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Chapter 1 – Scientific Measurements. I. Advancement of Scientific Ideas . A. Greeks – Great thinkers, poor experimenters… Performing Experiments was for slaves!. 1. Aristotle – 430 B.C. Student of Plato Considered Four Elements: Air, Earth, Water, Fire.

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I. Advancement of Scientific Ideas

A. Greeks – Great thinkers, poor experimenters…

Performing Experiments was for slaves!

1. Aristotle – 430 B.C. Student of Plato

Considered Four Elements: Air, Earth, Water, Fire

Every Element has its own Natural State:

Fire

Air

Water

Earth


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  • When you drop a rock in Water it returns to Earth

  • When you bubble air through Water, it returns to Air

  • Stones thrown up are given “Violent Motion” until they

  • run out and fall because of “Natural Motion”.

  • The more mass an object has the faster it falls to its

  • natural state – explains that heavy objects fall faster than light.


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b. Galileo Galilei – 1564-1642 A.D.

Doubted Aristolian ideas. He thought ideas should be proven through observation and experimentation.

He applied this idea to the concept of how things fall….do heavier objects fall faster than lighter objects?


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II. Introduction to Physics

  • Physics is the branch of physical science that deals

  • with the physical changes of objects.

  • The idealized models on which physics is based are often

  • expressed in numerical equations. Therefore, a strong background

  • in math is necessary.

Physics is considered to be the most basic form of science.


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B. The relationship between physics to the other branches of

Science:

  • Physics forms the foundation for other fields of science. It encompasses mechanics, thermodynamics, waves, optics, electromagnetism, relativity, and quantummechanics.

  • Chemistry is about how matter is put together, how atoms combine to form molecules, and how moleculesform the things around us.

  • Biology is the study of living matter.

Physics  Chemistry  Biology


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III. Structure of the Class

  • Concepts will be presented

  • These concepts will be put into a mathematical form

  • The concepts and equations will be applied to and investigation


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IV. Measurements in Science

A. S.I. – System International

An agreed upon base of units that can be understood by a range of audiences.

SI Standards: meter (m), kilogram (kg), second (s)


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B. Scientific Notation

Coefficient and Exponent – Shorthand for another

number

58000  5.8 X 104

0.0023  2.3 X 10-3

Ex: Convert the following into scientific notation

a.) 842,023

 8.42023 x 105

 1.8 x 10-5

b.) 0.000018

c.) 283,022,018

 2.83022018 x 108


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SI Units utilize prefixes that symbolize powers of 10

Important prefixes that you need to know:

milli-

centi-

deci-

kilo-

1x10-3

1x10-2

1x10-1

1x103


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0 0.1 0.2

(0.02/2.54) x 100 = 0.79%

– In any measurement, the last digit can

be guessed

C. Accuracy/Uncertainty

0.185

1.      Accuracy – can be reported as +/- half the last reported digit

2.      % Uncertainty = Uncertainty in measurement X 100

Value or Mean

% Error = (|Your Result - Accepted Value|) x 100

Accepted Value

Ex: What is the percent uncertainty in the measurement 2.54 ± 0.02 cm?


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D. Sig Figs – Answer can be no better than the weakest

measurement

.      - Rules for calculating sig figs

  • All non-zero digits are considered significant.

  • Example: the number 1 has one significant figure. In 20 and 300, the first figure is

  • significant while the others may or may not be. 123.45 has five significant

  • figures: 1, 2, 3, 4 and 5.

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

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

  • Leading zeros are not significant. Example, 0.00012 has two significant figures: 1 and 2.

  • Trailing zeros in a number containing a decimal point are significant.

  • 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.


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- Sig figs in calculations: Addition/Subtraction ,

Multiplication/Division

Measured quantities are often used in calculations.

The precision of the calculation is limited by the precision

of the measurements on which it is based.

Addition and SubtractionThe final answer should have the same number

of digits to the right of the decimal as the

measurement with the SMALLEST number of

digits to the right of the decimal.

Example: Add the following measurements: 32.01 m + 5.325 m + 12 m

Added together, you will get 49.335 m, but the sum should be reported as '49' meters.


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979.0 m

3.281 ft

x

=

3212 ft

1

1 m

Multiplication and DivisionThe final answer has the same number of significant figures as the measurement

Having the smallest number of significant figures.

Ex: If a density calculation is made in which 25.624 grams is divided by 25 mL,

the density should be reported as 1.0 g/mL, not as 1.0000 g/mL or 1.000 g/mL.

E. Dimensional Analysis – Multiply by conversion factor to

get answer (Unit Conversion)

Question: The highest waterfall in the world is Angel Falls, with a total drop of 979.0 m. Express this measurement in feet if 1 m = 3.281 ft.


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1. In all calculations, write down the units explicitly.

2. Treat all units as algebraic quantities.

3. Use the conversion factors.

4. Check to see that your calculations are correct by verifying

that the units combine algebraically to give the desired

answer.


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F. Finding mathematical relationships by Graphing

1. by using the mathematical relationship showing

the formula for a line, one can find how two variables

relate. (Linear Relationship)

y = mx + b

y= variable graphed on y-axis

m = slope; rise over run X2-X1

Y2-Y1

x= variable graphed on x-axis

b= Y intercept (value of where

graphed line crosses the y-axis)

this equals zero if line goes through

the origin.


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40

20

0

Length (cm)

1 2 3 4

Mass(kg)

Y=mX + b Y = Length

X = Mass

b = 0 (line goes through origin)

m= slope (rise/run)

20-0 = 6.7

3-0

If Y increases as X increases, then the slope is + and the line slopes upward;

If Y decreases as X increases, then the slope is – and the line slopes downward.


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2. Nonlinear Relationship – when the graph is NOT A

STRAIGHT LINE

  • Quadratic Relationships between 2 variables:

  • y = ax2 + bx + c

A quadratic relationship exists when one variable

depends on the square of another.

The graph of a quadratic

relationship results in a

parabola.Whether it opens up or down

depends on the value of the

Coefficient of the squared

term, a (+ or -)


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G. Phythagorean Theorem

( h )

a2 + b2 = c2

( ho )

OR

ho2 + ha2 = h2

( ha )


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H. Trigonometry

All students enrolled in this class must have a firm foundation

in math. In particular, algebra, geometry, and trigonometry.

Scientists use mathematics to help them describe how the

physical universe works.

Θ, Greek theta = angle

sine cosine tangent

cosθ = adjacent

hypotenuse

tanθ = opposite

adjacent

sinθ = opposite

hypotenuse

sin θ = ho

h

cos θ = ha

h

tan θ = ho

ha


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θ

The sine, cosine, and tangent

of an angle are numbers without

units, because each is expressed

as the RATIO of the lengths of

two sides of a right triangle.

THE CHOICE OF WHICH SIDE

OF THE TRIANGLE IS LABELED

AS OPPOSITE AND ADJACENT

CAN ONLY BE MADE AFTER THE

ANGLE, θ IS IDENTIFIED.

“What do Greeks chant?”

“SOH, CAH, TOA”

sin = opp/hyp; cos = adj/hyp; tan = opp/adj


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Ex: On a sunny day, a tall building casts a shadow that is 67.2 m

Long. The angle between the suns rays and the ground is θ=50.0º.

Determine the height of the building.

Given:

θ=50.0º.

ha = 67.2 m

Solve for ho,the height of the

building

tan θ = ho

ha

ho= ha tan θ

ho= 67.2 m (tan 50.0º)

ho= 67.2 m (1.19) =

80.0 m


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θ 67.2 m

1.83 m

32.0 m

Ex: An observer, whose eyes are 1.83 m above the ground,

Is standing 32.0 m away from a tree. The ground is level, and

The tree is growing perpendicular to it. The observer’s line

Of sight with the tree top makes an angle of 20.0° above the

Horizontal. How tall is the tree?

θ = 20.0°

ha= 32.0 m

h

ho

ho = ?

tan θ = ho

ha

ho = ha tan θ

ho = 32.0 m ( tan 20.0 °)

ha

ho = 32.0 m ( 0.364)

ho = 11.6 m

Total height = 1.83 m + ho = 1.83 m + 11.6 m =

13.4 m


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θ = sin 67.2 m -1ho

h

θ = cos-1ha

h

θ = tan-1ho

ha

( )

( )

( )

Inverse Trigonometric Functions

Often, the values for two sides of a right triangle are

available. Inverse trigonometric functions are used to

find the angle θ


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( ) 67.2 m

θ = tan-1ho

ha

θ = tan-12.25

14.0

( )

Ex: Wailoa River boat ramp drops off gradually at an angle θ. For fishing reasons,

it is necessary to know how deep the lake is at various distances from the shore.

To provide some information about the depth, Reyn rows straight out from the

shore at a distance of 14.0 m and drops a weighted fishing line. By measuring the

length of the line, Reyn determines the depth to be 2.25 m. A) What is the value

of θ? B) What would be the depth, d of the lake at a distance of 22.0 m from the

shore?

A)

ho = 2.25 m

ha = 14.0 m

θ = tan-1 (0.16)

θ =

9.13°


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B) 67.2 m

θ = 9.15°

ho = d

ha = 22.0 m

tan θ = ho

ha

ho= ha tan θ

ho= 22.0 m (tan 9.15°)

ho= 22.0 m (tan 9.15°)

ho=

3.54 m


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