Physical Quantities and Measurement

1. Quantum Field Theory Relativistic Mechanics speed Quantum Mechanics Classical Mechanics size Physical Quantities and Measurement • What is Physics? • Natural Philosophy • science of matter and energy • fundamental principles of engineering and technology • an experimental science: theoryexperiment • simplified models • range of validity

2. Quantifying predictions and observations • physical quantities: numbers used to describe physical phenomena • height, weight e.g. • may be defined operationally • standard units: International System (SI aka Metric) • defined units established in terms of a physical quantity • derived units established as algebraic combinations of other units

3. Scientific Notation: powers of 10 5,820 = 5.82x103 = 5.82E3 .000527 = 5.28x10-4 = 5.28E-4 note: 103 = 1x103 =1E3 not 10E3! Common prefixes How big (in terms of everyday life/other things) is a meter nanometer gram centimeter kilometer kilogram

4. Dimensional Analysis: consistency of units • Algebraic equations must always be dimensionally consistent. • You can’t add apples and oranges! • converting units • treat units as algebraic quantities • multiplying or dividing a quantity by 1 does not affect its value

5. Some Useful Conversion factors: • 1 inch = 2.54 cm • 1 m = 3.28 ft • 1 mile = 5280 ft • 1 mile = 1.61 km • Units Conversion Examples • Example 1-1 The world speed record, set in 1983 is 1019.5 km/hr. Express this speed in m/s • Example how man cubic inches are there in a 2 liter engine?

6. Significant Figures and Uncertainty • Every measurement of a physical quantity involves some error • random error • averages out • small random error  accurate measurement • systematic error • does not average out • small systematic error  precise measurement less precise less accurate

7. Indicating the accuracy of a number: x ± Dx or x± dx • nominal value: the indicated result of the measurement • numerical uncertainty: how much the “actual value” might be expected to differ from the nominal value • sometimes called the numerical error • 1 standard deviation • A measured length of 20.3 cm ± .5 cm means that the actual length is expected to lie between 19.8 cm and 20.8 cm. It has a nominal value of x = 20.3 cm with an uncertainty of Dx .5 cm. • fractional uncertainty: the fraction of the nominal value corresponding to the numerical uncertainty • percentage uncertainty: the percentage of the nominal value corresponding to the numerical uncertainty

8. Uncertainties in calculations • Adding and subtracting: add numerical uncertainty • Multiplying or Dividing: add fractional/percentage uncertainty • Powers are “multiple multiplications”

9. More complex algebraic expressions must be broken down operation by operation a = 3.13±.05 b = 7.14 ±.01 c = 14.44 ±.2%

10. Significant Figures: common way of implicitly indicating uncertainty • number is only expressed using meaningful digits (sig. figs.) • last digit (the least significant digit = lsd) is uncertain • 3 one digit • 3.0 two digits (two significant figures = 2 sig. figs.) • 3.00 three digits,etc. (300 how many digits?) • Combining numbers with significant digits • Addition and Subtraction: least significant digit determined by decimal places (result is rounded) • .57 + .3 = .87 =.9 11.2 - 17.63 = -6.43 = -6.4 • Multiplication and Division: number of significant figures is the number of sig. figs. of the factor with the fewest sig. figs. • 1.3x7.24 = 9.412 = 9.4 17.5/.3794 = 46.12546 = 46.1 • Integer factors and geometric factors (such as p) have infinite precision • p x 3.762 = 44.4145803 = 44.4

11. Estimates and Order of magnitude calculations • an order of magnitude is a (rounded) 1 sig fig calculation, whose answer is expressed as the nearest power of 10. • Estimates should be done “in your head” • check against calculator mistakes! • Additional Homework: with • a = 3.13±.05 b = 7.14 ±.01 c = 14.44 ±.2% • evaluate expressions (nominal value and uncertainty expressed as numerical uncertainty and percentage uncertainty)