PHYS 150 Physics I
Prerequisite:
Corequisite:
Instructor:
title:
office:
phone:
e-mail:
homepage:
MATH 131 Calculus I
PHYS 150L (lab)
Dr. Johnny B. Holmes
Professor of Physics
CW 103
321-3448
jholmes@cbu.edu
facstaff.cbu.edu/~jholmes/
PHYS 150: Physics I
A beginning course in physics covering the
topics of kinematics, dynamics, gravitation,
work, energy, momentum, rotational
kinematics and dynamics.
Prerequisites: algebra and basic trig, definition of
derivative and integral, ability to differentiate and
integrate power laws, sines and cosines.
Corequisite: PHYS 150 L (the lab)
Can substitute for PHYS 201 Introductory Physics I
Grading: (explained on syllabus)
• 4 tests, each counts as one grade
• 1 set of 9 regular collected homework
problems which counts as one grade
• 2 sets of 10 computer homework programs
where each set counts as one grade
• final exam, which counts as 3 grades
Total: 10 grades, final grade will be based on
the average of these 10 grades.
100 – A – 93 – B – 82 – C – 70 – D – 65 – F - 0
Absence policy
If you miss 3 or fewer classes, your lowest
single score will be dropped (not counting
homework scores). If the final is lowest, it
will count only 2 instead of 3 times. Thus,
if you have 3 or fewer class absences, the
total will be based on the remaining 9
grades.
Regular Homework
There are 9 regular collected homework
problems for the semester. These must be
done and written up using the 7 step
paradigm described in the syllabus. This
paradigm is not good for the problems with
obvious solutions, but is good for those
problems that do not have obvious
solutions. It is also a good way of
communicating your thinking.
7-step problem solving paradigm
1.
2.
3.
4.
I want to and I can (motivation)
What do you know (draw a diagram!)
What are you looking for (define symbols)
Brainstorm (how is what you are looking for
related to what you know; what laws apply)
5. Plan the solution
6. Execute the solution (be sure to include units)
7. Check your answer – is it reasonable?
Computer Homework
The 20 computer homework programs
(each program consists of a problem set)
are designed to give you graded practice.
They emphasize getting the answer right
the first time. If you get an answer wrong,
the computer will tell you right away, and
often tell you how to get it right. It is your
task to actually get them correct. A random
number generator will change the numbers
so you will have to learn how to do them
and not just remember the right answer.
Tests
• The 4 tests and final exam emphasize
familiarity, recognition and speed. The
material on the tests should be somewhat
familiar. You should be able to recognize
the type of problem, the basic principles
involved, and determine which techniques
to apply.
Study sheet
This course emphasizes basic principles
and problem solving, not memorization.
To reduce the perceived need to memorize,
you are permitted to bring to the tests one
8.5” x 11” sheet of paper with information
on one side.
You may bring two study sheets to the final
(writing on one side only).
Math Review
The first several computer homework problem
sets are reviews of the basic algebra that we
will use in this course. Included in this
review are relations, linear equations,
simultaneous equations, and quadratic
equations.
The first regular collected homework involves
a review of angles and basic trigonometry.
Math Review #1:
Linear Equations
Review: One Linear Equation:
ax + dy = c
(we know a, d, and c; we don’t know x and y)
This is one equation in two unknowns. There
are lots of correct answers to this. Example:
5x + 3y = 35. Several possible answers are:
(x=4, y=5), (x=7, y=0), (x=-5, y=20).
Review: One Equation:
ax + dy = c can be written in the normal form
y = mx + b where m = -a/d and b = +c/d.
Thus the equation 5x + 3y = 35 becomes
y = (-5/3)x + (35/3)
y
Every point on
this line satisfies
the equation
10
-5
(x=4, y=5), (x=7, y=0), (x=-5, y=20).
5
-10
x
Math Review #2:
Simultaneous Equations
• However, if we have two equations with
two unknowns, then there is usually just one
possible answer: Example:
5x + 3y = 35 AND 2x - y = 3 .
• In this case, we can solve for one unknown
(say y) in terms of x:
y = 2x - 3 (using the second equation).
Math Review #2:
Simultaneous Equations
5x + 3y = 35 AND 2x - y = 3 .
• Using this relation for y: y = 2x - 3 in the
other (first) equation yields:
5x + 3(2x - 3) = 35 , or
5x + 6x - 9 = 35, or 11x = 44, or
x = 44/11 = 4 = x. Now we can use this
value for x in the y = 2x - 3 to get
y = 2(4) - 3 = 5 = y.
Math Review
5x + 3y = 35 AND 2x - y = 3 .
• Check of our answer: (x=4, y=5)
5x + 3y = 35, or 5(4) + 3(5) = 35, or
20+15 = 35 which checks out.
2x - y = 3, or 2(4) -5 = 3, or 8 - 5 = 3
which also checks out. Hence we have our
solution.
Math Review #2:
Simultaneous Equations
5x + 3y = 35 AND 2x - y = 3 .
Graphically, each equation graphs as a straight
line, and the
y
single solution
(in our case, x=4, y=5)
is the intersection of
the two lines
x
10
-5
5
-10
Math Review
The computer homework program on
Simultaneous Equations has up to three
equations with three unknowns. You can
proceed the same way.
1. Use one equation to eliminate one of the three
unknowns in the other two equations.
2. Then use one of these two equations to
eliminate a second unknown from the last
equation.
3. Then use the last equation to solve for the
remaining one unknown.
Math Review #3a: Angles
Because space is three dimensional (we’ll talk
about this soon), we need angles.
What is an angle?
How do you measure an angle?
Math Review #3a: Angles
How do you measure an angle?
1) in circles (cycles, rotations, revolutions)
2) in degrees - but what is a degree?
Why do they break the circle into 360 equal degrees?
3) in radians - but why use a weird number
like 2p for a circle? Why does p have the weird
value of 3.1415926535… ?
Math Review #3a: Angles
Full circle = 360o (Comes from year - full cycle of
seasons is broken into 365 days; but 365 is
awkward number; use “nicer” number of 360.)
Full circle = 2p radians
(Comes from definition of
angle measured in radians:
q = arclength / radius
=s/r)
s
q
r
Math Review #4: Trig
The trig functions are based on a right
triangle:
• sin(q) = opposite/hypotenuse = y/r
• cos(q) = adjacent/hypotenuse = x/r
• tan(q) = opposite/adjacent = y/x
• (The hypotenuse is the side opposite the right angle.)
r
y
q
x
What is Physics?
First of all, Physics is a Science. So our first
question should be: What is a Science?
Science
•
•
•
•
•
•
What is a science?
Physics is a science. Biology is a science.
Is Psychology a science?
Is Political Science a science?
Is English a science?
What makes a field of inquiry into a
science?
Scientific Method
What makes a field of inquiry into a science?
• Any field that employs the scientific
method can be called a science.
• So what is the Scientific Method?
• What are the “steps” to this “method”?
Scientific Method
• 1. Define the “problem”: what are you
studying?
• 2. Gather information (data). This should be
repeatable (reproducible) by anyone else with the
proper equipment.
• 3. Hypothesize (try to make “sense” of the data
by trying to guess why it works or what law it
seems to obey). This hypothesis should suggest
how other things should work. So this leads to the
need to:
• 4. TEST, but this is really gathering more
information (really, back to step 2).
Scientific Method
Is the scientific method really a never ending
loop, or do we ever reach the “TRUTH” ?
Consider: can we “observe” or “measure”
perfectly? If not, then since observations
are not perfect, can we perfectly test our
theories? If not, can we ever be
“CERTAIN” that we’ve reached the whole
“TRUTH” ?
Scientific Method
If we can’t get to “THE TRUTH”, then why
do it at all?
We can make better and better observations,
so we should be able to know that we are
getting closer and closer to “THE TRUTH”.
Is it possible to get “close enough”?
Look at our applications (engineering): is our
current understanding “good enough” to
make air conditioners?
Physics
Now Physics is a science, but so are
Chemistry and Biology.
How does Physics differ from these others?
It differs in the first step of the method: what
it studies. Physics tries to find out how
things work at the most basic level. This
entails looking at: space, time, motion (how
location in space changes with time), forces
(causes of motion), and the concept of
energy.
Metric System
• Since physics is a science, and science deals
with observations, physics deals with
MEASUREMENTS.
• How do we MEASURE? What do we use
as the standard for our measurements?
• In this course we will look at common units
of measurement as well as the METRIC
units of measurement.
Metric System:
Basic quantities
• Some measurements are basic, and some are
combinations of other more basic ones:
• What are the basics: (MKS system)
– length (in Meters)
– amount of “stuff” called mass (in Kilograms)
– time (in Seconds)
• What are some of the combinations:
– speed (distance per time)
– area (distance times another distance)
– lots of other things
Prefixes
The metric unit for length is the meter. We
can indicate a multiple of meters or a
fraction of a meter by using prefixes:
• centi (cm) = .01 meters = 10-2 m
• milli (mm) = .001 meters = 10-3 m
• micro (m) = .000001 meters = 10-6 m
• nano (nm) = .000000001 meters = 10-9 m
• pico (pm) = .000000000001 meters = 10-12 m
Prefixes – cont.
(The prefixes on this page will not be used in this
course, but you may run into them in future
courses.)
•
•
•
•
femto:
atto:
zepto
yocto
(fg) = 10-15 grams
(ag) = 10-18 grams
(zg) = 10-21 grams
(yg) = 10-24 grams
Prefixes
For bigger values we have:
• kilo (km) = 1,000 meters = 103 m
• mega (Mm) = 1,000,000 meters = 106 m
• giga (Gm) = 1,000,000,000 meters = 109 m
• tera (Tm) = 1,000,000,000,000 meters = 1012 m
These prefixes can be applied to many
different units, not just meters. These will
be used throughout the course.
Time
How do we measure time?
year
month
week
day
hour
minute
second
Time
Year: time to make one cycle through seasons
Month: time for moon to make one cycle through
phases
Week: 7 days (one for each “planetary” body visible to
naked eye: SATURNday, SUNday, MOONday, etc.)
Day: time for sun to make one cycle across the sky
Hour: Break day into day and night; break each of
these into 12 parts - like year is broken into 12
months.
Minute: a minute piece of an hour (1/60th)
Second: a minute piece of a minute - or second
minute piece of an hour.
Length
What units do we use to measure length?
Length
Foot (whose foot?)
Inch (length of a section of your finger)
Mile (1,000 double paces of a Roman legion)
League (3 miles), fathom (6 feet) , chain (100 links either 20
yards or 100 feet, or 10 yards in football) , etc.
Meter The meter was originally defined as one ten-millionth
(0.0000001 or 10-7) of the distance, as measured over the earth's
surface in a great circle passing through Paris, France, from the
geographic north pole to the equator. One meter is now defined as the
distance traveled by a ray of electromagnetic (EM) energy through a
vacuum in 1/299,792,458 (3.33564095 x 10-9) of a second.
(We’ll worry about MASS later.)
Position
Before we can analyze motion, which is how
something’s position changes with time, we
need to analyze position.
How do we locate something (that is, indicate
its position)?
One Dimension
In one dimension, we can specify the position
with one number: the distance from some
specified starting place.
Example: A mark on a rope can be specified
by how far that mark is from one end of the
rope.
Two Dimensions
In two dimensions, we have more options in
specifying the location of an object.
Example: where is Memphis?
• We could use use a rectangular system (x,y)
that specifies how far North (y) and how far
East (x) it is from some specified location.
• We could also use a polar system (r,q) that
specifies how far it is (r, straight line
distance) along with the direction (q, angle
from due North).
Three Dimensions
Notice that both systems need TWO numbers
- hence the TWO DIMENSIONS.
With THREE dimensions, we need three
numbers and we have even more options.
Example: Locate an airplane in the sky.
Some of the options are: rectangular (x,y,z),
spherical (r,q,), cylindrical (r,,z).
Number of spatial dimensions
• It is easy to see the need for three dimensions. Do
we need FOUR?
• What would a fourth dimension be like?
• Experimentally, what do we find for the space that
we live in - how many spatial dimensions do we
have?
• Most of the time in this course we will work in
two dimensions - many cases can be reduced to
this and it is mathematically easier.
• Is space “flat”? Is the earth’s surface “flat”?
Open versus Closed Universe?
Vectors
How do we work with a quantity that needs
two (or more) numbers to specify it (like
position does)?
We can work either with a group of numbers
sometimes put in parenthesis, or we can
work with unit vectors that we add together:
(x,y), or x*x + y*y (where x indicates the x
direction and x indicates how far in that direction),
or x*i + y*j (where i indicates the x direction and j
the y direction).
Vectors
• The individual numbers in the vector are
called the components of the vector.
• There are two common ways of expressing
a vector in two dimensions: rectangular
and polar.
Vectors - Rectangular Form
• Rectangular (x,y) is often used on city
maps. Streets generally run East-West or
North-South. The distance East or West
along a street give one distance, and the
distance along the North-South street gives
the second distance - all measured from
some generally accepted origin.
Vectors
If you are at home (the origin), and travel four
blocks East and then three blocks North, you
will end up at position A.
Position A (relative to your house) is then
(4 blocks, 3 blocks) where the first number
indicates East (+) or West (-), and the second
number indicates North (+) or South (-).
Vectors
• If you had a helicopter or could walk
directly there, it would be shorter to actually
head straight there.
A
How do you specify
3 bl
the location of point
A this way?
4 bl
Vectors - Polar Form
• The distance can be calculated by the
Pythagorian Theorem:
r = [ x2 + y2 ] = 5 bl.
A
• The angle can be
5 bl 3 bl
calculated using the
tangent function:
4 bl
q = tan-1 (y/x) = 37o
Transformation Equations
• These two equations are called the
rectangular to polar transformation
equations:
r = [ x2 + y2 ]
q = tan-1 (y/x) .
• Do these work for all values of x and y,
including negative values?
Transformation Equations
r = [ x2 + y2 ]
• The r equation does work all the time since
when you square a positive or negative
value, you still end up with a positive value.
Thus r will always be positive (or zero),
never negative.
Transformation Equations
q = tan-1 (y/x) .
• However, the theta equation does depend on
the signs of x and y. From this equation you
get the same angle if x and y are both the same
sign (both positive or both negative), or if one
is positive and one negative - regardless of
which one is the positive one.
• How do we work with this?
Transformation Equations
q = tan-1 (y/x)
Some scientific calculators have a built-in
transformation button.
However, you should know how to do this the
“hard way” regardless of whether your
calculator does or does not have that button.
Transformation Equations
q = tan-1 (y/x)
Note: If x is positive, you must be in the first
or fourth quadrant (theta between -90o and
+90o. Your calculator will always give you
the right answer for theta if x>0.
If x is negative, you must be in the second or
third quadrant (theta between 900 and 270o).
All you have to do if x<0 is add 180o to
what your calculator gives you.
Inverse Transformations
• Can we go the other way? That is, if we
know (r,q) can we get (x,y) ?
• If we recall our trig functions,
we can relate x to r and q:
x = r cos(q), and similarly
r
y = r sin(q).
y
Do these work for all values
x
of r and q? YES!
Inverse Transformations
x = r cos(q), and y = r sin(q)
Example: If (r,q) = (5 bl, 37o), what do we
get for (x,y) ?
x = r cos(q) = 5 bl * cos(37o) = 4 bl.
y = r sin(q) = 5 bl * sin(37o) = 3 bl.
[Note that this is the (x,y) we started with to get the
(r,q) = (5 bl, 37o).]
Motion
Motion involves changing the position of an
object during a time interval.
If position is a vector, then the change in
position should also be a vector. The
change in position involves the difference
between the final and initial positions. But
before we concern ourselves with
subtraction (finding a difference), we need
to look at addition!
Addition of Vectors
Suppose you start from home (the origin), and
go 4 blocks East (x1=4 bl) and 3 blocks
North (y1=3 bl) to point A. Then you leave
point A and go 2 blocks West (x2=-2 bl) and
4 blocks South (y2=-4 bl) .
Where do you end up (relative to the origin where you started)?
Addition of Vectors
( 4 bl, 3 bl)
+ (-2 bl, -4 bl)
-------------= (2 bl, -1 bl) ???
(2 blocks East, 1 block South)
YES!
When we add vectors expressed
in rectangular form, we just
add the individual components!
Addition of Vectors
Does it work the same way in polar form?
( 4 bl, 3 bl) --> (5 bl, 37o)
+ (-2 bl, -4 bl) --> (4.5 bl, 243.5o)
------------------------------= (2 bl, -1 bl) --> (2.2 bl, -26.5o)
Note that 5 bl + 4.5 bl does NOT equal 2.2 bl,
and 37o + 243.5o does NOT equal -26.5o !
Adding the polar components does NOT
WORK!
Addition of Vectors
When we add vectors, we can only add them
when they are in RECTANGULAR form.
If they are in polar form, we must first
transform them into rectangular form, then
add them in rectangular form (by adding the
components), then convert them back into
polar form!
You will get practice with this in the Vector
Addition Computer Homework Program.
Subtraction of Vectors
If a + b = c , this can be re-written as a = c - b.
Can we do the same with vectors, that is,
if
(x1, y1) + (x2,y2) = (x3,y3)
then does
(x1, y1) = (x3, y3) - (x2, y2) ?
Subtraction of Vectors
If a + b = c , this can be re-written as a = c - b.
Can we do the same with vectors, that is,
if (x1, y1) + (x2,y2) = (x3,y3) then does
(x1, y1) = (x3, y3) - (x2, y2) ?
YES as long as the vectors are in rectangular
form!
Addition of Vectors
Do the previous rules for addition of location
vectors also work for other vector
quantities?
Consider the idea of FORCE. Is FORCE a
vector?
We can answer this by asking: does force
have a magnitude and a direction (can you
have a force acting sideways) ?
Vectors versus Scalars
The answer is YES, so FORCE is a vector,
and several forces acting on the same object
can be added together as vectors to get the
resultant force.
We will play with this idea in the first lab
experiment.
Is TIME a vector?
(Can you move sideways in time?)
No - time is not a vector; it is a SCALAR.