Download If and

Mathematics
Review
Mathematics is a language.
The Meaning
of Numbers
Algebra
It is used to describe the world around us.
Can you tell me what this means?
N

Trigonometry
Geometry
Choosing the
Correct
Equation(s)
i 1
i
 I
If you understand only “how to do” the math, then you will need to know
the numbers to see any meaning behind this equation.
However, if you understand the “meaning” of the math, then the
equation itself tells you a great deal about how nature works.
The equation says the following…
1
“The total torque acting on an object is the same as its moment of inertia
multiplied by its angular acceleration.”
Mathematics
Review
Mathematics is a language.
N

The Meaning
of Numbers
i 1
Algebra
Trigonometry
Geometry
Choosing the
Correct
Equation(s)
i
 I
means that
“The total torque acting on an object is the same as its moment of inertia
multiplied by its angular acceleration.”
You see i is any of N torques, I is moment of inertia and  is angular
acceleration.
N
 means to sum all of what is behind it for every value of i from 1 to N .
i1
2
But you still do not know the meaning of torque, moment of inertia and
angular acceleration.
Mathematics
Review
Mathematics is a language.
The Meaning
of Numbers
Algebra
Trigonometry
Geometry
Choosing the
Correct
Equation(s)
N

i 1
i
 I
means that
“The total torque acting on an object is the same as its moment of inertia
multiplied by its angular acceleration.”
But you still do not know the meaning of torque, moment of inertia and
angular acceleration.
Torque is a measure of how hard you are trying turn something. Moment
of inertia tells us how hard it is to change how fast it turns. And angular
acceleration measures how much it changes how fast it turns.
3
Mathematics
Review
Mathematics is a language.
The Meaning
of Numbers
Algebra
Trigonometry
Geometry
Choosing the
Correct
Equation(s)
N

i 1
i
 I
means that
“The total torque acting on an object is the same as its moment of inertia
multiplied by its angular acceleration.”
Torque is a measure of how hard you are trying turn something. Moment
of inertia tells us how hard it is to change how fast it turns. And angular
acceleration measures how much it changes how fast it turns.
This same equation can describe a grinding wheel, the hands of a clock,
the motion of a wrench, and an infinite number of other situations.
4
If I plug in the numbers as an example, I only learn ONE of the situations!!
Mathematics
Review
The Meaning
of Numbers
Algebra
Trigonometry
You need to remember algebra.
Here are some of the basics…
If A  B  C
then A  B  B  C  B
Distributive property
A  B  C    A  B   A  C 
ACB
Geometry
Choosing the
Correct
Equation(s)
If A  B  C
Commutative properties
then A B / B  C / B
A B  B  A
AC/B
A B  B  A
…and similarly for square roots, squares, subtraction and division except
that subtraction and division are NOT commutative!
5
Mathematics
Review
Some simple examples…
The Meaning
of Numbers
Algebra
Trigonometry
If F1x  F2 x  max
then F1x  F2 x  max
Distributive property
m  v1x  v2 x   mv1x  mv2 x
Geometry
Choosing the
Correct
Equation(s)
If maz  FTOTALz
then m  FTOTALz / az
Commutative properties
m1  m2  m2  m1
max  ax m
…and similarly for square roots, squares, subtraction and division except
that subtraction and division are NOT commutative!
m1  m2  m2  m1
6
m / ax  ax / m
Mathematics
Review
Here are some other helpful concepts…
The Meaning
of Numbers
Algebra
Trigonometry
If A  C
and B  C
then
Ratios
If AC  D
and BC  E
AB
…even if YOU DON’T KNOW C!
Geometry
Simultaneous Equations
Choosing the
Correct
Equation(s)
A D

then
B E
…even if YOU DON’T KNOW C!
a1 A  b1B  c1
+
a2 A  b2 B  c2
 da1  a2  A   db1  b2  B   dc1  c2 
7
usually we use this in such a way that one
of the coefficients is zero
Mathematics
Review
Here are some more simple examples…
The Meaning
of Numbers
Algebra
Trigonometry
If F1x  ma1
and F2 x  ma1
Ratios
If m1L  T1
and m2 L  K 2
then F1x  F2 x
…even if YOU DON’T KNOW m or a1!
Geometry
Simultaneous Equations
Choosing the
Correct
Equation(s)
m1 T1

then
m2 K 2
…even if YOU DON’T KNOW L!
2 F  6Q  10
+
12 F  5Q  3
 6  2 12 F   6 6  5 Q  6 10   3
8
usually we use this in such a way that one
of the coefficients is zero
0 F  35Q  57
57
Q
35
Mathematics
Review
This is the unit circle…
The Meaning
of Numbers
sine
The
second
Also
note
that to
The
distance
quadrant
has
the
is the
thecosine
right
on
negative
negative
ifisthe
cos-axiscosines
the and
positive
sines.
line
is drawn
cosine
of theto
and

It axes are
Algebra
Trigonometry
Geometry
the
left onIIthe
angle.
cos-axis.
1
2
We
speak of
Hereoften
are some
four
otherquadrants.
examples.
9
sin
3
2
cosine
Choosing the
Correct
Equation(s)
Note that the angle
The
thirdgoes
quadrant
always
from
has
thenegative
positivecosines
cos-axis
and
sines.
counterclockwise.
1
The first
quadrant
All lines
drawn
has positive
cosines
here have
a
and sines.
length of 1
and an angle
I equal to the
angle we are
working with.
150o
o
330
1
2
3
2
o
45
30o
1
12
2
11

1
1
2
3
2
III
IV
cos
Why
is the sine
The height
negative
along thehere?
sinaxis
the sine
The is
fourth
of
the angle.
quadrant
has
positive cosines
and negative
sines.
Mathematics
Review
Given a right triangle, the trigonometric functions for either
non-right angle are given by the following…
The Meaning
of Numbers
Algebra
Trigonometry
opposite
(o)
hypotenuse
(h)
Geometry
Choosing the
Correct
Equation(s)
θ
adjacent
(a)
o
sin  
h
h
csc  
o
a
cos  
h
h
sec  
a
o
tan  
a
a
cot  
o
The value of the angle can also be determine by using any two of the
sides. For example,
o
tan    
a
1
10
Mathematics
Review
Here is an example of how to use it…
The Meaning
of Numbers
Algebra
Trigonometry
Geometry
Choosing the
Correct
Equation(s)
3
5
csc  36.87  
5
3
4
5
h=5
cos  36.87   sec  36.87  
5
4
Note: This
is NOT
3
4
tan  36.87  
cot  36.87  
drawn to
4
3
scale!
o
θ=36.87
sin 36.87 
o=3
a=4
The value of the angle can also be determine by using any two of the
sides. For example,
3
tan    36.87
4
1
11
Mathematics
Review
Here are some useful angle relations…
a
The Meaning
of Numbers
a
Algebra
b
a b
b a
a b
b b
b
a  b  180 
a
Trigonometry
a  b  180
Geometry
a  b  180 

b
Choosing the
Correct
Equation(s)
a
a  b  c  180
a
a
12
C
c

a
b
A
c
B
A
B
C


sin a sin b sin c
a
Mathematics
Review
For example…
The Meaning
of Numbers
Algebra
Trigonometry
Geometry
Choosing the
Correct
Equation(s)
13
a
If a  30 then b  150
b
a  b  180

Mathematics
Review
The Meaning
of Numbers
Algebra
Trigonometry
Geometry
Choosing the
Correct
Equation(s)
Here are some basic geometric and trigonometric formulae which we
will use often in this and the next class…
Circumference of a Circle
Trigonometric Formulae
sin 2   cos 2   1
Area of a Circle
A  r
Surface Area of a Sphere
A  4r 2
Volume of a Sphere
4 3
V  r
3
Quadratic Formula
A  2rL
Ax 2  Bx  C  0
V  r 2 L
 B  B 2  4 AC
x
2A
Surface Area of a Cylinder
(not including end faces)
Volume of a Cylinder
14
C  2r
2
sin  A  B   sin A cos B  cos A sin B
cos A  B   cos A cos B  sin A sin B
Mathematics
Review
Finding out which equation or set of equations to use while solving a
problem in physics is the most difficult part of the process.
The Meaning
of Numbers
It is also the most crucial part!
Algebra
Trigonometry
Geometry
Choosing the
Correct
Equation(s)
15
Still, if you follow a few basic steps, the difficulty will be far less and
you will need to spend much less time on PreAssignments, Homework
and Exams.
An example solved by a naïve student (Bailey D. Wonderdog’s nemesis,
the neighbors cat, for instance) will help us see what the rules are and
how to apply them.
The velocity of a 40 g baseball is initially 10 m/s north. After it is hit by
a bat that is moving at 5 m/s south, the ball is now moving 10 m/s south.
The ball has a radius of 20 cm. What is the impulse that caused the ball
to change its velocity?
Mathematics
Review
An example of choosing the correct equations.
The Meaning
of Numbers
Algebra
The velocity of a 40 g baseball is initially 10 m/s north. After it is hit by
a bat that is moving at 5 m/s south, the ball is now moving 10 m/s south.
The ball has a radius of 20 cm. What is the impulse that caused the ball
to change its velocity?
Trigonometry
Geometry
If you look in your textbook, you will find the equation
V  IR
Choosing the
Correct
Equation(s)
At first this naively appears to be the simplest equation we can use for
this problem.
We might be tempted to guess that V is the velocity, I is the impulse ,
and R is the radius.
16
Let’s try this …
Mathematics
Review
An example of choosing the correct equations.
The Meaning
of Numbers
Algebra
The velocity of a 40 g baseball is initially 10 m/s north. After it is hit by
a bat that is moving at 5 m/s south, the ball is now moving 10 m/s south.
The ball has a radius of 20 cm. What is the impulse that caused the ball
to change its velocity?
V  IR
Trigonometry
Geometry
Choosing the
Correct
Equation(s)
When trying to plug in the numbers, we see our first challenge. There
two different objects and each have different velocities.
For this
problem,
the answers
are…
Which one do we choose?
To answer this, we must ask ourselves two things.
17
1.
2.
What physical quantity are we looking for?
What object is that physical variable related to?
1.
2.
Impulse
The ball
Mathematics
Review
An example of choosing the correct equations.
The Meaning
of Numbers
Algebra
Trigonometry
Geometry
Choosing the
Correct
Equation(s)
The velocity of a 40 g baseball is initially 10 m/s north. After it is hit by
a bat that is moving at 5 m/s south, the ball is now moving 10 m/s south.
The ball has a radius of 20 cm. What is the impulse that caused the ball
to change its velocity?
V  IR
Thus, we would use the quantities associated with the ball in this
problem.
Rule #1: We must know which object we are considering in a problem.
Plugging in the numbers, we see that I 
18
V 10

 0.5.
R 20
If we plug this into the homework software, it will tell us we are
incorrect. What went wrong.
Mathematics
Review
An example of choosing the correct equations.
The Meaning
of Numbers
Algebra
Trigonometry
Geometry
Choosing the
Correct
Equation(s)
The velocity of a 40 g baseball is initially 10 m/s north. After it is hit by
a bat that is moving at 5 m/s south, the ball is now moving 10 m/s south.
The ball has a radius of 20 cm. What is the impulse that caused the ball
to change its velocity?
V  IR
Well, first of all, the velocity is in m/s and the radius is in cm. So, we
have to convert one of the units to make them the same. You will learn
how to do this in the lecture called “Math for Physics”. If we do it
properly in this case, we find that…
m

10

V 
s   100 cm 
1
I 

  50 s
R
20 cm  1 m 
19
Rule #2: Use the proper units.
Mathematics
Review
An example of choosing the correct equations.
The Meaning
of Numbers
Algebra
Trigonometry
Geometry
Choosing the
Correct
Equation(s)
The velocity of a 40 g baseball is initially 10 m/s north. After it is hit by
a bat that is moving at 5 m/s south, the ball is now moving 10 m/s south.
The ball has a radius of 20 cm. What is the impulse that caused the ball
to change its velocity?
V  IR
Next we go back to the very first thing we learned in this lecture. The
variables of physics are words in the language of math.
If we look up the equation again and read carefully, we will find that it
means…
20
The voltage across a resistive element in a circuit is the same as the
current through it multiplied by its resistance. The variables are not
even close to what we wanted to use!!!!
Mathematics
Review
An example of choosing the correct equations.
The Meaning
of Numbers
Algebra
The velocity of a 40 g baseball is initially 10 m/s north. After it is hit by
a bat that is moving at 5 m/s south, the ball is now moving 10 m/s south.
The ball has a radius of 20 cm. What is the impulse that caused the ball
to change its velocity?
V  IR
Trigonometry
Geometry
Choosing the
Correct
Equation(s)
Rule #3: Know the
meaning of each
variable.
We now look up the word impulse in the index of our book or in the
notes and find that the variable that represents it is J.
We now find two equations that contain J on the website for the class.
tf
J 
 Fdt
t0
21
and
But which one should we use?
J  mv
Mathematics
Review
An example of choosing the correct equations.
The Meaning
of Numbers
Algebra
The velocity of a 40 g baseball is initially 10 m/s north. After it is hit by
a bat that is moving at 5 m/s south, the ball is now moving 10 m/s south.
The ball has a radius of 20 cm. What is the impulse that caused the ball
to change its velocity?
Rule #4: Use what is
Trigonometry
Geometry
Choosing the
Correct
Equation(s)
J  mv
known and unknown
to sort out equations
that are not useful.
is the only one of the two equations for which we have all of the
information to solve. It reads…
The impulse on an object is the same as its mass multiplied by the
change in its velocity.
We know the mass and the change in the velocity of the ball.
22
The other equation needed force and time, neither of which is known.
Mathematics
Review
An example of choosing the correct equations.
The Meaning
of Numbers
Algebra
The velocity of a 40 g baseball is initially 10 m/s north. After it is hit by
a bat that is moving at 5 m/s south, the ball is now moving 10 m/s south.
The ball has a radius of 20 cm. What is the impulse that caused the ball
to change its velocity?
Trigonometry
Geometry
Choosing the
Correct
Equation(s)
J  mv
Now we just need to plug in the ball’s mass (40 g) and its change in
velocity. It had 10 m/s to begin with and ended with 10 m/s as well.
Thus,
J  mv  10 g  0 m / s   0 g m / s
But the homework software
stills says that we are incorrect!
23
Mathematics
Review
An example of choosing the correct equations.
The Meaning
of Numbers
Algebra
The velocity of a 40 g baseball is initially 10 m/s north. After it is hit by
a bat that is moving at 5 m/s south, the ball is now moving 10 m/s south.
The ball has a radius of 20 cm. What is the impulse that caused the ball
to change its velocity?
Trigonometry
Geometry
Choosing the
Correct
Equation(s)
J  mv
Rule #5:
Vectors! Gotta
use vectors!!!
The arrows on top of the variables J and v tell us that they are vectors.
Which
is the
When we subtract the initial from the final velocity, we must also take
correct
into account their direction. (One is north, the other south).
J  mv  10 g  10 m / s    10 m / s    20 g m / s
24
answer
!!!!!!!!
Mathematics
Review
In summary....
The Meaning
of Numbers
Rule #1: We must know which object we are considering in a problem.
Algebra
Rule #2: Use the proper units.
Trigonometry
Rule #3: Know the meaning of each variable.
Geometry
Choosing the
Correct
Equation(s)
Rule #4: Use what is known and unknown to sort out equations that are
not useful.
Rule #5: Vectors! Gotta use vectors!!!
Follow these rules when solving problems and you will find that physics
is not so bad.
25
This is what DR. Mike means when he says you must use concepts to
solve problems in physics.