Introduction to Mechanics
Order of Magnitude
Vectors
Lana Sheridan
De Anza College
Jan 11, 2016
Last time
• precision and accuracy
• dimensional analysis
• unit conversions (non-SI units)
Overview
• order of magnitude calculations
• vectors
• trigonometry
Order of Magnitude Calculation
One way to get a hypothesis what an answer should be: do an
Order of Magnitude Calculation.
This is a useful tool for estimating the answer.
The goal is just to get an idea of how big the answer should be.
Order of magnitude examples
About how many times does your heart beat during your
life?
Order of magnitude examples
About how many times does your heart beat during your
life?
Your heart rate?
Order of magnitude examples
About how many times does your heart beat during your
life?
Your heart rate? Call it 100 (102 ) beats per minute for simplicity.
How many minutes in a life...?
years in a life × minutes in a year × beats in a minute
Order of magnitude examples
About how many times does your heart beat during your
life?
Your heart rate? Call it 100 (102 ) beats per minute for simplicity.
How many minutes in a life...?
years in a life × minutes in a year × beats in a minute
Years in a life?
Order of magnitude examples
About how many times does your heart beat during your
life?
Your heart rate? Call it 100 (102 ) beats per minute for simplicity.
How many minutes in a life...?
years in a life × minutes in a year × beats in a minute
Years in a life? Optimistic: 100 = 102 .
Minutes in a year:
365 × 24 × 60 ≈ 400 × 25 × 50 = 500, 000 = 5 × 105 min/year
Order of magnitude examples
About how many times does your heart beat during your
life?
Total heart beats in your life:
years in a life × minutes in a year × beats in a minute
(102 years) × (5 × 105 min/year) × (102 beats/min)
= 5 × 109 beats
= 5 billion beats
1
vendian.org
Order of magnitude examples
What is the radius of the Earth?
Order of magnitude examples
What is the radius of the Earth?
If you fly across the United States, how many time zones do you
cross?
Order of magnitude examples
What is the radius of the Earth?
If you fly across the United States, how many time zones do you
cross? Answer: 3.
Order of magnitude examples
What is the radius of the Earth?
If you fly across the United States, how many time zones do you
cross? Answer: 3.
What is the average distance across the US?
Order of magnitude examples
What is the radius of the Earth?
If you fly across the United States, how many time zones do you
cross? Answer: 3.
What is the average distance across the US? Answer: about 3000
miles.
On average, there are about 1000 miles of distance traveled per
time zone.
How many time zones are around the Earth?
Order of magnitude examples
What is the radius of the Earth?
If you fly across the United States, how many time zones do you
cross? Answer: 3.
What is the average distance across the US? Answer: about 3000
miles.
On average, there are about 1000 miles of distance traveled per
time zone.
How many time zones are around the Earth?
There must be 24 time zones around the earth in all since there
are 24 hours in the day.
Order of magnitude examples
What is the circumference of the Earth?
1
maa.org
Order of magnitude examples
What is the circumference of the Earth? Answer: about 24,000
miles.
The circumference of a circle is c = 2πr where r is the radius.
Take 2π ≈ 6. The radius of the Earth:
r=
1
maa.org
24, 000 mi
c
≈
= 4, 000 mi
2π
6
Order of magnitude examples
What is the circumference of the Earth? Answer: about 24,000
miles.
The circumference of a circle is c = 2πr where r is the radius.
Take 2π ≈ 6. The radius of the Earth:
r=
24, 000 mi
c
≈
= 4, 000 mi
2π
6
1 mi ≈ 1.6 km
Radius of the Earth in meters:
4, 000 mi × 1600 m/mi = 6, 400, 000 m = 6.4 × 106 m
1
maa.org
Order of magnitude examples
What is the circumference of the Earth? Answer: about 24,000
miles.
The circumference of a circle is c = 2πr where r is the radius.
Take 2π ≈ 6. The radius of the Earth:
r=
24, 000 mi
c
≈
= 4, 000 mi
2π
6
1 mi ≈ 1.6 km
Radius of the Earth in meters:
4, 000 mi × 1600 m/mi = 6, 400, 000 m = 6.4 × 106 m
Actual answer: 6.37 × 106 m
1
maa.org
Pretty close!
Vectors and Scalars
scalar
A scalar quantity indicates an amount. It is represented by a real
number. (Assuming it is a physical quantity.)
Vectors and Scalars
scalar
A scalar quantity indicates an amount. It is represented by a real
number. (Assuming it is a physical quantity.)
vector
A vector quantity indicates both an amount (magnitude) and a
direction. It is represented by a real number for each possible
direction, or a real number and (an) angle(s).
Vectors and Scalars
scalar
A scalar quantity indicates an amount. It is represented by a real
number. (Assuming it is a physical quantity.)
vector
A vector quantity indicates both an amount (magnitude) and a
direction. It is represented by a real number for each possible
direction, or a real number and (an) angle(s).
In the lecture notes vectors are represented using bold variables.
Notation for Vectors
In the lecture notes vector variables are represented using bold
variables.
Example:
k is a scalar
x is a vector
In the textbook and in writing, vectors are often represented with
an over-arrow: ~x
The magnitude of a vector, x is written:
|x| = x
Examples of Scalars and Vectors
Some physical quantities that are scalars are
• temperature
• mass
• pressure
Some physical quantities that are vectors are
• velocity
• force
How can we write vectors? - with angles
Bearing angles
Example, a plane flies at a bearing of 70◦
N
E
Generic reference angles
A baseball is thrown at 10 m s−1 at 30◦ above the horizontal.
How can we write vectors? - as a list
A vector in the x, y -plane could be written
2, 1 or
2 1
or
2
1
(In some textbooks it is written h2, 1i, but there are reasons not to
write it this way.)
along coordinate axes. These projections are called the components of the vector or its
rectangular
components.
Any vector
cana
be list
completely described by its
How
can
we write
vectors?
- as
components.
A vector ainvector
the x,S
-plane
written
Aylying
Consider
in could
the xybe
plane
and making an arbitrary angle u
with the positive x axis as shown in S
Figure 3.12a. Thisvector
as the
can be expressed
S
is parallel to
2 the x axis, and A y , which
sum of two other component vectors
A x , which
1 Figure
or 23.12b,
1 we
or see that the three vectors form a
is parallel to the y axis.2,
From
1
S
S
S
right triangle
and
that
A
5
A
1
A
.
We
shall
often
refer to the “components
x
y
S
of a vector A ,” written Ax and Ay (without
the
boldface
notation). The compoS
nent
A
represents
the
projection
of
A
along
the
x
axis,
and reasons
the component
(In xsome textbooks it isS written h2, 1i, but there are
not to Ay
represents the projection of A along the y axis. These components can
be
positive
S
it this
or write
negative.
The way.)
component Ax is positiveSif the component vector A x points in
the positive x direction and is negative if A x points in the negative x direction. A
similar
statement
forythe
component
.
When
drawn isinmade
the x,
-plane
it looksAylike:
y
y
F
1
l
r
1
S
S
S
A
Ay
u
O
a
S
Ax
S
A
2
x
Ay
u
O
b
S
Ax
2
x
v
r
v
v
f
along coordinate axes. These projections are called the components of the vector or its rectangular
components. Any vector can be completely described by its
Vector
Components
components.
A vector ainvector
the x,S
-plane
written
Aylying
Consider
in could
the xybe
plane
and making an arbitrary angle u
with the positive x axis as shown in S
Figure 3.12a. Thisvector
as the
can be expressed
S
is parallel to
2 the x axis, and A y , which
sum of two other component vectors
A x , which
2,From
1 Figure
or 23.12b,
1 we
or see that the three vectors form a
is parallel to the y axis. S
1
S
S
right triangle
and
that
A
5
A
1
A
.
We
shall
often
refer to the “components
x
y
S
of a vector A ,” written Ax and Ay (without
the boldface notation). The compoS
nent Ax represents the projection
of A along the x axis, and the component Ay
S
represents
projection
A along the y axis.
These
components
can
be positive
We saythe
that
2 is theofx-component
of the
vector
(2, 1) and
S1 is the
or negative. The component Ax is positiveSif the component vector A x points in
y -component.
the positive x direction and is negative if A x points in the negative x direction. A
similar statement is made for the component Ay.
y
y
1
F
1
S
S
S
A
Ay
O
a
S
Ax
S
A
u
2
x
Ay
u
O
b
S
Ax
l
r
2
x
v
r
v
v
f
Vectors Properties and Operations
Equality
Vectors A = B if and only if the magnitudes and directions are the
same. (Each component is the same.)
When two vectors are added, the sum is indep
tion. (This fact may seem trivial, but as you will
important when vectors are multiplied. Procedure
cussed in Chapters 7 and 11.) This property, which
construction in Figure 3.8, is known as the commu
Vectors Properties and Operations
Equality
S
S
S
S
1 Bthe
5 B 1 A
Commutative
lawifofand
addition
Vectors
A=B
only if the magnitudes and directionsA are
same. (Each component is the same.)
Addition
S
A+B
S
A "
S
C
S
S
B
S
R!
S
B
A"
S
R !
S
B "
S
C "
D
S
D
S
B
S
S
A
A
S
Figure 3.6
When vector B is
S
S
added to vector A , the resultant R is
the vector that runs from the tail of
S
S
Figure 3.7 Geometric construction for summing four vectors. The
S
resultant vector R is by definition
Vectors Properties and Operations
Doing addition:
To add vectors, break each vector into components and sum each
component independently.
Example
w = 5 m at 36.9◦ above the horizontal.
u = 17 m at 28.1◦ above the horizontal.
Vectors Properties and Operations
Doing addition:
To add vectors, break each vector into components and sum each
component independently.
Example
w = 5 m at 36.9◦ above the horizontal.
u = 17 m at 28.1◦ above the horizontal.
This means w = 4 i + 3 j m and u = 15 i + 8 j m.
w+u = ?
Vectors Properties and Operations
Doing addition:
To add vectors, break each vector into components and sum each
component independently.
Example
w = 5 m at 36.9◦ above the horizontal.
u = 17 m at 28.1◦ above the horizontal.
This means w = 4 i + 3 j m and u = 15 i + 8 j m.
w+u = ?
= (4 + 15)i + (3 + 8)j
= (19 i + 11 j) m
or 22.0 m at 30.1◦ above the horizontal.
he bookkeeping
nents
separately.
Vectors
Properties and Operations
Doing addition:
To add vectors, break each vector into components and sum each
component independently.
y
(3.14)
ant vector are
(3.15)
s, we add all the
ctor and use the
omponents with
Ry
S
By
R
Ay
S
B
S
A
x
Bx
Ax
Rx
btained from its
Figure 3.16
This geometric
construction in Figure 3.8, is known as the commutative law of addition:
S
S
S
S
A 1 BOperations
5 B 1 A
Vectors Properties and
Properties of Addition
f addition
(3.5)
S
Draw B ,
S
then add A .
S
S
S
B "
S
S
C
S
S
S
R !
B
S
A
S
B
S
Draw A ,
S
then add B .
S
A
• (A + B)
+3.7
C Geometric
= A +construc(B + C) (associative)
Figure
Figure 3.8 This construction
S
S
!
S
C
S
C
S
S
S
(A S
!
S
S
A!B
S
S
B
A
S
B)
!
(B S
!
C)
S
C
B!C
S
S
3.3S Some Properties of Vectors
shows that A 1 B 5 B 1 A or, in
other words, that vector addition is
commutative.
S
S
B;
Add A and
S
then add C to
the result.
tion for summing four vectors. The
S
resultant vector R is by definition
the one that completes the polygon.
S
S
Add B and C ;
then add the
S
result to A.
S
vector B is
S
he resultant R is
from the tail of
A
S
B
R! S
B"
S
C "
B
A "
B
S
S
D
S
A!
S
D
S
A
A"
S
B
• A + B = B + A (commutative)
S
B
S
A
Figure 3.9 Geo
tions for verifyin
law of addition.
S
Consider a vector A lying in the xy plane and making an arbitrary angle
ithComponents
the positive x axis as shown in S
Figure 3.12a. This vector can be expressed
as th
S
um of two other component vectors A x , which is parallel to the x axis, and A y , whic
Consider the 2 dimensional vector A.
parallel to the y axis. S
From SFigure
3.12b, we see that the three vectors form
S
ight triangle
and
that
A
5
A
1
A
shall
often the
referhead
to the
“component
y. Weby
SinceSthe two vectors addx together
attaching
of one
to
f a vector
A
,”
written
A
and
A
(without
the
boldface
notation).
The compo
x whichy is the
S same as adding the components,
the tail of the other,
ent Ax represents the projection
of A along the x axis, and the component A
we can always write aSvector in the x, y -plane as the sum of two
epresents the projection of A along the y axis. These components can
be positiv
S
component
vectors.
r negative. The component Ax is positiveSif the component vector A x points i
he positive x direction and is negative if A x points in the negative x direction.
A = Ax + A
milar statement is made for the component
Ayy.
y
y
S
S
S
A
Ay
u
O
a
S
Ax
A
x
u
O
b
S
Ax
S
Ay
x
um of two other component vectors A x , which is parallel to the x axis, and A y , whic
parallel to the y axis. S
From SFigure
3.12b, we see that the three vectors form
S
Components
ight triangle
and
that
A
5
A
1
A
x
y. We shall often refer to the “component
S
f a vector A ,” written Ax and Ay (without
the boldface notation). The compo
S
ent Ax represents the projection
of A along the x axis, and the component A
S
Forthe
example,
the y axis.
components can
epresents
projection of A along
These
be positiv
S
1 = 2, 0 + 0, 1
r negative. The component A2,
x is positiveSif the component vector A x points i
he positive
x direction
negative
if A x points
the negative
x direction.
We then
say thatand
2 isisthe
x-component
of theinvector
(2, 1) and
1
milar statement
is
made
for
the
component
A
.
y
is the y -component.
y
y
S
S
S
A
Ay
u
O
a
S
Ax
A
x
u
O
b
S
Ax
S
Ay
x
Representing Vectors: Unit Vectors
We can write a vector in the x, y -plane as the sum of two
component vectors.
To indicate the components we define unit vectors.
Representing Vectors: Unit Vectors
We can write a vector in the x, y -plane as the sum of two
component vectors.
To indicate the components we define unit vectors.
Unit vectors have a magnitude of one unit.
Representing Vectors: Unit Vectors
We can write a vector in the x, y -plane as the sum of two
component vectors.
To indicate the components we define unit vectors.
Unit vectors have a magnitude of one unit.
In two dimensions, a pair of perpendicular unit vectors are usually
denoted i and j (or sometimes x̂, ŷ).
Representing Vectors: Unit Vectors
We can write a vector in the x, y -plane as the sum of two
component vectors.
To indicate the components we define unit vectors.
Unit vectors have a magnitude of one unit.
In two dimensions, a pair of perpendicular unit vectors are usually
denoted i and j (or sometimes x̂, ŷ).
A 2 dimensional vector can be written as v = (2, 1) = 2i + j.
um of two other component vectors A x , which is parallel to the x axis, and A y , whic
parallel
to the y axis. S
From SFigure
3.12b, we see that the three vectors form
Components
S
ight triangle
and
that
A
5
A
1
A
. We shall often refer to the “component
x
y
S
f a vector A ,” written Ax and Ay (without
the boldface notation). The compo
Vector A is the sum of a piece Salong x and a piece along y :
ent Ax represents the projection
of
A
along
the x axis, and the component A
S
A = A i + Ay j.
epresents thexprojection
of A along the y axis. These components can
be positiv
S
r negative. The component Ax is positiveSif the component vector A x points i
Ax is the i-component (or x-component) of A and
he positive x direction and is negative if A x points in the negative x direction.
Ay is the j-component (or y -component) of A.
milar statement
is made for the component Ay.
y
y
S
S
S
A
Ay
u
O
a
S
Ax
A
x
u
O
b
Notice that Ax = A cos θ and Ay = A sin θ.
S
Ax
S
Ay
x
Summary
• order of magnitude calculations
• vectors
• trigonometry
Homework (will not be collected)
Walker Physics:
• Ch 1, onward from page 14. Probs: 37, 39
• Ch 3, onward from page 76. Questions: 2, 4, 11. Problems:
1, 5, 7, 11, 13 (Some of these require trigonometry. We will
discuss this tomorrow.)