Dimensions of Physics

This usually involves mathematical formulas. The essence of physics is to measure the observable world and describe the principles that underlie everything in creation. This usually involves mathematical formulas.

The Metric System first established in France and followed voluntarily in other countries renamed in 1960 as the SI (Système International d’Unités) seven fundamental units

Dimension can refer to the number of spatial coordinates required to describe an object can refer to a kind of measurable physical quantity

Dimension the universe consists of three fundamental dimensions: space time matter

Length the meter is the metric unit of length definition of a meter: the distance light travels in a vacuum in exactly 1/299,792,458 second.

Time defined as a nonphysical continuum that orders the sequence of events and phenomena SI unit is the second

Mass a measure of the tendency of matter to resist a change in motion mass has gravitational attraction

The Seven Fundamental SI Units length time mass thermodynamic temperature meter second kilogram kelvin

The Seven Fundamental SI Units amount of substance electric current luminous intensity mole ampere candela

SI Derived Units involve combinations of SI units examples include: area and volume force (N = kg • m/s²) work (J = N • m)

Measurements - Quantitative Unit of measurement – the unit being measured Pure number – the number of units determined by the act of measuring Measurement – the product of the pure number and the unit of measurement

Your Turn to Decide 4 feet extra large Hot 100 ºF Sunny 96

Conversion Factors any factor equal to 1 that consists of a ratio of two units You can find many conversion factors in Appendix C of your textbook.

Unit Analysis First, write the value that you already know. 18 m

Note that the old unit goes in the denominator. Unit Analysis Next, multiply by the conversion factor, which should be written as a fraction. 100 cm 1 18 m × m Note that the old unit goes in the denominator.

Remember that this method is called unit analysis. Then cancel your units. 100 cm 1 18 m × m Remember that this method is called unit analysis.

Unit Analysis Finally, calculate the answer by multiplying and dividing. 100 cm 1 18 m × = 1800 cm m

Unit Analysis Bridge

Sample Problem #1 Convert 13400 m to km. × 1 km 1000 m 13400 m =

Sample Problem #2 How many seconds are in a week? × 7 d 1 wk × 24 h 60 min 1 h 1 wk × 60 s 1 min = 604,800 s

Sample Problem #3 Convert 35 km to mi, if 1.6 km ≈ 1 mi. ≈ × 1 mi

Dimensional Analysis Converts from one unit to another Conversion Factor – a fraction (ratio) comparing two units Examples: 12 inches = 1 foot 3 feet = 1 yard 100 cm = 1 m 60 s = 1 min Conversion factors can be inverted!!

Grid Method for Conversions Given Information Conversion Requested Information (Understood 1) Factor

Grid Method for Conversions Convert 8.4 miles to feet. 8.4 miles 5280 feet 44,352 feet (Understood 1) 1 mile

Grid Method for Conversions Convert 8.4 miles to feet. 8.4 miles 5280 feet 44,352 feet (Understood 1) 1 mile

Dimensional Analysis 14 x 1018 s How many days is equal to 14 x 1018 s? 14 x 1018 s 1 min 1 hour 1 day 60 s 60 min 24 hours

Types of measurement Quantitative- use numbers to describe Easy to verify Easy to agree upon, no personal bias The measuring instrument limits how good the measurement is Qualitative- use description without numbers

How good are the measurements? Scientists use two words to describe how good the measurements are Accuracy- how close the measurement is to the actual value Precision- how well can the measurement be repeated

Differences Accuracy can be true of an individual measurement or the average of several Precision requires several measurements before anything can be said about it Example

Let’s use a golf analogy.

Accurate? No Precise? Yes

Accurate? Yes Precise? Yes

Precise? No Accurate? Maybe?

Accurate? Yes Precise? We cant say!

Principles of Measurement

In terms of measurement Three students measure the room to be 10.2 m, 10.3 m and 10.4 m across. Were they precise? Were they accurate?

The Metric System An easy way to measure

Converting k h D d c m how far you have to move on this chart, tells you how far, and which direction to move the decimal place. The box is the base unit, meters, Liters, grams, etc.

Conversions k h D d c m Change 5.6 km to millimeters

Significant figures (sig figs) How many numbers mean anything When we measure something, we can (and do) always estimate between the smallest marks. 2 1 3 4 5

Significant figures (sig figs) The better marks the better we can estimate. Scientist always understand that the last number measured is actually an estimate 1 2 3 4 5

Sig Figs All nonzero numbers are significant! So….what do we do with zeros?? We follow the rules!

Significant Digit Rules Only applied to measured data. Counting numbers are infinitely significant! All nonzero digits are significant! All zeros between nonzero digits are significant! In a decimal number all zeros to the right of the last nonzero digit are significant!

Significant Digit Rules In a decimal number all zeros to the left of the first nonzero digit are NOT significant! In a number WITHOUT a decimal all trailing zeros (zeros to the right of the last nonzero digit) are NOT significant!

Remember…. Scientific Notation only shows significant digits in the decimal part of the expression. A decimal point following a zero at the end of the number indicates that the zero is significant.

Problems 50 has only 1 significant figure if it should have two, how can I write it? A decimal following a zero at the end. 50. A line over the significant zero 50 Scientific notation 5.0 x 101 now the zero counts

Sig figs. How many sig figs in the following measurements? 458 g

Sig Figs. 405.0 g 4050 g 0.450 g 4050.05 g 0.0500060 g Next we learn the rules for calculations

Adding and Subtracting with Sig Figs The last sig fig in a measurement is an estimate. Your answer when you add or subtract can not be better than your worst estimate. You must round the answer to the least precise place of the measurement in the problem

For example 27.93 6.4 + First line up the decimal places 27.93 6.4 + 27.93 6.4 Then do the adding Find the estimated numbers in the problem 34.33 This answer must be rounded to the tenths place

Rounding rules look at the number to the right of the one you’re rounding. If it is 0 to 4 don’t change it If it is 5 to 9 make it one bigger round 45.462 to four sig figs to three sig figs to two sig figs to one sig fig

Practice 4.8 + 6.8765 520 + 94.98 0.0045 + 2.113 6.0 x 102 - 3.8 x 103 5.4 - 3.28 6.7 - 0.542 500 -126 6.0 x 10-2 - 3.8 x 10-3

Multiplication and Division Rule is simpler: Same number of sig figs in the answer as the one with the least number of sig figs in the question 3.6 x 653 2350.8 3.6 has 2 s.f. 653 has 3 s.f. answer can only have 2 s.f. 2400

Practice 5.345 • 3.9 4.5 ÷ 6.245 4.50 • 6.245 9.8764 • 0.043 3.876 ÷ 1983 16047 ÷ 700 2400 • 123 4.5 x 102 • 1.45 x 103

Truth in Measurements and Calculations

...and be careful when using your calculator!

Compound Calculations Rule 1: If the operations are all of the same kind, complete them before rounding to the correct significant digits.

Compound Calculations Rule 2: If the solution to a problem requires a combination of both addition/subtraction and multiplication/division...

Compound Calculations (1) For intermediate calculations, underline the estimated digit in the result and retain at least one extra digit beyond the estimated digit. Drop any remaining digits.

Compound Calculations (2) Round the final calculation to the correct significant digits according to the applicable math rules, taking into account the underlined estimated digits in the intermediate answers.

What about angles and trigonometry?

Angles in the SI The SI uses radians. A radian is the plane angle that subtends a circular arc equal in length to the radius of the circle.

Angles in the SI 2π radians = 360° Angles measured with a protractor should be reported to the nearest 0.1 degree.

Multiply the number of degrees by π/180. Conversions Degrees to Radians: Multiply the number of degrees by π/180.

Multiply the number of radians by 180/π. Conversions Radians to Degrees: Multiply the number of radians by 180/π.

Angles in the SI Report angles resulting from trigonometric calculations to the lowest precision of any angles given in the problem.

Angles in the SI Assume that trigonometric ratios for angles given are pure numbers; SD restrictions do not apply.

Instruments tools used to measure critical to modern scientific research man-made

When you use a mechanical metric instrument (one with scale subdivisions based on tenths), measurements should be estimated to the nearest 1/10 of the smallest decimal increment.

The last digit that has any significance in a measurement is estimated.

Error the simple difference of the observed and accepted values may be positive or negative

Error absolute error—the absolute value of the difference

Percent Error observed – accepted accepted × 100%

Problem Solving

Problem Solving Read the exercise carefully! What information is given? What information is sought? Make a basic sketch

Problem Solving Determine the method of solution Substitute and solve Check your answer for reasonableness

Reasonable Answers Does it have the expected order of magnitude and units? Make a mental estimate Be sure to simplify units Express results to the correct number of SDs