Transcript Example

Ch. 1: Introduction, Measurement,
Estimating
Chapter 1 Outline
1. The Nature of Science
2. Models, Theories, & Laws
3. Measurement & Uncertainty
Significant Figures
4. Units, Standards, & the SI System
5. Converting Units
6. Order of Magnitude: Rapid Estimating
7. Dimensions & Dimensional Analysis
Physics:
The most basic of all sciences!
• Physics: The “Parent” of all sciences!
• Physics = The study of the behavior of and
the structure of matter and energy and of
the interaction between matter and energy.
Sub Areas of Physics
• This course (Phys. 1408, the Physics of the 16th & 17th Centuries):
– Motion (MECHANICS) (most of our time!)
– Fluids & Waves
• Next course (Phys. 2401, the Physics of 18th & the 19th Centuries):
– Electricity & magnetism
– Light & optics
• Advanced courses (Phys. 2402 & others. The Physics of the 20th Century!):
– Relativity, atomic structure, condensed matter, nuclear physics, ….
These are the most interesting Physics topics
& the topics which are the most relevant to
modern technology!
Mechanics: “Classical” Mechanics
Mechanics: “Classical” Mechanics
• “Classical” Physics:
“Classical”   Before the 20th Century
The foundation of pure & applied macroscopic physics & engineering!
– Newton’s Laws + Boltzmann’s Statistical Mechanics
(& Thermodynamics):  Describe most of macroscopic world!
– However, at high speeds (v ~ c) we need
Special Relativity: (Early 20th Century: 1905) Ch. 14 of M&T
– Also, for small sizes (atomic & smaller) we need
Quantum Mechanics: (1900 through ~ 1930) Physics 4307!
“Classical” Mechanics: (17th & 18th Centuries) Still useful today!
“Classical” Mechanics
The mechanics in this course is limited to macroscopic objects
moving at speeds v much, much smaller than the speed of light
c = 3  108 m/s. As long as v << c, our discussion will be valid.
So, we will work
exclusively in the
gray region in the
figure.
FYI: The Structure of Physics
SUMMARY: THE STRUCTURE OF PHYSICS
Low Speed
High Speed
v << c
v<~c
Large size
Classical Mechanics
Special Relativity
>> atomic size (Newton, Hamilton,
(Einstein)
Lagrange)
Small size Quantum Mechanics
< ~ atomic size
(Schrodinger,
Heisenberg)
Atomic Physics
Molecular
Physics
Solid State
Physics
Nuclear & Particle Physics
Relativistic Quantum
Mechanics
(Dirac)
Quantum Field Theory
(Feynman, Schwinger)
Quantum Electrodynamics
(Photons, Weak Nuclear Force)
Quantum Chromodynamics
(Gluons, Quarks, Leptons
Strong Nuclear Force)
Mechanics
• The science of HOW objects move (behave)
under given forces.
• (Usually) Does not deal with the sources of
forces. Answers the question: “Given the
forces, how do objects move”?
Physics: General Discussion
• The Goal of Physics (& all of science): To quantitatively
and qualitatively describe the “world around us”.
• Physics IS NOT merely a collection of facts & formulas!
• Physics IS a creative activity!
• Physics
Observation
• Requires IMAGINATION!!
Explanation.
Physics & Its Relation to Other Fields
• The “Parent” of all Sciences!
• The foundation for and is connected to ALL
branches of science and engineering.
• Also useful in everyday life and in MANY professions
–
–
–
–
–
Chemistry
Life Sciences (Medicine also!!)
Architecture
Engineering
Various technological fields
Physics Principles are used in many practical
applications, including construction.
Communication between Architects & Engineers is
essential if disaster is to be avoided.
The Nature of Science
• Physics is an EXPERIMENTAL science!
• Experiments & Observations:
– Important first steps toward scientific theory.
– It requires imagination to tell what is important
• Theories:
– Created to explain experiments & observations. Will
also make predictions
• Experiments & Observations:
– Will tell if predictions are accurate.
• No theory can be absolutely verified
– But a theory CAN be proven false!!!
Theory
• Quantitative (mathematical) description of
experimental observations.
• Not just WHAT is observed but WHY it is observed
as it is and HOW it works the way it does.
• Tests of theories:
– Experimental observations:
More experiments, more observation.
– Predictions:
Made before observations & experiments.
Model, Theory, Law
• Model: An analogy of a physical phenomenon to
something we are familiar with.
• Theory: More detailed than a model. Puts the
model into mathematical language.
• Law: Concise & general statement about how
nature behaves. Must be verified by many,
many experiments! Only a few laws.
– Not comparable to laws of government!
• How does a new theory get accepted?
• Predictions agree better with data than old theory
• Explains a greater range of phenomena than old theory
• Example:
– Aristotle believed that objects would return to a
state of rest once put in motion.
– Galileo realized that an object put in motion would
stay in motion until some force stopped it.
Measurement & Uncertainty.
Significant Figures
No measurement is exact; there is always some
uncertainty due to limited instrument accuracy and
difficulty reading results.
The photograph to the left
illustrates this – it would be
difficult to measure the width
of this 2  4 to better than a
millimeter.
Measurement & Uncertainty
• Physics is an EXPERIMENTAL science!
– Finds relations between physical quantities.
– Expresses those relations in the language of
mathematics. (LAWS & THEORIES)
• Experiments are NEVER 100% accurate.
– Always have uncertainty in final result.
• Experimental error.
– Common to state this precision (when known).
• Consider a simple measurement of the width of a
board. Find 23.2 cm.
• However, measurement is only accurate to 0.1
cm (estimated).
 Write width as (23.2  0.1) cm
 0.1 cm  Experimental uncertainty
• Percent Uncertainty:
 (0.1/23.2)  100   0.4%
Significant Figures
Significant figures (“sig figs”):
The number of significant figures is the number of
reliably known digits in a number. It is usually possible to
tell the number of significant figures by the way the number is
written:
23.21 cm has 4 significant figures
0.062 cm has 2 significant figures (initial zeroes don’t count)
80 km is ambiguous: it could have 1 or 2 significant
figures. If it has 3, it should be written 80.0 km.
Calculations Involving Several Numbers
When multiplying or dividing numbers:
The number of sig figs in the result  the same
number of sig figs as the number used in the
calculation with the fewest sig figs.
When adding or subtracting numbers:
The answer is no more accurate than the least
accurate number used in the calculation.
• Example:
(Not to scale!)
– Area of board, dimensions 11.3 cm  6.8 cm
– Area = (11.3)  (6.8) = 76.84 cm2
11.3 has 3 sig figs , 6.8 has 2 sig figs
 76.84 has too many sig figs!
– Proper number of sig figs in answer = 2
 Round off 76.84 & keep only 2 sig figs
 Reliable answer for area = 77 cm2
Sig Figs
• General Rule: The final result of a
multiplication or division should have only
as many sig figs as the number used in the
calculation which has the with least number
of sig figs.
• NOTE!!!! All digits on your calculator are
NOT significant!!
Calculators will not give you the right
number of significant figures; they usually
give too many, but sometimes give too few
(especially if there are trailing zeroes after a
decimal point).
The top calculator shows the result of
2.0 / 3.0.
The bottom calculator shows the result of
2.5 x 3.2.
Conceptual Example 1-1:
Significant figures
• Using a protractor, you measure an angle of 30°.
(a) How many significant figures should you quote in this
measurement?
(b) Use a calculator to find the cosine of the angle you
measured.
• (a) Precision ~ 1° (not
0.1°). So 2 sig figs &
angle is 30° (not 30.0°).
• (b) Calculator: cos(30°)
= 0.866025403. But angle
precision is 2 sig figs so
answer should also be 2 sig
figs. So cos(30°) = 0.87
Powers of 10
(Scientific Notation)
• READ Appendices A-2 & A-3
• Common to express very large or very small
numbers using power of 10 notation.
• Examples:
39,600 = 3.96  104
(moved decimal 4 places to left)
0.0021 = 2.1  10-3
(moved decimal 3 places to right)
PLEASE USE SCIENTIFIC NOTATION!!
Powers of 10
(Scientific Notation)
PLEASE USE SCIENTIFIC NOTATION!!
• This is more than a request!! I’m making it a
requirement!! I want to see powers of 10
notation on exams!! For example:
• For large numbers, like 39,600, I want to see
3.96  104 & NOT 39,600!!
• For small numbers, like 0.0021, I want to see
2.1  10-3 & NOT 0.0021!!
• On the exams, you will lose points if you don’t do this!!
Accuracy vs. Precision
• Accuracy is how close a measurement
comes to the accepted (true) value.
• Precision is the repeatability of the
measurement using the same
instrument & getting the same result!
It is possible to be accurate without
being precise and to be precise
without being accurate!
Units, Standards, SI System
• All measured physical quantities have units.
• Units are VITAL in physics!!
• In this course (and in most of the modern
world, except the USA!) we will use (almost)
exclusively the SI system of units.
SI = “Systéme International” (French)
More commonly called the “MKS system”
(meter-kilogram-second) or more simply,
“the metric system”
SI or MKS System
• Defined in terms of standards for length, mass, and time.
• Length unit: Meter (m) (kilometer = km = 1000 m)
– Standard Meter.
Newest definition in terms of speed of light  Length of path
traveled by light in vacuum in (1/299,792,458) of a second!
• Time unit: Second (s)
– Standard Second.
Newest definition  time required for 9,192,631,770 oscillations of
radiation emitted by cesium atoms!
• Mass unit: Kilogram (kg)
– Standard Kilogram
Mass of a specific platinum-iridium alloy cylinder kept at Intl
Bureau of Weights & Measures in France
Larger & Smaller Units are
Defined from SI standards by
Powers of 10 & Greek
Prefixes
These are the standard SI
prefixes for indicating powers of
10. Many (k, c, m, μ) are
familiar; Others (Y, Z, E, h, da,
a, z, y) are rarely used.
 __
 __
 __
 __
Typical Lengths (approx.)


Typical Times (approx.)


Typical Masses (approx.)


Units, Standards, and the SI System
We will work (almost)
exclusively in the
SI System,
where the basic units
are kilograms, meters,
& seconds.
Other Systems of Units
• CGS (centimeter-gram-second) system
– Centimeter = 0.01 meter
– Gram = 0.001 kilogram
• British (Engineering) System
(foot-pound-second; or
US Customary system)
– “Everyday life” system of units
– Only used by USA & some third
world countries. Rest of world
(including Britain!) uses SI system.
We will not use the British System!
– Conversions exist between the
British & SI systems. We will not use them in this course!
In this class, we will NOT do unit conversions!
We will work exclusively in SI (MKS) units!
Basic & Derived Quantities
• Basic Quantity  Must be defined in terms
of a standard (meter, kilogram, second).
• Derived Quantity  Defined in terms of
combinations of basic quantities
– Unit of speed (v = distance/time) = meter/second = m/s
– Unit of density (ρ = m/V) = kg/m3
Units and Equations
• In dealing with equations, remember that the
units must be the same on both sides of an
equation (otherwise, it is not an equation)!
• Example: You go 90 km/hr for 40 minutes.
How far did you go?
Units and Equations
• In dealing with equations, remember that the
units must be the same on both sides of an
equation (otherwise, it is not an equation)!
• Example: You go 90 km/hr for 40 minutes.
How far did you go?
– Equation from Ch. 2: x = vt, v = 90 km/hr, t = 40
min. To use this equation, first convert t to hours:
t = (⅔)hr so, x = (90 km/hr)  [(⅔)hr] = 60 km
The hour unit (hr) has (literally) cancelled out in the
numerator & denominator!
Converting Units
• As in the example, units in the numerator & the
denominator can cancel out (as in algebra)
• Illustration: Convert 80 km/hr to m/s
Conversions: 1 km = 1000 m; 1hr = 3600 s
Converting Units
• As in the example, units in the numerator & the
denominator can cancel out (as in algebra)
• Illustration: Convert 80 km/hr to m/s
Conversions: 1 km = 1000 m; 1hr = 3600 s
 80 km/hr =
(80 km/hr) (1000 m/km) (1hr/3600 s)
(Cancel units!)
80 km/hr  22 m/s
• Useful conversions:
(22.222…m/s)
1 m/s  3.6 km/hr; 1 km/hr  (1/3.6) m/s
Order of Magnitude; Rapid Estimating
• Sometimes, we are interested in only an
approximate value for a quantity. We are
interested in obtaining rough or order of
magnitude estimates.
• Order of magnitude estimates: Made by
rounding off all numbers in a calculation to 1 sig
fig, along with power of 10.
– Can be accurate to within a factor of 10 (often better)
Example: V = πr2d
• Example: Estimate!
Estimate how much
water there is in a
particular lake, which is
roughly circular, about
1 km across, & you
guess it has an average
depth of about 10 m.
Example 1-6: Thickness of a page.
Estimate the thickness of
a page of your textbook.
Hint: You don’t need
one of these!
Example 1-7: Height
by triangulation.
Estimate the height of
the building shown by
“triangulation,” with the
help of a bus-stop pole
and a friend. (See how
useful the diagram is!)
Example 1-8: Estimate the Earth radius.
If you have ever been on the shore of a large
lake, you may have noticed that you cannot see
the beaches, piers, or rocks at water level
across the lake on the opposite shore. The lake
seems to bulge out between you and the
opposite shore—a good clue that the Earth is
round. Suppose you climb a stepladder and
discover that when your eyes are 10 ft (3.0 m)
above the water, you can just see the rocks at
water level on the opposite shore.
On a map, you estimate the distance to the
opposite shore as d ≈ 6.1 km. Use h = 3.0 m
to estimate the radius R of the Earth.
Dimensions & Dimensional Analysis
The dimensions of a quantity are the base
units that make it up; generally written using
square brackets.
Example: Speed = distance/time
Dimensions of speed: [L/T]
Quantities that are being added or subtracted must have
the same dimensions. In addition, a quantity calculated
as the solution to a problem should have the correct
dimensions.
Dimensional Analysis
• If the formula for a physical quantity is known
 The correct units can easily be found!
• Examples: Volume: V = L3  Volume unit = m3
Cube with L =1 mm  V = 1 mm3 = 10-9 m3
Density: ρ = m/V Density unit = kg/m3
ρ = 5.3 kg/m3 = 5.3 10-6 g/mm3
• If the units of a physical quantity are known
 The correct formula can be “guessed”!
• Examples: Velocity: Car velocity is 60 km/h
Velocity unit = km/h
 Formula: v = d/t (d = distance, t = time)
Acceleration: Car acceleration is 5 m/s2
Acceleration unit = m/s2
 Formula: a = v/t (v = velocity, t = time)
Dimensional analysis is the checking of dimensions
of all quantities in an equation to ensure that those
which are added, subtracted, or equated have the same
dimensions.
Example: Is this the correct equation for velocity?
Check the dimensions:
Wrong!
Summary, Ch. 1
1. Physics = Measurements (Units) + Mathematics
(Algebra, Trig, Calculus) + Physical Principles (discussed as we
go) + Common Sense!
2. SI (mks) system of units:
Basic Units: Length  m, Time  s, Mass  kg
Unit conversions: 1000m = 1 km, 10-3 kg = 1g, ….
Derived units: ρ = m/V  Density unit = kg/m3
3. Dimensional Analysis
If the formula for a physical quantity is known
 The correct units can be found!
If the units of a physical quantity are known
 The correct formula can be “guessed”!
4. Theories are created to explain observations, & then tested
based on their predictions.
5. A model is like an analogy; it is not intended to be a true
picture, but to provide a familiar way of envisioning a
quantity.
6. A theory is much more well developed than a model, &
can make testable predictions; a law is a theory that can be
explained simply, & that is widely applicable.
7. Measurements can never be exact; there is always some
uncertainty. It is important to write them, as well as other
quantities, with the correct number of significant figures.
8. When converting units, check dimensions to see that the
conversion has been done properly.
9. Order-of-magnitude estimates can be very helpful
Comparison of 3
Fields of Study