PHYS 101 Chapter 1, Spring 2016

Measurement Chapter 1 Copyright © 2014 John Wiley & Sons, Inc. All rights reserved. © 2014 John Wiley & Sons, Inc. All rights reserved. 1-1 Measuring Things, Including Lengths 1.01 Identify the base quantities in the SI system. 1.02 Name the most frequently used prefixes for SI units. 1.03 Change units (here for length, area, and volume) by using chain-link conversions. 1.04 Explain that the meter is defined in terms of the speed of light in a vacuum. Learning Objectives 1-1 Measuring Things, Including Lengths  Science is based on the precise measurement and exploration of physical phenomena described by physical quantities. We discover science by learning how to measure these quantities (length, mass, time, pressure, temperature,…). Physics (most basic of the sciences)  Therefore, we need: 1. Rules for measurement and comparison 2. Units for measurement  A unit:  Is the unique name (kg, second, meter, Pascal, etc.) assigned to the measure of a quantity (mass, time, length, pressure, etc.) by comparison with a standard.  Standard corresponds to exactly 1.0 unit of the quantity (e.g. standard for length is 1.0 meter; any other length whatsoever must now be expressed in terms of that standard) © 2014 John Wiley & Sons, Inc. All rights reserved. 1-1 Measuring Things, Including Lengths  There are many different physical quantities, but not all are independent: distance vs. speed (distance/time)  Base quantities (international agreement):  Are seven fundamental quantities such as length, time  Three are needed for mechanics: length, time, mass  All have been assigned standards  Are used to define all other physical quantities  Base standards must be:  Accessible, so precise measurements can be taken  Invariable, so measurements do not change over time © 2014 John Wiley & Sons, Inc. All rights reserved. 1-1 Measuring Things, Including Lengths  SI units (the metric system) form the International System of Units (agreed upon in 1971, consisting of seven quantities: m, Kg, s, A, K, mol, cd)  SI base units include  Meters (length)  Seconds (time)  Kilograms (mass)  SI has many derived units, which are written in terms of base units  Joules (work-energy): 1 J = 1 kg m2/s2  Watts (power): 1 W = 1 J/s = 1 kg m2/s3 1-1 Measuring Things, Including Lengths  Scientific notation employs powers of 10 to write large or small numbers ( also written as E9, E-7 on most calculators)  A conversion factor (changing unit is often needed!)  A chain-link conversion method: multiply the original measurement by a conversion factor (a ratio of units that is equal to 1) to cancel unwanted units  Used to convert between units  Units obey the same algebraic rules as variables and numbers 1-1 Measuring Things, Including Lengths Reasoning Strategy: Converting Between Units 1. In all calculations, write down the units explicitly. 2. Treat all units as algebraic quantities. When identical units are divided, they are eliminated algebraically. 3. Be guided by the fact that multiplying or dividing an equation by a factor of 1 does not alter the equation. 1-1 Measuring Things, Including Lengths Examples Chain-link conversions: o1.3 km x (1000 m)/(1 km) = 1300 m = 1.3 x 103 o 0.8 km x (1000 m)/(1 km) x (100 cm)/(1 m) = 80 000 cm = 8 x 104 o 2845 mm x (1 m)/(1000 mm) x (3.281 ft)/(1 m) = 9.334 ft    second feet 95 s 3600 1 hour mile feet 5280 hour 65 miles 1 1 hour Speed 65 miles                          second 29 meters feet 3.281 meter 1 second feet 95 1 second feet Speed 95                    1 ft = 0.3048 m 1 mi = 1.609 km 1 hp = 746 W 1 liter = 10-3 m3 © 2014 John Wiley & Sons, Inc. All rights reserved. 1-1 Measuring Things, Including Lengths  Needs for accuracy and precisions in science have driven changes in the standards for units  In the past, 1 meter has been defined by: 1. One ten-millionth of the distance from the North pole to the equator 2. Later, for practical reasons, a platinum-iridium standard meter bar kept in France 3. In 1960, 1 650 763.73 wavelengths of an emission line of Kr-86 Today,  In each transition, the new distance was chosen so that the approximate length of 1 meter was preserved © 2014 John Wiley & Sons, Inc. All rights reserved. 1-1 Measuring Things, Including Lengths  Significant figures are meaningful digits whose values are known with certainty: 1.78 m vs 1.780 m;  In general, trailing zeros are not significant In other words, 3000 may have 4 significant figures but usually 3000 will have only 1 significant figure! When in doubt, use scientific notation 3.000 x 103 (4 significant figures) or 3 x 103 (1 significant figure)  Generally, round to the least number of significant figures of the given data  25 x 18 = 450 → 2 significant figures;  25 x 18.975 = 474.375→ the final answer should consist of only 2 significant figures → 470  Significant figures are not decimal places  0.00356 has 5 decimal places, 3 significant figures 3.56 x 10-3 © 2014 John Wiley & Sons, Inc. All rights reserved.  As a rule, when numbers are multiplied or divided, the number of significant figures in the final answer equals the smallest number of significant figures in any of the original factors.  As a rule, when numbers are added or subtracted, the last significant figure in the final answer occurs in the last column containing a number that results from a combination of digits that are all significant.  2.5 + 11 + 5.26 = 18.76  19 1-1 Measuring Things, Including Lengths 1-2 Time  Any standard of time needs to be able to answer: o When did a thing happen? o What was its duration?  Times follow the same conversion process as lengths  Standards of time in the past have included (any phenomenon that repeats itself is a candidate): 1. Average time for the Earth to rotate once: 1 day being set to 86,400 seconds 2. Quartz clock (piezo-electric material that uses an electronic oscillator that is regulated by a quartz crystal to keep time) 3. Cesium Atomic clocks (uncertainty of 1 sec in 60 millions years) © 2014 John Wiley & Sons, Inc. All rights reserved. 1-2 Time  The variation in the length of a day as measured by an atomic clock: Figure 1-2  The vertical scale here amounts to only 3 ms, or 0.003 s.  This shows the precision of atomic clocks, and the relative imprecision of Earth's rotation (affected by tides, winds) © 2014 John Wiley & Sons, Inc. All rights reserved. 1-3 Mass  The standard kilogram is a cylinder of platinum and iridium stored in France.  Accurate copies have been sent around the world, other masses can be measured by comparing them against these copies  The atomic mass unit (u) is a second mass standard  1 atom of Carbon-12 is assigned a mass 12 u (international agreement)  Used for measuring masses of atoms and molecules  1 u = 1.660 538 86 x 10-27 kg (+/- 10 x 10-35 kg)  Masses follow the same conversion process as lengths and times © 2014 John Wiley & Sons, Inc. All rights reserved. 1-3 Mass  For uniformly (homogeneous) system, we define the Mass per unit volume of that system to be the density Eq. (1-8) Examples Calculate . . . o Density of material: (18 kg) / (0.032 m3) = 560 kg/m3 o Mass of object: (380 kg/m3) x (0.0040 m3) = 1.5 kg o Volume of object: (250 kg) / (1280 kg/m3) = 0.20 m3 © 2014 John Wiley & Sons, Inc. All rights reserved. Measurement  Defined by relationships to base quantities  Each defined by a standard, and given a unit SI Units  International System of Units  Each base unit has an accessible standard of measurement Length  Meter is defined by the distance traveled by light in a vacuum in a specified time interval Changing Units  Use chain-link conversions  Write conversion factors as unity  Manipulate units as algebraic quantities 1 Summary © 2014 John Wiley & Sons, Inc. All rights reserved. Time  Second is defined in terms of oscillations of light emitted by a cesium-133 source  Atomic clocks are used as the time standard Mass  Kilogram is defined in terms of a platinum-iridium standard mass  Atomic-scale masses are measured in u, defined as mass of a carbon-12 atom Density  Mass/volume Eq. (1-8) 1 Summary DIMENSIONAL ANALYSIS DIMENSIONAL ANALYSIS Distance has the dimension of length, [L] Dimension of mass is [M] Dimension of time is [T] All physical quantities are expressed by specifying both a number and a unit. Dimension refers to the physical nature of a physical quantity and the type of unit used to specify it. Many physical quantities (derived units) are in terms of a combination of fundamental dimensions (base units) such as [L], [M], and [T] 2 2 1 vt x  Is the following equation dimensionally correct?        L T T T L L 2       DIMENSIONAL ANALYSIS is used to check mathematical relations for the consistency of their dimensions DIMENSIONAL ANALYSIS Is the following equation dimensionally correct? vt x        L T T L L       DIMENSIONAL ANALYSIS alone can not identify the correct the relation DIMENSIONAL ANALYSIS