College Physics

Chapter 1

Introduction

Science

Science

• is a body of knowledge

• is an ongoing human activity

• has beginnings that precede recorded history

Astronomy

•1500 BC- Stonehenge used to track the sun and mark the solstice •1200 BC- Babylonians study 'astrology' & invent the 12 signs used today •280 BC- Aristarchus (Greek) stated that the Sun was the center of the 'solar system'. It was almost 1800 yrs later that his theory would be widely accepted.

Early man studied the moon & the sun to learn when to sow their crops and when to harvest. We still use this method to this day.

Mathematics—The Language of Science

Integration of science and mathematics

• occurred some four centuries ago

• ideas of science are unambiguous when expressed in mathematical terms

• equations of science provide expressions of relationships between concepts

• equations are “guides to thinking”

Early Mathematics

• 1800 BC – Babylonian clay tablets cover topics which include fractions, algebra, quadratic and cubic equation

• Babylonian mathematics were written using a base-60 numeral system. This derives the modern day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 x 6) degrees in a circle.

Algebra

• The word algebra is a Latin variant of the Arabic word al-jabr. This came from the title of a book, Hidab al-jabr wal-muqubala, written in Baghdad about 825 A.D. by the Arab mathematician Mohamad ibn-Musa al-Khowarizmi

Mathematics as Mechanized Thinking

• Use mathematics to change scientific statements. If the original statement is correct, and you follow the rules faithfully, your final statement will also be correct.

This is a new statement about nature - derived using the rules of mathematics. Using mathematics, physicists can discover new relationships among physical quantities - mathematics mechanizes thinking.

Scientific Measurements

Measurements

• relate to how much you know about something

• of pinhole images of the Sun nicely lead to a calculation of the Sun’s diameter

Measurement

• Measurement is one of the first intellectual achievements of early humans.

• People learned to measure before they learned how to write and it was through measurement that people learned to count.

• Since humans have ten fingers, we learned to count by tens

Scientific Methods

Scientific Methods

• There’s no one method in which scientists do their work.

• Common steps of most methods:

1. observe

2. question

3. predict

4. test predictions

5. draw a conclusion

The Scientific Attitude

The scientific attitude is one of

• inquiry

• experimentation

• willingness to admit error

The Scientific Attitude

Scientists

• are experts at changing their minds

• must accept experimental findings

– test for erroneous beliefs

– understand objections and positions of antagonists

– Subject work to peer reviews

The Scientific Attitude Fact

a close agreement by observers about the same phenomenon

Hypothesis • an educated guess presumed to be factual until

supported by experiment

• scientific if there is a test to prove it wrong

The Scientific Attitude

Law or principle

• a hypothesis that has been tested repeatedly and has not been contradicted

Theory

• a synthesis of a large body of information that encompasses well-tested and verified hypotheses about certain aspects of the natural world

Theories and Experiments

• The goal of physics is to develop theories based on experiments

• A theory is a “guess,” expressed mathematically, about how a system works

• The theory makes predictions about how a system should work

• Experiments check the theories’ predictions

• Every theory is a work in progress

Fundamental Quantities and Their Dimension

• Length [L]

• Mass [M]

• Time [T]

– other physical quantities can be constructed from these three

Units

• To communicate the result of a measurement for a quantity, a unit must be defined

• Defining units allows everyone to relate to the same fundamental amount

Early Measurement Units

• inch - the width of the thumb.

• digit - the width of the middle finger (about 3/4 inch)

• palm - the width of four fingers (about 3 inches)

• span - the distance covered by the spread hand (about 9 inches)

• foot - the length of the foot. Later expressed as the length of 36 -barleycorns taken from the middle of the ear (about 12 inches).

• cubit - distance from the elbow to the tip of the middle finger (about 18 inches). The cubit is supposedly derived from the distance between a Pharaoh's elbow to the farthest fingertip of his extended hand.

• yard - distance from the center of the body to the fingertips of the outstretched arm (about 36 inches).

• fathom - distance spanned by the outstretched arms (about 72 inches).

Systems of Measurement

• Standardized systems

– agreed upon by some authority, usually a governmental body

• SI -- Systéme International

– agreed to in 1960 by an international committee

– main system used in this text

– also called mks for the first letters in the units of the fundamental quantities

Systems of Measurements, cont

• cgs – Gaussian system

– named for the first letters of the units it uses for fundamental quantities

• US Customary

– everyday units

– often uses weight, in pounds, instead of mass as a fundamental quantity

Length

• Units

– SI – meter, m

– cgs – centimeter, cm

– US Customary – foot, ft

• Defined in terms of a meter – the distance traveled by light in a vacuum during a given time

Mass

• Units – SI – kilogram, kg

– cgs – gram, g

– USC – slug, slug

• Defined in terms of kilogram, based on a specific cylinder kept at the International Bureau of Weights and Measures

Other Measurements

• Acceleration – SI – m/sec2, m/sec2

– cgs – galileo, cm/sec2

– USC – ft/sec2, ft/sec2 • Force

– SI – Newton, Kg.m/sec2

– cgs – Dyne, g.cm/sec2

– USC – Pound, slug. ft/sec2

• Power – SI – Watt, Kg.m2/sec3

– cgs – erg/s, g.cm2/sec3

– USC – Horsepower, slug. ft2/sec3

Standard Kilogram

Time

• Units

– seconds, s in all three systems

• Defined in terms of the oscillation of radiation from a cesium atom

US “Official” Atomic Clock http://www.youtube.com/watch?v=5pWhPufvE1E

Approximate Values

• Various tables in the text show approximate values for length, mass, and time

– Note the wide range of values

– Lengths – Table 1.1

– Masses – Table 1.2

– Time intervals – Table 1.3

Prefixes

• Prefixes correspond to powers of 10

• Each prefix has a specific name

• Each prefix has a specific abbreviation

• See table 1.4

Table 1.1, p.2

Table 1.2, p.2

Table 1.3, p.2

Table 1.4, p.3

Structure of Matter

• Matter is made up of molecules

– the smallest division that is identifiable as a substance

• Molecules are made up of atoms

– correspond to elements

More structure of matter

• Atoms are made up of

–nucleus, very dense, contains • protons, positively charged, “heavy”

• neutrons, no charge, about same mass as protons – protons and neutrons are made up of quarks

– orbited by

• electrons, negatively charges, “light” – fundamental particle, no structure

Structure of Matter

Powers of 10

Dimensional Analysis

• Technique to check the correctness of an equation

• Dimensions (length, mass, time, combinations) can be treated as algebraic quantities – add, subtract, multiply, divide

• Both sides of equation must have the same dimensions

Dimensional Analysis, cont.

• Cannot give numerical factors: this is its limitation

• Dimensions of some common quantities are listed in Table 1.5

Volume examples

Uncertainty in Measurements

• There is uncertainty in every measurement, this uncertainty carries over through the calculations

– need a technique to account for this uncertainty

• We will use rules for significant figures to approximate the uncertainty in results of calculations

Significant Figures

• A significant figure is one that is reliably known

• All non-zero digits are significant

• Zeros are significant when

– between other non-zero digits

– after the decimal point and another significant figure

– can be clarified by using scientific notation

Operations with Significant Figures

• Accuracy – number of significant figures

• When multiplying or dividing two or more quantities, the number of significant figures in the final result is the same as the number of significant figures in the least accurate of the factors being combined

Operations with Significant Figures, cont.

• When adding or subtracting, round the result to the smallest number of decimal places of any term in the sum

• If the last digit to be dropped is less than 5, drop the digit

• If the last digit dropped is greater than or equal to 5, raise the last retained digit by 1

Identifying significant figures

1. All non-zero digits are considered significant.

– For example, 91 has two significant figures (9 and 1), while 123.45 has five significant figures (1, 2, 3, 4 and 5).

2. Zeros appearing anywhere between two non-zero digits are significant.

– Example: 101.12 has five significant figures: 1, 0, 1, 1 and 2.

3. Leading zeros are not significant.

– For example, 0.00052 has two significant figures: 5 and 2.

Identifying significant figures

4. Trailing zeros in a number containing a decimal point are significant. – For example, 12.2300 has six significant figures: 1, 2, 2, 3, 0 and 0. The

number 0.000122300 still has only six significant figures (the zeros before the 1 are not significant). In addition, 120.00 has five significant figures since it has three trailing zeros.

5. The number 0 has one significant figure.

6. The significance of trailing zeros in a number not containing a decimal point can be ambiguous. – For example, it may not always be clear if a number like 1300 is precise

to the nearest unit (and just happens coincidentally to be an exact multiple of a hundred) or if it is only shown to the nearest hundred due to rounding or uncertainty.

Significant Figures using Scientific Notation

• Generally, the same rules apply to numbers expressed in scientific notation. However, in the normalized form of that notation, placeholder leading and trailing digits do not occur, so all digits are significant. For example, 0.00012 (two significant figures) becomes 1.2×10−4, and 0.00122300 (six significant figures) becomes 1.22300×10−3. In particular, the potential ambiguity about the significance of trailing zeros is eliminated. For example, 1300 to four significant figures is written as 1.300×103, while 1300 to two significant figures is written as 1.3×103.

Significant Figures in Arithmetic

• For multiplication and division, the result should have as many significant figures as the measured number with the smallest number of significant figures.

• For addition and subtraction, the result should have as many decimal places as the measured number with the smallest number of decimal places (for example, 100.0 + 1.111 = 101.1).

Significant Figures with Calculators

• In a long calculation involving mixed operations, carry as many digits as possible through the entire set of calculations and then round the final result appropriately. For example,

(5.00 / 1.235) + 3.000 + (6.35 / 4.0)=4.04858... + 3.000 + 1.5875=8.630829...

• The first division should result in 3 significant figures. The last division should result in 2 significant figures so the answer is 8.6

Conversions

• When units are not consistent, you may need to convert to appropriate ones

• Units can be treated like algebraic quantities that can “cancel” each other

• See the inside of the front cover for an extensive list of conversion factors

• Example:

2.5415.0 38.1

1

cmin cm

in

Examples of various units measuring a quantity

Order of Magnitude

• Approximation based on a number of assumptions

– may need to modify assumptions if more precise results are needed

• Order of magnitude is the power of 10 that applies

Estimate Number of Bicycle Commuters in the US

• Current US population – 311 million

• % of US population with bikes ~20% (1 in every 5 people own a bike)

• % of those with bikes who commute ~2% (1 cyclist in 50 commute)

• Answer = (3x108)x(2x10-1)x(2x10-2) = 1.2x106

Estimate Number of Bicycle Commuters in the US

• Answer = (3x108)x(5x10-3) = 1.5x106

Coordinate Systems

• Used to describe the position of a point in space

• Coordinate system consists of

– a fixed reference point called the origin

– specific axes with scales and labels

– instructions on how to label a point relative to the origin and the axes

Types of Coordinate Systems

• Cartesian

• Plane polar

Cartesian coordinate system

• Also called rectangular coordinate system

• x- and y- axes

• Points are labeled (x,y)

Plane polar coordinate system

• Origin and reference line are noted

• Point is distance r from the origin in the direction of angle , ccw from reference line

• Points are labeled (r,)

Trigonometry Review

sin

cos

tan

opposite side

hypotenuse

adjacent side

hypotenuse

opposite side

adjacent side

More Trigonometry

• Pythagorean Theorem

• To find an angle, you need the inverse trig function – for example,

• Be sure your calculator is set appropriately for degrees or radians

2 2 2r x y

1sin 0.707 45

Problem Solving Strategy

Problem Solving Strategy

• Read the problem

– Identify the nature of the problem

• Draw a diagram

– Some types of problems require very specific types of diagrams

Problem Solving cont.

• Label the physical quantities – Can label on the diagram

– Use letters that remind you of the quantity • Many quantities have specific letters

– Choose a coordinate system and label it

• Identify principles and list data – Identify the principle involved

– List the data (given information)

– Indicate the unknown (what you are looking for)

Problem Solving, cont.

• Choose equation(s) – Based on the principle, choose an equation or

set of equations to apply to the problem

• Substitute into the equation(s) – Solve for the unknown quantity

– Substitute the data into the equation

– Obtain a result

– Include units

Problem Solving, final

• Check the answer

– Do the units match?

• Are the units correct for the quantity being found?

– Does the answer seem reasonable?

• Check order of magnitude

– Are signs appropriate and meaningful?

Problem Solving Summary

• Equations are the tools of physics

– Understand what the equations mean and how to use them

• Carry through the algebra as far as possible

– Substitute numbers at the end

• Be organized

Precision versus Accuracy

Accuracy

• Accuracy is how close a measured value is to the actual (true) value.

Precision

• Precision is how close the measured values are to each other.

Precision or Accuracy?