces must balance the rightward forces.

Fx = 0 Fy = 0 at equilibrium

Solution of Problems in Static's:

1) Isolate a body. What point or object are you going to talk about?

2) Draw the forces acting on the body you have isolated, and label them. (If their value is not know, give them a symbol such as F1, FP, T1, etc.) Remember…this is a free-body diag.

3) Split each of the forces into its x and y components, and label the components in terms of the symbols given in rule 2 and the proper sines and cosines.

4) Write down your summation equations for the first condition of equilibrium.

5) Solve the equations for the unknowns.

Cont.

Equilibrium problem

10 kg

300 N

Fy

Solve for Fx & Fy?

Fx

20º

Ans: Fx = 102.61 N Fy = 183.91N

Equilibrium problem

Solve for F?10 kg

200 N

50 NF ө 30º

Answer: F = 88.34 N @ 60.65º S of W

Tension Warm-up

• Find the tension in each rope if the weight (W) is 50 N.

• Be sure to pick an appropriate point to draw your free-body-diagram. Then sum your x forces and then your y-forces.

W

37º 45º

Answers: T1= N, T2= N

When the strongman suspends the 10N telephone book with the rope held vertically the tension in each strand of rope is 5 N. If the strongman could suspend the book from the strands pulled horizontally as shown, the tension in each strand would be:

a)About 5 N

b)About 10 N

c)About 20 N

d)More than a million Newtons... basically impossible.

The two cars below are both accelerated by a 10 N force. Which car experiences the larger acceleration?

a) Car 1

b) Car 2

c) Both cars accelerate similarly.

Explain

10

10

VROOM!