PHYS-207 for Scientists

1. Note: course website is http://mail.sci.ccny.cuny.edu/~jtu/Teaching.htm PHYS-207 for Scientists Lecture 2: Measurements and Units HW1 (problems): 1.18, 1.27, 2.11, 2.17, 2.21, 2.35, 2.51, 2.67 Due Monday, Sept. 8.

2. Objectives of Physics • To find the limited number of fundamental laws that govern natural phenomena • To use these laws to develop theories that can predict the results of future experiments • Express the laws in the language of mathematics • Mathematics provides the bridge between theory and experiment

3. Theory and Experiments • Should complement each other • When a discrepancy occurs, theory may be modified • Theory may apply to limited conditions • Example: Newtonian Mechanics is confined to objects traveling slowly with respect to the speed of light • Try to develop a more general theory

4. 1.2 Measuring things We measure each quantity by its own “unit” or by comparison with a standard. A unit is a measure of a quantity that scientists around the world can refer to. This has to be both accessible and invariable. For example • 1 meter (m) is a unit of length. Any other length can be expressed in terms of 1 meter. A variable length, such as the length of a person’s nose is not appropriate.

5. 1.3 International System of Units The SI system, or the International System of Units, is also called the metric system. Three basic quantities are the length, time, and mass. Many units are derived from this set, as in speed, which is distance divided by time.

6. US Customary System • Still used in the US, but text will use SI

7. Prefixes • Prefixes correspond to powers of 10 • Each prefix has a specific name • Each prefix has a specific abbreviation

8. 1.3 International System of Units Scientific notation uses the power of 10. Example: 3 560 000 000 m = 3.56 x 109 m. Sometimes special names are used to describe very large or very small quantities (as shown in Table 1-2). For example, 2.35 x 10-9 s = 2.35 nanoseconds (ns)

9. Basic Quantities and Their Dimension • Dimension has a specific meaning – it denotes the physical nature of a quantity • Dimensions are denoted with square brackets • Length [L] • Mass [M] • Time [T]

10. Dimensions and Units • Each dimension can have many actual units • Table for the dimensions and units of some derived quantities

11. Dimensional Analysis • Technique to check the correctness of an equation or to assist in deriving an equation • Dimensions (length, mass, time, combinations) can be treated as algebraic quantities • add, subtract, multiply, divide • Both sides of equation must have the same dimensions • Any relationship can be correct only if the dimensions on both sides of the equation are the same • Cannot give numerical factors: this is its limitation

12. Symbols • The symbol used in an equation is not necessarily the symbol used for its dimension • Some quantities have one symbol used consistently • For example, time is t virtually all the time • Some quantities have many symbols used, depending upon the specific situation • For example, lengths may be x, y, z, r, d, h, etc.

13. Conversion of Units • When units are not consistent, you may need to convert to appropriate ones • Units can be treated like algebraic quantities that can cancel each other out

14. 1.4 Changing units Based of the base units, we may need to change the units of a given quantity using the chain-link conversion. For example, since there are 60 seconds in one minute, Conversion between one system of units and another can therefore be easily figured out as shown. The first equation above is often called the “Conversion Factor”.

15. 1.5 Length Redefining the meter: In 1792 the unit of length, the meter, was defined as one-millionth the distance from the north pole to the equator. Later, the meter was defined as the distance between two finely engraved lines near the ends of a standard platinum-iridium bar, the standard meter bar. This bar is placed in the International Bureau of Weights and Measures near Paris, France. In 1960, the meter was defined to be 1 650 763.73 wavelengths of a particular orange-red light emitted by krypton-86 in a discharge tube that can be set anywhere in the world. In 1983, the meter was defined as the length of the path traveled by light in a vacuum during the time interval of 1/299 792 458 of a second. The speed of light is then exactly 299 792 458 m/s.

16. 1.5 Length Some examples of lengths

17. Reasonableness of Results • When solving a problem, you need to check your answer to see if it seems reasonable • Reviewing the tables of approximate values for length, mass, and time will help you test for reasonableness

18. 1.5 Length: Estimation PROBLEM: The world’s largest ball of string is about 2 m in radius. To the nearest order of magnitude, what is the total length, L, of the string of the ball? SETUP: Assume that the ball is a sphere of radius 2 m. In order to get a simple estimate, assume that the cross section of the string is a square with a side edge of 4 mm. This overestimate will account for the loosely packed string with air gaps. CALCULATE: The total volume of the string is roughly the volume of the sphere. Therefore,

19. 1.5 Time Atomic clocks give very precise time measurements. An atomic clock at the National Institute of Standards and Technology in Boulder, CO, is the standard, and signals are available by shortwave radio stations. In 1967 the standard second was defined to be the time taken by 9 192 631 770 oscillations of the light emitted by cesium-133 atom.

20. 1.5 Mass A platinum-iridium cylinder, kept at the International Bureau of Weights and Measures near Paris, France, has the standard mass of 1 kg. Another unit of mass is used for atomic mass measurements. Carbon-12 atom has a mass of 12 atomic mass units, defined as

21. Standard Kilogram

22. 1.5 Density Density is typically expressed in kg/m3, and is often expressed as the Greek letter, rho (r).Example, Density and Liquefaction:

23. Uncertainty in Measurements • There is uncertainty in every measurement – this uncertainty carries over through the calculations • May be due to the apparatus, the experimenter, and/or the number of measurements made • Need a technique to account for this uncertainty • We will use rules for significant figures to approximate the uncertainty in results of calculations

24. Significant Figures • A significant figure is one that is reliably known • Zeros may or may not be significant • Those used to position the decimal point are not significant • To remove ambiguity, use scientific notation • In a measurement, the significant figures include the first estimated digit

25. Significant Figures, examples • 0.0075 m has 2 significant figures • The leading zeros are placeholders only • Can write in scientific notation to show more clearly: 7.5 x 10-3 m for 2 significant figures • 10.0 m has 3 significant figures • The decimal point gives information about the reliability of the measurement • 1500 m is ambiguous • Use 1.5 x 103 m for 2 significant figures • Use 1.50 x 103 m for 3 significant figures • Use 1.500 x 103 m for 4 significant figures

26. Note: course website is http://mail.sci.ccny.cuny.edu/~jtu/Teaching.htm PHYS-207 for Scientists Lecture 2: Measurements and Units HW1 (problems): 1.18, 1.27, 2.11, 2.17, 2.21, 2.35, 2.51, 2.67 Due Monday, Sept. 8.