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 Every Element has its own Natural State: Fire Air Water Earth
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.
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?
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.
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
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
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)
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
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
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?
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.
- 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.
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.
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.
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.
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.
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 -)
G. Phythagorean Theorem ( h ) a2 + b2 = c2 ( ho ) OR ho2 + ha2 = h2 ( ha )
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
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
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
θ 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
θ = sin-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
( ) 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°
B) 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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