Transcript Physics - The Mathematics of Basic Physics

Chapter 1
Introduction and Mathematical Concepts
Table of Contents
1.
2.
3.
4.
5.
6.
7.
8.
9.
The Nature of Physics
Units
Role of Units in Problem Solving
Trigonometry
Scalars and Vectors
Vector Addition and Subtraction
Components of a Vector
Addition of Vectors by Means of Components
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
Chapter 1: Introduction and
Mathematical Concepts
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)
Chapter 1: Introduction and
Mathematical Concepts
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 × 103 km2
c) 8.26 × 103 km2
e) 3.58 × 102 km2
b) 8.26 km2
d) 3.58 km2
Question #4
 If one inch is equal to 2.54 cm, express
9.68 inches in meters.
a) 0.262 m
c) 0.0508 m
e) 0.246 m
b) 0.0381 m
d) 0.114 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
Is the following equation dimensionally correct?
2
1
2
x  vt
L 2
L   T  LT
T 
Question #5
Using the dimensions given for the variables in the table,
determine which one of the following expressions is
correct.
g
a) f  2  l
b)
c)
d)
f 
2l
g
2 f 
g
l
2 f 
l
g
e)
f  2  gl
Question #6
Given 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 #7
Approximately 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
Chapter 1: Introduction and
Mathematical Concepts
Section 4 - Trigonometry
Basics you should remember…
ho
sin  
h
ha
cos  
h
ho
tan 
ha
Basics you should remember…
 ho 
  sin  
h
1  ha 
  cos  
h
1  ho 
  tan  
 ha 
1
h  h h
2
2
o
2
a
Question #8
Determine the angle  in the right triangle shown.
a) 54.5
b) 62.0
c) 35.5
d) 28.0
e) 41.3
Question #9
Determine 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 #10
Determine 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
Chapter 1: Introduction and
Mathematical Concepts
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 lb
4 lb
Question #11
Which 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.
Chapter 1: Introduction and
Mathematical Concepts
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
  
AB R

R


A

B
Graphical Subtraction of Vectors
 Same as addition, multiply value by (-1)
 Resultant is still tail to tip
  
AB R

A


R

B

-B
Question #12
Which expression is false concerning the vectors
shown in the sketch?
a)
C  A  B
b)
A B C  0
c)
C  A B
d)
C <A+B
e)
A2 + B2 = C2
Chapter 1: Introduction and
Mathematical Concepts
Section 7
Components of a Vector
Vector Component

T hevectorcomponentsof A are two perpendicular


vectorsA x and A y thatare parallelto the x and y axes,
 

and add togethervectorially so thatA  A x  A y .
Scalar Components
It is often easier to work with the scalar components
rather than the vector components.
Ax and Ay are thescalar components

of A.
xˆ and yˆ are unit vectors with magnitude1.

A  Ax xˆ  Ay yˆ
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.
cos  x r
sin   y r


y  r sin   175msin 50.0   134m
x  r cos  175m cos50.0  112m


r  112mxˆ  134myˆ
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 a 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
Question #15
Vector a 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
Chapter 1: Introduction and
Mathematical Concepts
Section 8
Addition of Vectors by Means of
Components
Addition using components

C

A

B
Ay
Ax
  
C AB

A  Ax xˆ  Ay yˆ

B  Bx xˆ  By yˆ
By

C
By
Bx
Ay
Ax
Bx

C  Ax xˆ  Ay yˆ  Bx xˆ  B y yˆ
  Ax  Bx xˆ  Ay  B y yˆ
Cx  Ax  Bx
Cy  Ay  By
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 xand 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:

A= 56 km, east

B = 11 km, 22° south of east

C = 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
Adding Multiple Vectors
Adding Vectors
F2
F1
n 

FR   Fk
k 1

 


FR  F1  F2  F3  F4
F4
F3
Adding Multiple Vectors
2
F2
F1
1
3
F3
4
F4
F1 = 50 N 1 = 30o
F2 = 100 N 2 = 135o
F3 = 30 N 3 = 250o
F4 = 40 N
4 = 300o
Fk cosθk 
43.3
70.7
10.3
20.0
17.7
FR 
Fk sinθk 
25.0
70.7
28.2
34.6
32.9
 17.7 2  32.92
FR  37.4 N
Adding Multiple Vectors
FR = 37.4 N

R
32.9
17.7
32.9
tan  
 17.7
1  32.9 
  tan 
  61.7o
  17.7 
R  180    180   61.7 
FR 
 17.7 2  32.92
FR  37.4 N
 R  118o
Now You Try:
2
F2
F1
1
3
F3
F1 = 90 N
F2 = 80 N
F3 = 50 N
F4 = 70 N
4
F4
1 = 45o
2 = 150o
3 = 220o
4 = 340o
Chapter 1: Introduction and
Mathematical Concepts
“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
 Add opposite (“-”)
 Multiplication
 Multiply by inverse (“ “)
 Square
 Square root
 Sine
 Arcsine
 log
 10x
 ln
 ex
Graphing
 Linear equations
y = mx + b
 Quadratic equations
y = ax2 + bx + c
y = a(x-h)2 + k
 Wave equations
 y = A sin (w x + ) + d
Effect of slope on a line
25
20
15
y=-2x
10
y=-x
5
y=-x/2
0
y=0*x
-5
-10
-15
-20
-25
0
2
4
6
8
10
12
y=x/2
y=x
y=2x
Effect of y-intercept
14
12
y=x-3
10
y=x-2
8
y=x-1
6
y=x
4
y=x+1
2
y=x+2
0
-2 0
-4
2
4
6
8
10
12
y=x+3
Effect of "a"
250
200
150
100
50
0
-50
-100
-150
-200
-250
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
-15
-10
-5
0
5
10
15