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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
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
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)
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
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
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.
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
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
Chapter 1: Introduction and Mathematical Concepts
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
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
Adding Multiple Vectors
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
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