PH 221-1D Spring 2013

Transcription

1 PH 221-1D Spring 2013 Introduction and Measurement Lecture 1 Chapter 1 (Halliday/Resnick/Walker, Fundamentals of Physics 9 th edition)

2 The Nature of Physics The science of physics has developed out of the efforts of men and women to explain our physical environment. These efforts have been so successful that the laws of physics now encompass a remarkable variety of phenomena, including planetary orbits, radio and TV waves, magnetism and lasers, to name just a few. The exciting feature of physics is its capacity for predicting how nature will behave in one situation on the basis of experimental data obtained in another situation. Such predictions place physics at the heart of modern technology and, therefore, can have a tremendous impact on our lives (e.g. the medical profession uses X-rays, ultrasonic, and magnetic resonance methods for the obtaining images of the interior of the human body, and physics lies at the core of all these). Because physics is so fundamental, it is a required course for students in a wide range of major areas. During studies of physics you will learn how to see the world through the eyes of physics, reason as a physicist does, and will learn how to apply physics principles to a wide range of problems. 2

3 Chapter 1 Measurement In this chapter we will explore the following concepts: 1. Measurement of a physical parameter 2. Units, Systems of units 3. Basic Units in Mechanics 4. Changing units 5. Significant figures 3

4 In Physics we carry out experiments in which we measure physical parameters. We then try to deduce the relationship between the measured quantities. We usually express this relationship in the form of a mathematical equation which we call the physical law that describes the phenomenon under study. A familiar example is Ohm s law. The experiment in this case consists of measuring the electric voltage difference V applied across a conductor and the resulting electric current I that flows through the conductor If we plot I versus V we get a straight line. This is expressed in the form: The equation is known as: Ohm s Law R V I Constant R is known as the resistance of the conductor 4

5 Assume that you step on your bathroom scale and that it reads 120 The number alone is meaningless. It must be accompanied by the units 120 lb is a very different reading from 120 kg! Conclusion: We measure each physical quantity in its own units, by comparison with a standard. The unit is a unique name we assign to measures of that quantity (meter unit of length). The standard corresponds to exactly 1.0 unit of the quantity. Does this mean that we have to define units for all parameters? The answer is no. In mechanics we need to define only three parameters: These parameters are: Length, Time, and Mass They are known as: base quantities Note: For the rest of the non-mechanical parameters we need to define only one more unit, that of the electric current In this book we use the International System of Units (SI) In this system the units for the base quantities are: Parameter Unit Name Symbol Length meter m Time second s Mass kilogram kg 5

6 Powers of Ten and Scientific Notation In science, very large and very small decimal numbers are conveniently expressed in terms of powers of ten = = = 1/10 = = = 1/10 2 = = = 1/10 3 = Example: Earth s radius = 6,380,000 m = 6.38 x 10 6 m The factor of ten raised to the six power is ten multiplied by itself six times. The decimal place should be moved six places to the right to obtain the number without the power of ten. Numbers expressed with the aid of powers of ten are said to be in scientific notation. Calculations that involve the multiplication and division of powers of ten are carried out according to the following general rules: 1/10 n = 10 -n 10 n x 10 m = 10 n+m 10 n /10 m = 10 n-m Where n and m are any positive or negative numbers. 6

7 earth A The meter In 1792 the meter was defined to be one ten-millionth of the distance from the north pole to the equator. C equator B AB 1 m 10 7 For practical reasons the meter was later defined as the distance between two fine lines on a standard meter bar made of platinum-iridium. Since 1983 the meter is defined as the length traveled by light in vacuum during the time interval of 1/ of a second. The reason why this definition was adapted was that the measurement of the speed of light had become extremely precise 7

8 Length Metric System (decimal approach) All quantities are divided into 10, 100, 1000 Multiples and submultiples of the meter (m) Kilometer 1 km = 10 3 m Meter 1 m Centimeter 1 cm = 10-2 m Millimeter 1 mm = 10-3 m Micron 1 µm = 10-6 m Nanometer 1 nm = 10-9 m Angstrom 1 Å = m Fermi 1 fm = m Multiples and submultiples of the foot Mile 1 mile = 5280 feet = m Yard 1 yard = 3 ft = m Foot 1 ft = m Inch 1 in. = cm The meter is the length traveled by a light wave in a vacuum in a time interval of 1/299,792,458 s 8

9 Some distances and sizes 9

10 Time Conceptually, time is the measure of the rate at which change occurs. In 1967, the SI unit of time, the second (s), was defined as the interval required for 9,192,631,770 vibrations of the cesium-133 atom measured via an atomic beam clock years ago the Egyptians divided day and night into 12 equal hours. Babylonian arithmetic used 60 at its number base began tradition of subdividing things into 60 equal parts. 14 th century mechanical clock hour was divided into 60 minutes, 60 sec. = 1 minute 10

11 The Second Initially the second was defined as follows: (1-5) 1 1 second of the time it takes the earth to complete a full rotation about its axis The problem with this definition is that the length of the day is not constant as is shown in the figure. For this reason since 1967 the second is defined as the time taken by light oscillations of a particular wavelength emitted by a cesium-133 atom. This definition is so precise that it would take two cesium clocks 6000 years before their readings would differ more than 1 second. 11

12 Multiples and Submultiples of the Second 12

13 The kilogram The SI standard of mass is a platinum-iridium cylinder shown in the figure. The cylinder is kept at the International Bureau of Weights and Measures near Paris and assigned a mass of 1 kilogram. Accurate copies have been sent to other countries. 13

14 The meter, the second, and the kilogram are the fundamental units or base units for length, time and mass. Any other physical quantity can be measured by introducing a derived unit constructed from some combination of the base units. 14

15 Significant Figures (1-8) A certain parameter, for example the length L of an object, can be determined with a varying degree of accuracy. The accuracy depends on the measurement method and the measuring instrument. If I measure L with a ruler (smallest division = 1 mm) I can write L as: L = m. The length L is given with four significant figures. It would be meaningless to write L as: L = m because my ruler cannot measure a fraction of a millimeter. If on the other hand I use calipers that can measure with an accuracy of 0.1 mm, then I can write L = m, and L is given with five significant figures. In a calculation the number of significant figures cannot be larger than the number of significant figures of the parameters used in the measurement Example: A car traveling with constant speed v covers a distance d = 123 m d 123 m in a time t = 7.89 s. The speed v is given by: v = = = m/s t 7.89 s It is meaningless to use 9 significant figures to express v because d and t used to determine v are known with an accuracy of only 3 significant figures The correct way to express v is: v = 15.6 m/s i.e. with 3 significant figures. 15

16 Ruler Calipers 16

17 Significant Figures Measurement is very different from counting, although both associate numbers with notions. We can count the number of beans in a jar and know it exactly. But we cannot measure the height of a jar exactly. There is no such thing as an exact measurement. Practically, measurements are made to some desired precision that suits the experimenter s purposes (often determined by the limitations of the available instruments). Example: A rod whose length is being measured with a ruler. The object is between 4.1 and 4.2 cm long. Because there are no finer divisions than 1 mm on the scale, fractions of a millimeter will have to be estimated. We conclude that the rod is 4.15 cm long. The 4 is certain, as is the 1, but the 5 might be in error as much as ±1 and trying to arrive at any more figures would be meaningless. 17

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24 Changing Units Quite often we have to change the units of a physical parameter. To do that we must have the conversion factor between one unit and the other. Appendix D lists several conversion factors between SI units and other units Example: Express the highway speed limit of 65 miles per hour in meters per second. 1 mile 1609 m 1 mile = 1609 m. The converion factors can be written as : m 1 mile 1 hour 3600 s 1 hour = 3600 s. The converion factors can be written as : s 1 hour The method is called chain link conversion We use one of the two forms of the conversion factor that eliminates the units we wish to change and introduces the new units. In our example: miles miles 1609 m 1 hour m hour hour 1 mile 3600 s s 24

25 The role of units in problem solving Since any quantity, such as length, can be measured in several different units, it is important to know how to convert from one unit to another. Example: Someone is 2.00 yard tall. Using the fact that 1 inch is exactly 2.54 cm, how tall is the person in centimeters? Reasoning: When converting between units we write down the units explicitly in the calculations and treat them like any algebraic quantity. In particular we will take advantage of the following algebraic fact: Multiplying or dividing an equation by a factor of 1 does not alter the equation. Solution: -Since 3.00 feet = 1.00 yard, it follows that (3.00 feet)/(1.00 yard) = 1 -Since 12.0 inches = 1.00 foot, it follows that (12.0 inches)/(1.00 foot) = 1 -Since 2.54 cm = 1.00 inch, it follows that (2.54 cm)/(1.00 inch) = 1 -Using these factor to multiply the equation: Height = h = 2.00 yards we find that h = (2.00 yd)(1)(1)(1) = (2.00 yd) * (3.00 ft)/(1.00 yd) * (12.0 in)/(1.00 ft) * (2.54 cm)(1 inch) = cm After rounding to three significant figures: h = 183 cm 25

26 Guides the use of conversion factors Units as a problem solving aid Provides an internal check to eliminate certain kinds of errors if the units are carried along during each step of the calculation and treated like any algebraic factor. Example: The tank of a car contains 2.0 gallons of gas to start with and that gas is added at a rate of 7.0 gallons/minute. The total amount of gas in the tank 96 seconds later can be obtained by adding the amount put into the tank to the amount present initially. Total amount = Initial amount of gas + Gas added = 2.0 gallons + (7.0 gallons/min)(96 sec) = 2.0 gallons gallons * sec/min = 674 because the units for the two added terms are not the same Solution: Total amount = 2.0 gallons gallons/min * 1 min/60 sec * 96 sec = 2.0 gallons + 11 gallons = 13 gallons Only quantities that have exactly the same units can be added or subtracted. The procedure of carrying along the units serves as automatic reminder to convert all data in calculations into a consistent set of units. 26

27 Dimensional Analysis In physics, the term dimensional analysis is used to refer to the physical nature of a quantity and the type of unit used to specify that quantity. Dimensional analysis is used to check mathematical relations for the consistency of their dimensions. Example: A car starts from rest and accelerates to a speed v in a time t. We wish to calculate distance x traveled by the car, but we are not sure whether the correct relation is x = ½vt 2 or x = ½vt. We can determine the correct relation by checking the quantities on both sides of the equation to see if they have the same dimensions. If the dimensions are not the same, the relation is incorrect. x = ½vt 2 cannot be correct [L] = [L/T] * [T] 2 = [L][T] x = ½vt [L] = [L/T] * [T] = [L] 27