PHYS 150 Physics I
Prerequisite: MATH 131 Calculus I
Corequisite: PHYS 150L (lab)
Instructor: Dr. Johnny B. Holmes
title: Professor of Physics
office: CW 103
phone: 321-3448
e-mail: [email protected]
homepage: 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 gradesTotal: 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. I want to and I can (motivation)
2. What do you know (draw a diagram!)
3. What are you looking for (define symbols)
4. 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 HomeworkThe 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).
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(x=4, y=5), (x=7, y=0), (x=-5, y=20).
Review: One Equation:
-5 5
-10
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 , or5x + 6x - 9 = 35, or 11x = 44, orx = 44/11 = 4 = x. Now we can use
this value for x in the y = 2x - 3 to gety = 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
-5 5
-10
10
Math ReviewThe 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 2 for a circle? Why does 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 = 2 radians (Comes from definition of angle measured in radians:
= arclength / radius = s / r ) r
s
Math Review #4: TrigThe trig functions are based on a right triangle:
• sin() = opposite/hypotenuse = y/r
• cos() = adjacent/hypotenuse = x/r
• tan() = opposite/adjacent = y/x• (The hypotenuse is the side opposite the right angle.)
r y
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: (fg) = 10-15 grams
• atto: (ag) = 10-18 grams
• zepto(zg) = 10-21 grams
• yocto(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?yearmonthweekdayhourminutesecond
TimeYear: 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?
LengthFoot (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 DimensionsIn 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,) that specifies how far it is (r, straight line distance) along with the direction (, 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,,), 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?
VectorsHow 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.
How do you specify 3 bl
the location of point
A this way? 4 bl
A
Vectors - Polar Form
• The distance can be calculated by the Pythagorian Theorem:
r = [ x2 + y2 ] = 5 bl.
• The angle can be 5 bl 3 bl
calculated using the tangent function: 4 bl
= tan-1 (y/x) = 37o
A
Transformation Equations
• These two equations are called the rectangular to polar transformation equations:
r = [ x2 + y2 ]
= 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
= 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
= 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
= 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,) can we get (x,y) ?
• If we recall our trig functions,we can relate x to r and :x = r cos(), and similarly ry = r sin(). yDo these work for all values xof r and ? YES!
Inverse Transformations
x = r cos(), and y = r sin()
Example: If (r,) = (5 bl, 37o), what do we get for (x,y) ?
x = r cos() = 5 bl * cos(37o) = 4 bl.
y = r sin() = 5 bl * sin(37o) = 3 bl.[Note that this is the (x,y) we started with to get the
(r,) = (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 expressedin rectangular form, we justadd 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.
