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Chapter 3B - Vectors. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University. © 2007. Surveyors use accurate measures of magnitudes and directions to create scaled maps of large regions. Vectors.

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chapter 3b vectors

Chapter 3B - Vectors

A PowerPoint Presentation by

Paul E. Tippens, Professor of Physics

Southern Polytechnic State University

© 2007

objectives after completing this module you should be able to
Objectives: After completing this module, you should be able to:
  • Demonstrate that you meet mathematics expectations: unit analysis, algebra, scientific notation, and right-triangle trigonometry.
  • Define and give examples of scalar and vector quantities.
  • Determine the components of a given vector.
  • Find the resultant of two or more vectors.
expectations

m

s

1 km

1000 m

3600 s

1 h

40--- x ---------- x -------- = 144 km/h

Expectations
  • You must be able convert units of measure for physical quantities.

Convert 40 m/s into kilometers per hour.

expectations continued

Example:

Solve for vo

Expectations (Continued):
  • College algebra and simple formula manipulation are assumed.
expectations continued6

(6.67 x 10-11)(4 x 10-3)(2)

(8.77 x 10-3)2

Gmm’

r2

F = -------- = ------------

Expectations (Continued)
  • You must be able to work in scientific notation.

Evaluate the following:

F = 6.94 x 10-9 N = 6.94 nN

expectations continued7
Expectations (Continued)
  • You must be familiar with SI prefixes

The meter (m) 1 m = 1 x 100 m

1 Gm = 1 x 109 m 1 nm = 1 x 10-9 m

1 Mm = 1 x 106 m 1 mm = 1 x 10-6 m

1 km = 1 x 103 m 1 mm = 1 x 10-3 m

expectations continued8

R

y

q

x

Expectations (Continued)
  • You must have mastered right-triangle trigonometry.

y = R sin q

x = R cos q

R2 = x2 + y2

mathematics review

If you feel you need to brush up on your mathematics skills, try the tutorial from Chap. 2 on Mathematics. Trig is reviewed along with vectors in this module.

Mathematics Review

Select Chap. 2 from the On-Line Learning Center in Tippens—Student Edition

physics is the science of measurement

Weight

Time

Physics is the Science of Measurement

Length

We begin with the measurement of length: its magnitude and its direction.

distance a scalar quantity

B

A

Distance: A Scalar Quantity
  • Distance is the length of the actual path taken by an object.

A scalar quantity:

Contains magnitude only and consists of a number and a unit.

(20 m, 40 mi/h, 10 gal)

s = 20 m

displacement a vector quantity

B

D = 12 m, 20o

A

q

Displacement—A Vector Quantity
  • Displacement is the straight-line separation of two points in a specified direction.

A vector quantity:

Contains magnitude AND direction, a number,unit & angle.

(12 m, 300; 8 km/h, N)

distance and displacement

D

4 m,E

6 m,W

Distance and Displacement
  • Displacement is the x or y coordinate of position. Consider a car that travels 4 m, E then 6 m, W.

Net displacement:

D= 2 m, W

What is the distance traveled?

x= -2

x= +4

10 m !!

identifying direction

N

60o

50o

W

E

60o

60o

S

Identifying Direction

A common way of identifying direction is by reference to East, North, West, and South. (Locate points below.)

Length = 40 m

40 m, 50o N of E

40 m, 60o N of W

40 m, 60o W of S

40 m, 60o S of E

identifying direction15

N

45o

N

W

E

W

E

50o

S

S

Identifying Direction

Write the angles shown below by using references to east, south, west, north.

500 S of E

Click to see the Answers . . .

450 W of N

vectors and polar coordinates

90o

90o

R

180o

180o

50o

q

0o

0o

270o

270o

Vectors and Polar Coordinates

Polar coordinates (R,q) are an excellent way to express vectors. Consider the vector 40 m, 500 N of E, for example.

40 m

R is the magnitude and q is the direction.

vectors and polar coordinates17

90o

180o

0o

60o

50o

60o

60o

3000

210o

270o

120o

Vectors and Polar Coordinates

Polar coordinates (R,q) are given for each of four possible quadrants:

(R,q) = 40 m, 50o

(R,q) = 40 m, 120o

(R,q) = 40 m, 210o

(R,q) = 40 m, 300o

rectangular coordinates

y

(-2, +3)

(+3, +2)

+

+

x

-

Right, up = (+,+)

Left, down = (-,-)

(x,y) = (?, ?)

-

(-1, -3)

(+4, -3)

Rectangular Coordinates

Reference is made to x and y axes, with + and -numbers to indicate position in space.

trigonometry review

R

y

q

x

Trigonometry Review
  • Application of Trigonometry to Vectors

Trigonometry

y = R sin q

x = R cos q

R2 = x2 + y2

slide20

300

90 m

Example 1:Find the height of a building if it casts a shadow 90 m long and the indicated angle is 30o.

The height h is opposite 300 and the known adjacent side is 90 m.

h

h = (90 m) tan 30o

h = 57.7 m

finding components of vectors

R

y

q

x

Finding Components of Vectors

A component is the effect of a vector along other directions. The x and y components of the vector (R,q) are illustrated below.

x = R cos q

y = R sin q

Finding components:

Polar to Rectangular Conversions

slide22

N

400 m

y = ?

30o

E

x = ?

R

y

q

x

Example 2:A person walks 400 m in a direction of 30o N of E. How far is the displacement east and how far north?

N

E

The x-component (E) is ADJ:

x = R cosq

The y-component (N) is OPP:

y = R sinq

slide23

N

400 m

y = ?

30o

E

x = ?

The x-component is:

Rx = +346 m

Example 2 (Cont.):A 400-m walk in a direction of 30o N of E. How far is the displacement east and how far north?

Note:x is the side adjacent to angle 300

ADJ = HYP x Cos 300

x = R cosq

x = (400 m)cos30o

= +346 m, E

slide24

N

400 m

y = ?

30o

E

x = ?

The y-component is:

Ry = +200 m

Example 2 (Cont.):A 400-m walk in a direction of 30o N of E. How far is the displacement east and how far north?

Note:y is the side opposite to angle 300

OPP = HYP x Sin 300

y = R sinq

y = (400 m) sin 30o

= + 200 m, N

slide25

N

The x- and y- components are each + in the first quadrant

400 m

Ry = +200 m

30o

E

Rx = +346 m

Example 2 (Cont.):A 400-m walk in a direction of 30o N of E. How far is the displacement east and how far north?

Solution: The person is displaced 346 m east and 200 m north of the original position.

signs for rectangular coordinates

x = R cos q

y = R sinq

Signs for Rectangular Coordinates

90o

First Quadrant:

R is positive (+)

0o > q < 90o

x = +; y = +

R

+

q

0o

+

signs for rectangular coordinates27

x = R cosq

y = R sinq

Signs for Rectangular Coordinates

90o

Second Quadrant:

R is positive (+)

90o > q < 180o

x = - ; y = +

R

+

q

180o

signs for rectangular coordinates28

x = R cosq

y = R sin q

Signs for Rectangular Coordinates

Third Quadrant:

R is positive (+)

180o > q < 270o

x = - y = -

q

180o

-

R

270o

signs for rectangular coordinates29

x = R cos q

y = R sin q

Signs for Rectangular Coordinates

Fourth Quadrant:

R is positive (+)

270o > q < 360o

x = + y = -

q

+

360o

R

270o

resultant of perpendicular vectors
Resultant of Perpendicular Vectors

Finding resultant of two perpendicular vectors is like changing from rectangular to polar coord.

R

y

q

x

R is always positive; q is from + x axis

slide31

40 lb

40 lb

30 lb

30 lb

Example 3:A 30-lb southward force and a 40-lb eastward force act on a donkey at the same time. What is the NET or resultant force on the donkey?

Draw a rough sketch.

Choose rough scale:

Ex:1 cm = 10 lb

Note: Force has direction just like length does. We can treat force vectors just as we have length vectors to find the resultant force. The procedure is the same!

4 cm = 40 lb

3 cm = 30 lb

finding resultant cont

q

40 lb

f

40 lb

30 lb

R = (40)2 + (30)2 = 50 lb

R = x2 + y2

-30

40

30 lb

tan f =

Finding Resultant: (Cont.)

Finding (R,q) from given (x,y) = (+40, -30)

Rx

Ry

R

q = 323.1o

f = -36.9o

four quadrants cont

30 lb

30 lb

R

R

R = 50 lb

q

40 lb

Ry

q

Ry

f

f

q

Rx

Rx

40 lb

40 lb

R

30 lb

Rx

Rx

q

40 lb

f

Ry

Ry

R = 50 lb

30 lb

R

Four Quadrants: (Cont.)

f = 36.9o; q = 36.9o; 143.1o; 216.9o; 323.1o

unit vector notation i j k

y

j

i

x

k

z

Unit vector notation (i,j,k)

Consider 3D axes (x, y, z)

Define unit vectors, i, j, k

Examples of Use:

40 m, E = 40 i 40 m, W = -40 i

30 m, N = 30 j 30 m, S = -30 j

20 m, out = 20 k 20 m, in = -20 k

slide35

R

+40 m

f

-30 m

Example 4:A woman walks 30 m, W; then 40 m, N. Write her displacementini,jnotation and inR,qnotation.

In i,j notation, we have:

R = Rxi + Ry j

Rx = - 30 m

Ry = + 40 m

R = -30 i + 40 j

Displacement is 30 m west and 40 m north of the starting position.

example 4 cont next we find her displacement in r q notation

R

+40 m

f

-30 m

Example 4 (Cont.):Next we find her displacementinR,qnotation.

q= 1800 – 59.10

q = 126.9o

R = 50 m

(R,q) = (50 m, 126.9o)

slide37

46 km

f=?

B

35 km

R = ?

A

q = 1800 + 52.70

q = 232.70

Example 6:Town A is 35 km south and 46 km west of Town B. Find length and direction of highway between towns.

R = -46 i – 35 j

R = 57.8 km

f = 52.70 S. of W.

slide38

F = 240 N

Fy

F

280

Or in i,j notation:

Fy

F = -(212 N)i+ (113 N)j

Fx

Example 7. Find the components of the 240-N force exerted by the boy on the girl if his arm makes an angle of 280 with the ground.

Fx= -|(240 N) cos 280|= -212 N

Fy= +|(240 N) sin 280|= +113 N

slide39

F = 300 N

32o

32o

Fx

320

Fy

Fy

F

Or in i,j notation:

F = -(254 N)i- (159 N)j

Example 8. Find the components of a 300-N force acting along the handle of a lawn-mower. The angle with the ground is 320.

Fx= -|(300 N) cos 320|= -254 N

Fy= -|(300 N) sin 320|= -159 N

component method

1. Start at origin. Draw each vector to scale with tip of 1st to tail of 2nd, tip of 2nd to tail 3rd, and so on for others.

Component Method

2. Draw resultant from origin to tip of last vector, noting the quadrant of the resultant.

3. Write each vector in i,j notation.

4. Add vectors algebraically to get resultant in i,j notation. Then convert to (R,q).

slide41

N

B

3 km, W

4 km, N

C

E

A

D

2 km, E

2 km, S

Example 9.A boat moves 2.0 km east then 4.0 km north, then 3.0 km west, and finally 2.0 km south. Find resultant displacement.

1. Start at origin. Draw each vector to scale with tip of 1st to tail of 2nd, tip of 2nd to tail 3rd, and so on for others.

2. Draw resultant from origin to tip of last vector, noting the quadrant of the resultant.

Note: The scale is approximate, but it is still clear that the resultant is in the fourth quadrant.

example 9 cont find resultant displacement

N

B

3 km, W

4 km, N

C

E

A

D

2 km, S

2 km, E

5. Convert to R,q notation See next page.

Example 9 (Cont.)Find resultant displacement.

3.Write each vector ini,jnotation:

A = +2 i

B = + 4 j

C = -3 i

D = - 2 j

4.Add vectors A,B,C,D algebraically to get resultant ini,jnotation.

-1 i

+ 2 j

R =

1 km, west and 2 km north of origin.

example 9 cont find resultant displacement43

Resultant Sum is:

R = -1 i + 2 j

N

B

3 km, W

4 km, N

C

E

D

A

2 km, S

2 km, E

Ry= +2 km

R

f

Rx = -1 km

Example 9 (Cont.)Find resultant displacement.

Now, We Find R, 

R = 2.24 km

 = 63.40 N or W

reminder of significant units

N

For convenience, we follow the practice of assuming three (3) significant figures for all data in problems.

D

3 km

2 km

C

B

4 km

E

A

2 km

Reminder of Significant Units:

In the previous example, we assume that the distances are 2.00 km, 4.00 km, and 3.00 km.

Thus, the answer must be reported as:

R = 2.24 km, 63.40 N of W

significant digits for angles

30 lb

R

q

40 lb

Ry

f

q

Rx

40 lb

R

30 lb

Rx

Ry

Significant Digits for Angles

Since a tenthof a degree can often be significant, sometimes a fourth digit is needed.

Rule:Write angles to the nearest tenth of a degree. See the two examples below:

q = 36.9o; 323.1o

example 10 find r q for the three vector displacements below

A = 5 m, 00

C = 0.5 m

R

B = 2.1 m, 200

B

q

C = 0.5 m, 900

200

A = 5 m

B = 2.1 m

Example 10: Find R,q for the three vector displacements below:

1. First draw vectors A, B, and C to approximate scale and indicate angles. (Rough drawing)

2. Draw resultant from origin to tip of last vector; noting the quadrant of the resultant. (R,q)

3. Write each vector in i,j notation. (Continued ...)

example 10 cont find i j for three vectors a 5 m 0 0 b 2 1 m 20 0 c 0 5 m 90 0

A = 5.00 i + 0 j

B = 1.97 i + 0.718 j

C = 0 i + 0.50 j

6.97 i

+ 1.22 j

Example 10 (Cont.): Find i,j for three vectors: A = 5 m,00; B = 2.1 m, 200; C = 0.5 m, 900.

4. Add vectors to get resultant R in i,j notation.

R =

example 10 cont find i j for three vectors a 5 m 0 0 b 2 1 m 20 0 c 0 5 m 90 049

R = 6.97 i + 1.22 j

Diagram for finding R,q:

R

q

Ry 1.22 m

Rx= 6.97 m

Example 10 (Cont.): Find i,j for three vectors: A = 5 m,00; B = 2.1 m, 200; C = 0.5 m, 900.

5. Determine R,q from x,y:

R = 7.08 m

q = 9.930 N. of E.

slide50

q

Example 11:A bike travels 20 m, E then 40 m at 60o N of W, and finally 30 m at 210o. What is the resultant displacement graphically?

C = 30 m

Graphically, we use ruler and protractor to draw components, then measure the Resultant R,q

B = 40 m

30o

R

60o

f

A = 20 m, E

R = (32.6 m, 143.0o)

Let 1 cm = 10 m

a graphical understanding of the components and of the resultant is given below

Cy

30o

R

Ry

60o

f

0

q

Rx

Ax

Cx

Bx

A Graphical Understanding of the Components and of the Resultant is given below:

Note: Rx = Ax + Bx + Cx

By

B

Ry = Ay + By + Cy

C

A

example 11 cont using the component method to solve for the resultant

Cy

By

30o

B

C

R

Ry

A

60

f

q

Rx

Ax

Cx

Bx

Example 11 (Cont.)Using the Component Method to solve for the Resultant.

Write each vector in i,j notation.

Ax = 20 m, Ay = 0

A = 20 i

Bx = -40 cos 60o = -20 m

By= 40 sin 60o = +34.6 m

B = -20 i + 34.6 j

Cx = -30 cos 30o = -26 m

C = -26 i - 15 j

Cy = -30 sin 60o = -15 m

example 11 cont the component method

Cy

By

30o

B

C

R

Ry

A

60

f

q

Rx

Ax

R= (-26)2 + (19.6)2 = 32.6 m

Cx

Bx

R

19.6

-26

+19.6

tan f =

f

-26

Example 11 (Cont.)The Component Method

Add algebraically:

A = 20 i

B = -20 i + 34.6 j

C = -26 i - 15 j

R= -26 i + 19.6 j

q = 143o

example 11 cont find the resultant

Cy

By

30o

B

C

R

Ry

A

60

f

q

Rx

Ax

Cx

Bx

R

+19.6

f

-26

Example 11 (Cont.)Find the Resultant.

R = -26 i + 19.6 j

The Resultant Displacement of the bike is best given by its polar coordinates R and q.

R = 32.6 m; q = 1430

example 12 find a b c for vectors shown below

Cx

A = 5 m, 900

B

Cy

B = 12 m, 00

350

A

C = 20 m, -350

C

q

R

A = 0 i + 5.00 j

B = 12 i + 0 j

C = 16.4 i – 11.5 j

28.4 i

- 6.47 j

Example 12.Find A + B + C for Vectors Shown below.

Ax = 0; Ay = +5 m

Bx = +12 m; By = 0

Cx = (20 m) cos 350

Cy = -(20 m) sin -350

R =

example 12 continued find a b c

B

350

A

C

q

q

R

R

Example 12 (Continued).Find A + B + C

Rx = 28.4 m

Ry = -6.47 m

R = 29.1 m

q = 12.80 S. of E.

vector difference

First Consider A + B Graphically:

B

B

R

B

A

A

Vector Difference

For vectors, signs are indicators of direction. Thus, when a vector is subtracted, the sign (direction) must be changed before adding.

R = A + B

vector difference58

Now A – B: First change sign (direction) of B, then add the negative vector.

B

B

-B

A

R’

-B

A

A

Vector Difference

For vectors, signs are indicators of direction. Thus, when a vector is subtracted, the sign (direction) must be changed before adding.

addition and subtraction

Subtraction results in a significant difference both in the magnitude and the direction of the resultant vector. |(A – B)| = |A| - |B|

Comparison of addition and subtraction of B

B

B

A

R

R’

-B

B

A

A

Addition and Subtraction

R = A + B

R’ = A - B

example 13 given a 2 4 km n and b 7 8 km n find a b and b a

A – B; B - A

+A

-A

+B

-B

A 2.43 N

B 7.74 N

Example 13.Given A = 2.4 km, N and B = 7.8 km, N: find A – B and B – A.

A - B

B - A

R

R

(2.43 N – 7.74 S)

(7.74 N – 2.43 S)

5.31 km, S

5.31 km, N

summary for vectors

Components of R:

R

Ry

Rx = R cosq

q

Ry = R sin q

Rx

Summary for Vectors
  • A scalar quantity is completely specified by its magnitude only. (40 m, 10 gal)
  • A vector quantity is completely specified by its magnitude and direction. (40 m, 300)
summary continued

Resultant of Vectors:

R

Ry

q

Rx

Summary Continued:
  • Finding the resultant of two perpendicular vectors is like converting from polar (R, q) to the rectangular (Rx, Ry) coordinates.
component method for vectors
Component Method for Vectors
  • Start at origin and draw each vector in succession forming a labeled polygon.
  • Draw resultant from origin to tip of last vector, noting the quadrant of resultant.
  • Write each vector in i,j notation (Rx,Ry).
  • Add vectors algebraically to get resultant in i,j notation. Then convert to (R,q).
vector difference64

Now A – B: First change sign (direction) of B, then add the negative vector.

B

B

-B

A

R’

-B

A

A

Vector Difference

For vectors, signs are indicators of direction. Thus, when a vector is subtracted, the sign (direction) must be changed before adding.