Physics and Measurement, Summaries of Physics

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Raymond A Serway ,

Robert J Beichner

authors

Physics and Measurement

Like all other sciences, physics is based on experimental

observations and quantitative measurements.

Classical physics, which means all of the physics developed before

1900 , includes the theories, concepts, laws, and experiments in

classical mechanics, thermodynamics, and electromagnetism

Modern physics, began near the end of the 19 th century. Modern

physics developed mainly because of the discovery that many

physical phenomena could not be explained by classical physics.

The two most important developments in modern physics were the

theories of relativity and quantum mechanics. Einstein’s theory of

relativity revolutionized the traditional concepts of space, time, and

energy; quantum mechanics, which applies to both the microscopic

and macroscopic worlds, was originally formulated by a number of

distinguished scientists to provide descriptions of physical

phenomena at the atomic level

Physics is interested in materials, energy and relation between them

Mass

The basic SI unit of mass, the kilogram (kg), is defined as the

mass of a specific platinum–iridium alloy cylinder kept at the

International Bureau of Weights and Measures at Sèvres,

France. This mass standard was established in 1887 and has not

been changed since that time because platinum–iridium is an

unusually stable alloy (Fig. 1. 1 a). A duplicate of the Sèvres

cylinder is kept at the National Institute of Standards and

Technology (NIST) in Gaithersburg, Maryland.

Time

Before 1960 , the standard of time was defined in terms of the mean solar day for the year 1900. 2 The mean solar second was originally defined as of a mean solar day. The rotation of the Earth is now known to vary slightly with time, however, and therefore this motion is not a good one to use for defining a standard. In 1967 , consequently, the second was redefined to take advantage of the high precision obtainable in a device known as an atomic clock (Fig. 1. 1 b). In this device, the frequencies associated with certain atomic transitions can be measured to a precision of one part in

1012. This is equivalent to an uncertainty of less than one second every 30 000 years. Thus, in 1967 the SI unit of time, the second, was redefined using the characteristic frequency of a particular kind of cesium atom as the “reference clock.” The basic SI unit of time, the second (s), is defined as 9 192 631 770 times the period of vibration of radiation from the cesium- 133 atom. To keep these atomic clocks— and therefore all common clocks and watches that are set to them—synchronized, it has sometimes been necessary to add leap seconds to our clocks.

1.2 THE BUILDING BLOCKS OF MATTER

All ordinary matter consists of atoms, and each atom is made up of electrons surrounding a central nucleus. Following the discovery of the nucleus in 1911 , the question arose: Does it have structure? That is, is the nucleus a single particle or a collection of particles? The exact composition of the nucleus is not known completely even today, but by the early 1930 s a model evolved that helped us understand how the nucleus behaves. Specifically, scientists determined that occupying the nucleus are two basic entities, protons and neutrons. The proton carries a positive charge, and a specific element is identified by the number of protons in its nucleus. This number is called the atomic number of the element. For instance, the nucleus of a hydrogen atom contains one proton (and so the atomic number of hydrogen is 1 ), the nucleus of a helium atom contains two protons (atomic number 2 ), and the nucleus of a uranium atom contains 92 protons (atomic number 92 ). In addition to atomic number, there is a second number characterizing atoms—mass number, defined as the number of protons plus neutrons in a nucleus. As we shall see, the atomic number of an element never varies (i.e., the number of protons does not vary) but the mass number can vary (i.e., the number of neutrons varies). Two or more atoms of the same element having different mass numbers are isotopes of one another. Protons, neutrons, and a host of other exotic particles are now known to be composed of six different varieties of particles called quarks, which have been given the names of up, down, strange, charm, bottom, and top. The up, charm, and top quarks have charges of + 2 / 3 that of the proton, whereas the down, strange, and bottom quarks have charges of - 1 / 3 that of the proton. The proton consists of two up quarks and one down quark which you can easily show leads to the correct charge for the proton. Likewise, the neutron consists of two down quarks and one up quark, giving a net charge of zero.

1.3 DENSITY

A property of any substance is its density 𝜌 (Greek

letter rho), defined as the amount of mass contained

in a unit volume , which we usually express as mass

per unit volume:

The difference in density between aluminum and lead is due, in part, to their different atomic masses. The atomic mass of an element is the average mass of one atom in a sample of the element that contains all the element’s isotopes, where the relative amounts of isotopes are the same as the relative amounts found in nature. The unit for atomic mass is the atomic mass unit (u), where 1 u 1. 660 540 2 10 -^27 kg. The atomic mass of lead is 207 u, and that of aluminum is 27. 0 u. However, the ratio of atomic masses, 207 u/ 27. 0 u 7. 67 , does not correspond to the ratio of densities, ( 11. 3 g/cm 3 )/( 2. 70 g/cm 3 ) 4. 19. The discrepancy is due to the difference in atomic separations and atomic arrangements in the crystal structure of these two substances. The mass of a nucleus is measured relative to the mass of the nucleus of the carbon- 12 isotope, often written as 12 C. (This isotope of carbon has six protons and six neutrons. Other carbon isotopes have six protons but different numbers of neutrons.) Practically all of the mass of an atom is contained within the nucleus. Because the atomic mass of (^12) C is defined to be exactly 12 u, the proton and neutron each have a mass of about 1 u. One mole (mol) of a substance is that amount of the substance that contains as many particles (atoms, molecules, or other particles) as there are atoms in 12 g of the carbon- 12 isotope. One mole of substance A contains the same number of particles as there are in 1 mol of any other substance B. For example, 1 mol of aluminum contains the same number of atoms as 1 mol of lead.

Experiments have shown that this number, known as

Avogadro’s number, N A , is

N A =6.022 137 x 1023 particles/mol

Avogadro’s number is defined so that 1 mol of carbon- 12 atoms has a mass of exactly 12 g. In general, the mass in 1 mol of any element is the element’s atomic mass expressed in grams. For example, 1 mol of iron (atomic mass 55. 85 u) has a mass of 55. 85 g (we say its molar mass is 55. 85 g/mol), and 1 mol of lead (atomic mass 207 u) has a mass of 207 g (its molar mass is 207 g/mol). Because there are 6. 02 1023 particles in 1 mol of any element, the mass per atom for a given element is

N A

1.4 DIMENSIONAL ANALYSIS

The word dimension has a special meaning in physics. It usually denotes the physical nature of a quantity. Whether a distance is measured in the length unit feet or the length unit meters, it is still a distance. We say the dimension—the physical nature—of distance is length. The symbols we use in this book to specify length, mass, and time are L, M, and T , respectively. We shall often use brackets [ ] to denote the dimensions of a physical quantity. For example, the symbol we use for speed in this book is v , and in our notation the dimensions of speed are written As another example, the dimensions of area, for which we use the symbol A , are The dimensions of area, volume, speed, and acceleration are listed in Table 1. 6. In solving problems in physics, there is a useful and powerful procedure called dimensional analysis. This procedure, which should always be used, will help minimize the need for rote memorization of equations. Dimensional analysis makes use of the fact that dimensions can be treated as algebraic quantities. That is, quantities can be added or subtracted only if they have the same dimensions. Furthermore, the terms on both sides of an equation must have the same dimensions. By following these simple rules, you can use dimensional analysis to help determine whether an expression has the correct form. The relationship can be correct only if the dimensions are the same on both sides of the equation.

• 1. 7 Significant Figures
• The number of significant figures in a measurement can be used to express something about the uncertainty.
• Assume that the accuracy to which we can measure the length of the label is ± 0. 1 cm. If the length is measured to be

5. 5 cm, we can claim only that its length lies somewhere between 5. 4 cm and 5. 6 cm. In this case, we say that the

measured value has two significant figures. Note that the significant figures include the first estimated digit.

• Now suppose we want to find the area of the label by multiplying the two measured values. If we were to claim the

area is ( 5. 5 cm)( 6. 4 cm)= 35. 2 cm^2 , our answer would be unjustifiable because it contains three significant figures, which

is greater than the number of significant figures in either of the measured quantities.

• Rule: When multiplying several quantities, the number of significant figures in the final answer is the same as the

number of significant figures in the quantity having the lowest number of significant figures. The same rule applies to

division.

• Applying this rule to the previous multiplication example, we see that the answer for the area can have only two

significant figures because our measured quantities have only two significant figures. Thus, all we can claim is that the

area is 35 cm^2 , realizing that the value can range between ( 5. 4 cm)( 6. 3 cm)= 34 cm^2 and ( 5. 6 cm)( 6. 5 cm)= 36 cm^2.

• Zeros may or may not be significant figures. Those used to position the decimal point in such numbers as 0. 03

and 0. 0075 are not significant. Thus, there are one and two significant figures, respectively, in these two values.

When the zeros come after other digits, however, there is the possibility of misinterpretation. For example,

suppose the mass of an object is given as 1500 g. This value is ambiguous because we do not know whether the

last two zeros are being used to locate the decimal point or whether they represent significant figures in the

measurement. To remove this ambiguity, it is common to use scientific notation to indicate the number of

significant figures. In this case, we would express the mass as 1. 5 x 103 g if there are two significant figures in the

measured value, 1. 50 x 103 g if there are three significant figures, and 1. 500 x 103 g if there are four. The same

rule holds for numbers less than 1 , so that 2. 3 x 10

• 4

has two significant figures (and so could be written 0. 00023 )

and 2. 30 x 10 -^4 has three significant figures (also written 0. 000230 ).

• In general, a significant figure in a measurement is a reliably known digit (other than a zero used to locate the

decimal point) or the first estimated digit.

• Rule: When numbers are added or subtracted, the number of decimal places in the result should equal the

smallest number of decimal places of any term in the sum.

• 123 + 5. 35 = 128 , not 128. 35.
• 1. 0001 + 0. 0003 = 1. 0004 (the result has five significant figures, even though one of the terms in the sum, 0. 0003 ,

has only one significant figure.)

• Likewise, 1. 002 - 0. 998 = 0. 004 the result has only one significant figure even though one term has four

significant figures and the other has three.