Post on 13-Jan-2016
PhysicsTopic #1
MEASUREMENT & MATHEMATICS
Scientific Method
• Problem to Investigate
• Observations
• Hypothesis
• Test Hypothesis
• Theory
• Test Theory
• Scientific Law Mathematical proof
Measurement & Uncertainty
• Uncertainty:• No measurement is absolutely precise
• Estimated Uncertainty:• Width of a board is 8.8cm +/- 0.1cm
• 0.1cm represents the estimated uncertainty in the measurement
• Actual width between 8.7-8.9cm
Measurement & Uncertainty
• Percent Uncertainty:• Ratio of the uncertainty to the measured
value, x 100• Example:
• Measurement = 8.8 cm
• Uncertainty = 0.1 cm
• Percent Uncertainty =
Is the diamond yours?
A friend asks to borrow your precious diamond for a day to show her family. You are a bit worried, so you carefully have your diamond weighed on a scale which reads 8.17 grams. The scale’s accuracy is claimed to be +/- 0.05 grams. The next day you weigh the returned diamond again, getting 8.09 grams. Is this your diamond?
Scale Readings
- Measurements do not necessarily give the “true” value of the mass
- Each measurement could have a high or low by up to 0.05g
- Actual mass of your diamond between 8.12g and 8.22g
Reasoning: (8.17g – 0.05g = 8.12g) (8.17g + 0.05g= 8.22g)
* Actual mass your diamond
- Between 8.12g and 8.22g
* Actual mass of the returned diamond
- 8.09g +/- 0.05g Between 8.04g and 8.14g
** These two ranges overlap not a strong reason to doubt that the returned diamond is yours, at least based on the scale readings
ACCURACY-
How close a measurement comes to the TRUE value
PRECISION-
How close a SERIES of measurements are to ONE ANOTHER
PERCENT (%) ERROR- Absolute value of the theoretical minus the experimental, divided by the theoretical, multiplied by 100
Theoretical - Experimental / Theoretical x 100
Accuracy, Precision, and Percent Error
Metric System
• Expanded & updated version of the metric system:
Systeme International d’Unites
Fundamental SI Units
Physical Quantity Name Abbreviation
Length meter m
Mass kilogram kg
Time second s
Temperature Kelvin K
Electric current ampere A
Amt of Substance mole mol
Luminous Intensity candela cd
Metric System
kilokilo kk 10103 3 = 1000= 1000
hectohecto hh 10102 2 = 100= 100
dekadeka dada 10101 1 = 10= 10
meter, liter, meter, liter, gram (Base)gram (Base)
m, l, gm, l, g 10100 0 = 1= 1
decideci dd 1010-1-1 = 0.1= 0.1
centicenti cc 1010-2 -2 = 0.01= 0.01
millimilli mm 1010-3 -3 = 0.001= 0.001
SI Prefixes
pico p 10-12
nano n 10-9
micro µ 10-6
milli m 10-3
centi c 10-2
kilo k 103
mega M 106
giga G 109
tera T 1012
Little GuysLittle Guys Big GuysBig Guys
Reference Table
Scientific Notation• Alternative way to express very large or very
small numbers
• Number is expressed as the product of a number between 1 and 10 and the appropriate power of 10.
Large Number: 238,000. =
Decimal placed between 1st and 2nd digit
Small Number : 0.00043 =
2.38 x 105
4.3 x 10-4
Scientific Notation
Express the following numbers in Scientific Notation
1. 3,570
2. 0.0055
3. 98,784 x 104
4. 45
Scientific Notation
• “Scientific Notation” or “Powers of Ten”• Allows the number of significant figures to
be clearly expressed• Example:
• 56, 800 5.68 x 104
• 0.0034 3.4 x 10-3
• 6.78 x 104 Number is known to an accuracy of 3 significant figures
• 6.780 x 104 Number is known to an accuracy of 4 significant figures
Scientific Notation• Multiplying Numbers in Scientific
Notation• Multiply leading values• Add exponents• Adjust final answer, so leading value is
between 1 and 10
• Dividing Numbers in Scientific Notation• Divide leading values• Subtract exponents• Adjust final answer, so leading value is
between 1 and 10
Scientific Notation• Adding & Subtracting Numbers in
Scientific Notation• Adjust so exponents match• Then, add or subtract leading values
only• Adjust final answer, so leading value is
between 1 and 10
Significant Figures
• All of the important/necessary or reliably known numbers
• GUIDELINES• Non-zero digits always significant• Zeros at the beginning of a number Not
significant (Decimal point holders)• 0.0578 m 3 Significant Figures (5, 7, 8)
• Zeros within the number Significant• 108.7 m 4 Significant Figures (1, 0, 8, 7)
• Zeros at the end of a number, after a decimal point Significant• 8709.0 m 5 Significant Figures (8, 7, 0, 9, 0)
Significant Figures
• Non-zero integers • Always counted as significant figures
** How many significant figures are there in 3,456?
4 Significant Figures
Significant Figures
ZEROS* Leading Zeros
- Never significant
0.0486 3 Significant Figures
0.003 1 Significant Figure
Significant Figures
ZEROS* Captive zeros
- Always significant
16.07 4 Significant Figures10.98 4 Significant Figures70.8 3 Significant Figures
Significant Figures
ZEROS* Trailing Zeros
- Significant only if the number contains a decimal point
9.300 4 Significant Figures1.5000 5 Significant Figures
Converting Units
• Physics problems require the use of the correct units
• Conversion factors• Allow you to change from one unit of
measurement to another• Ex: 1 foot = 12 inches
• Converting units• Choose the appropriate conversion factor• Multiply by the conversion factor as a fraction• Make sure units cancel!
Units for length, mass, and time (aswell as a few others), are regarded asbase SI units
These units are used in combination to define additional units for other important physical quantities, such as force and energy Derived Units
Derived Units
Derived UnitsDerived Unitswebsite
• Units that are created based on formulas and Units that are created based on formulas and equationsequations– VolumeVolume
– VV = length= length··widthwidth··height = mheight = m·m·m = m·m·m = m33
– AreaArea – A = length·width = m·m = mA = length·width = m·m = m22
– ForceForce• F = mass·acceleration = kg·m·sF = mass·acceleration = kg·m·s-2 -2 = Newton, N= Newton, N
– WorkWork• W = Force·distance = N·m = Joule, JW = Force·distance = N·m = Joule, J
– PressurePressure• P = Force/Area = N·mP = Force/Area = N·m-2-2 = Pascal, Pa = Pascal, Pa
Dimensional Analysis
• Useful tool utilized to check the dimensional consistency of any equation to check whether calculations make sense
• Length is represented by L
• Mass is represented by M
• Time is represented by T
• For an equation to be valid, the left dimension must equal the right dimension
Trigonometry
• Pythagorean Theorem• Used to find the length of any side of a right
triangle when you know the lengths of the other two sides• Right triangle Triangle with a 90° angle
• c2 = a2 + b2
• c = Length of the hypotenuse• a, b, = Lengths of the legs
Trigonometric Functions
• sin θ = opposite/hypotenuse
• cos θ = adjacent/hypotenuse
• tan θ = opposite/adjacent
Trigonometric Functions
• If you know the ratio of lengths of 2 sides of a right triangle, you can use inverse functions to determine the angles of that triangle
• θ = arcsin (opposite/hypotenuse)• θ = arccos (adjacent/hypotenuse)• θ = arctan (opposite/adjacent)
• Often written: sin−1, cos−1, tan−1