Ch 1: Introduction and Math Concepts

Chapter 1

Introduction and Mathematical Concepts

Table of Contents

1. The Nature of Physics2. Units3. Role of Units in Problem Solving4. Trigonometry5. Scalars and Vectors6. Vector Addition and Subtraction7. Components of a Vector8. Addition of Vectors by Means of Components9. Other Stuff

Chapter 1: Introduction and Mathematical Concepts

Section 1 – The Nature of Physics

What is Physics? The “Fundamental Science”

Study of matter and how it moves through space-time

Applications of concepts such as Energy, and Force

The general analysis of the natural world

“understand” and predict how our universe behaves


Section 2 - Units

Units

To “understand” nature, we must first study what it does

Must have/use a universal way of describing what nature does

Systems of measurement “British” (American) Metric SI

Base Units

Most fundament forms of measurement Mass – kilogram (kg) Length – meter (m) Time – second (s) Count – mole (mol) Temperature – kelvin (K) Current – ampere (A) Luminous Intensity – candela (cd)

SI Features

Derived Units

Common combinations of base units

e.g.: area, force, pressure

Prefixes

Adjust scale of measurement

Metric – powers of 10

SI – powers of 1000

SI Prefixes 1024 yotta (Y) 1021 zetta (Z) 1018 exa (E) 1015 peta (P) 1012 tera (T) 109 giga (G) 106 mega (M) 103 kilo (k)

10-3 milli (m) 10-6 micro (µ) 10-9 nano (n) 10-12 pico (p) 10-15 femto (f) 10-18 atto (a) 10-21 zepto (z) 10-24 yocto (y)


Section 3 The Role of Units in Problem Solving

Conversion of Units

Remember from algebra…

Multiplying by 1 does not change number

If 1 m = 1000 mm, then

1 m/1000 mm = 1

Question #1

When we measure physical quantities, the units may be anything that is reasonable as long as they are well defined. It’s usually best to use the international standard units. Density may be defined as the mass of an object divided by its volume. Which of the following units would probably not be acceptable units of density?

a)gallons/liter b)kilograms/m3 c)pounds/ft3 d)slugs/yd3 e)grams/milliliter

Question #2

A car starts from rest on a circular track with a radius of 150 m. Relative to the starting position, what angle has the car swept out when it has traveled 150 m along the circular track?

a) 1 radian b) /2 radians c) radians d) 3/2 radians e) 2 radians

Question #3

A section of a river can be approximated as a rectangle that is 48 m wide and 172 m long. Express the area of this river in square kilometers.

a) 8.26 × 103 km2 b) 8.26 km2 c) 8.26 × 103 km2 d) 3.58 km2 e) 3.58 × 102 km2

Question #4 If one inch is equal to 2.54 cm, express

9.68 inches in meters.

a) 0.262 m b) 0.0381 m c) 0.0508 m d) 0.114 m e) 0.246 m

Dimensional Analysis When in doubt, look at the units Since units are part of the number, units

must balance out for a valid equation By analyzing the units, you can determine if

your solution is correct. If the units from your calculation do not

give you the units you need, you have an error

Example

DIMENSIONAL ANALYSIS

[L] = length [M] = mass [T] = time

221 vtx

Is the following equation dimensionally correct?

TLTT

LL 2

Question #5Using the dimensions given for the variables in the table, determine which one of the following expressions is correct.

a)

b)

c)

d) e)

f g

2l

2f g

l

2f l

gf 2 gl

g

lf

2

Question #6Given the following equation: y = cnat2, where n is

an integer with no units, c is a number between zero and one with no units, the variable t has units of seconds and y is expressed in meters, determine which of the following statements is true.

a) a has units of m/s and n =1.b) a has units of m/s and n =2.c) a has units of m/s2 and n =1.d) a has units of m/s2 and n =2.e) a has units of m/s2, but value of n cannot be

determined through dimensional analysis.

Question #7Approximately how many seconds are there in

a century?

a) 86,400 s

b) 5.0 × 106 s

c) 3.3 × 1018 s

d) 3.2 × 109 s

e) 8.6 × 104 s


Section 4 - Trigonometry

Basics you should remember…

h

hosin

h

hacos

a

o

h

htan

Basics you should remember…

h

ho1sin

h

ha1cos

a

o

h

h1tan

222ao hhh

Question #8Determine the angle in the right triangle shown.

a) 54.5

b) 62.0

c) 35.5

d) 28.0

e) 41.3

Question #9Determine the length of the side of the right triangle

labeled x.

a) 2.22 m

b) 1.73 m

c) 1.80 m

d) 2.14 m

e) 1.95 m

Question #10Determine the length of the side of the right triangle

labeled x.

a) 0.79 km

b) 0.93 km

c) 1.51 km

d) 1.77 km

e) 2.83 km


Section 5 – Scalar & Vectors

Scalar & Vector

A scalar quantity is one that can be

described by a single number:

temperature, speed, mass

A vector quantity deals inherently with both

magnitude and direction:

velocity, force, displacement

More on Vectors Arrows are used to represent vectors. The

direction of the arrow gives the direction of the vector.

By convention, the length of a vector arrow is proportional to the magnitude of the vector.

8 lb4 lb

Question #11Which one of the following statements is true

concerning scalar quantities?a) Scalar quantities must be represented by base

units.b) Scalar quantities have both magnitude and

direction.c) Scalar quantities can be added to vector quantities

using rules of trigonometry.d) Scalar quantities can be added to other scalar

quantities using rules of trigonometry.e) Scalar quantities can be added to other scalar

quantities using rules of ordinary addition.


Section 6 Vector Addition and Subtraction

Graphical Addition of vectors Remember length of arrow is proportional to

magnitude Angle of arrow proportional to direction Place tail of 2nd vector at tip of 1st Resultant starts at 1st and ends at 2nd

R

B

A

RBA

Graphical Subtraction of Vectors

Same as addition, multiply value by (-1) Resultant is still tail to tip

R

B-

A

RBA

B

Question #12Which expression is false concerning the vectors shown

in the sketch?

a)

b)

c)

d) C < A + B

e) A2 + B2 = C2

C A B

C A B

0A B C


Section 7 Components of a Vector

Vector Component

.AAA

AA

A

yx

that soy vectoriall together add and

axes, and the toparallel are that and vectors

larperpendicu twoare of components vector The

yxyx

Scalar Components

It is often easier to work with the scalar components rather than the vector components.

. of

componentsscalar theare and

A

yx AA

1. magnitude with rsunit vecto are ˆ and ˆ yx

yxA ˆˆ yx AA

In math, they are called i and j

Example Problem

A displacement vector has a magnitude of 175 m and points at an angle of 50.0 degrees relative to the x axis. Find the x and y components of this vector.

rysin

m 1340.50sinm 175sin ry

rxcos

m 1120.50cosm 175cos rx

yxr ˆm 134ˆm 112

Question #13

During the execution of a play, a football player carries the ball for a distance of 33 m in the direction 76° north of east. To determine the number of meters gained on the play, find the northward component of the ball’s displacement.

a) 8.0 m b) 16 m

c) 24 m d) 28 m

e) 32 m

Question #14

Vector has components ax = 15.0 and ay = 9.0. What is the approximate magnitude of vector ?

a) 12.0 b) 24.0

c) 10.9 d) 6.87

e) 17.5

a

Question #15

Vector has a horizontal component ax = 15.0 m and makes an angle = 38.0 with respect to the positive x direction. What is the magnitude of ay, the vertical component of vector ?

a) 4.46 m b) 11.7 m

c) 5.02 m d) 7.97 m

e) 14.3 m

a


Section 8 Addition of Vectors by Means of Components

Addition using components

BAC

yxA ˆˆ yx AA

yxB ˆˆ yx BB

yx

yxyxC

ˆˆ

ˆˆˆˆ

yyxx

yxyx

BABA

BBAA

xxx BAC yyy BAC

A

BC

xAyA xB

yB

xA xByA

yBC

Quesiton #16,17

The drawing above shows two vectors A and B, and the drawing on the right shows their components. Each of the angles θ = 31°.

When the vectors A and B are added, the resultant vector is R, so that R = A + B. What are the values for Rx and Ry, the x- and y-components of R?

Rx = m

Ry = m

Question #18,19

The displacement vectors A and B, when added together, give the resultant vector R, so that R = A + B. Use the data in the drawing and the fact that φ = 27° to find the magnitude R of the resultant vector and the angle θ that it makes with the +x axis.

R = m

θ = degrees

Question #20

Use the component method of vector addition to find the resultant of the following three vectors: = 56 km, east = 11 km, 22° south of east = 88 km, 44° west of south

A) 66 km, 7.1° west of south B) 97 km, 62° south of east

C) 68 km, 86° south of east D) 52 km, 66° south of east

E) 81 km, 14° west of south

C

B

A

Adding Multiple Vectors

n

1kkR FF

4321R FFFFF

F2

F3

F4

F1

Adding Vectors


F1F2

F3

F4

1

2

3 4

F1 = 50 N 1 = 30o

F2 = 100 N 2 = 135o

F3 = 30 N 3 = 250o

F4 = 40 N 4 = 300o

θsinF θcosF kkkk

43.3 25.070.7 70.710.3 28.220.0 34.6

17.7 32.9

22R 9.327.17F

N 4.37FR


22R 9.327.17F

N 4.37FR

7.17

9.32tan

7.17

9.32tan 1 o7.61

FR = 37.4 N

R

17.7

32.9

180R 7.61180

oR 118

Now You Try:

F1F2

F3

F4

1

2

3 4

F1 = 90 N 1 = 45o

F2 = 80 N 2 = 150o

F3 = 50 N 3 = 220o

F4 = 70 N 4 = 340o


“Section 9” Additional Stuff You Should Know

Basic Rules

Multiplication of 1 Multiplying a number by 1 doesn’t change

it Addition Property of Equality

Add the same thing to both sides Multiplication Property of Equality

Multiply both sides of equation by same thing

“undo” function on both sides

Inverse “Functions” for algebra

Addition

Multiplication

Square

Sine

log

ln

Add opposite (“-”)

Multiply by inverse

Square root

Arcsine

10x

ex

(“ “)

Graphing

Linear equations

y = mx + b

Quadratic equations

y = ax2 + bx + c

y = a(x-h)2 + k

Wave equations

y = A sin (x + ) + d

Effect of slope on a line

-25

-20

-15

-10

-5

0

5

10

15

20

25

0 2 4 6 8 10 12

y=-2x

y=-x

y=-x/2

y=0*x

y=x/2

y=x

y=2x

Effect of y-intercept

-4

-2

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12

y=x-3

y=x-2

y=x-1

y=x

y=x+1

y=x+2

y=x+3

Effect of "a"

-250-200-150-100-50

050

100150200250

-15 -10 -5 0 5 10 15

y=-2x2+x+1

y=-x2+x+1

y=-x2/2+x+1

y=0x2+x+1

y=x2/2+x+1

y=x2+x+1

y=2x2+x+1

Education

Ch 1: Introduction and Math Concepts