PHYSICS AND MEASUREMENTS
1.1 Standards Of Length, Mass, And Time
1.2 Matter And Model Building
1.3 Density And Atomic Mass
1.4 Dimensional Analysis
1.5 Conversion Unit
1.6 Order Of Magnitude
1.7 Significant Figures
1.1 Standards of Length, Mass, and Time
In 1960, an international committee established a set of standards for the fundamental quantities of science. It is called the SI (Système International), and its units of length, mass, and time are the meter, kilogram, and second, respectively. Other SI standards established by the committee are those for temperature (the kelvin), electric current (the ampere), luminous intensity (the candela), and the amount of substance (the mole).
As recently as 1960, the length of the meterwas defined as the distance between two lines on a specific platinum–iridium barstored under controlled conditions in France. This standard was abandoned for several reasons, a principal one being that the limited accuracy with which the separation between the lines on the bar can be determined does not meet the current requirements of science and technology.
In the 1960s and 1970s, the meter was definedas 1 650 763.73 wavelengths of orange-red light emitted from a krypton-86lamp. However, in October 1983, the meter (m) was redefined as the distance traveled by light in vacuum during a time of 1/299 792 458 second
The SI unit of mass, the kilogram (kg), is defined as the mass of a specificplatinum–iridium alloy cylinder kept at the International Bureau of Weightsand 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. A duplicate of the Sèvres cylinder is kept at the National Institute of Standards andTechnology (NIST) in Gaithersburg, Maryland.
In 1967, the second was redefined to take advantage of the high precision attainable in a device known as an atomic clock , which uses the characteristic frequency of the cesium-133 atom as the “reference clock.” The second (s) is now defined as 9.192.631.770 times the period of vibration of radiation from the cesium atom.
1.2 Matter and Model Building
In 1897, J. J. Thomson identified the electron as a charged particle and as a constituent of the atom. Following the discovery of the nucleus in 1911, a model was developed in which each atom is made up of electrons surrounding a central nucleus.
By the early 1930s 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 electric charge, and a specific chemical element is identified by the number of protons in its nucleus. This number is called the atomic number of the element
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.
The existence of neutrons was verified conclusively in 1932. A neutron has no charge and a mass that is about equal to that of a proton. One of its primary purposes is to act as a “glue” that holds the nucleus together.
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, charmed, bottom, and top.
The up, charmed, and top quarks have electric charges of that of the proton, whereas the down, strange, and bottom quarks have charges of that of the proton.
1.3 Density and Atomic Mass
Let us look now at an example of a derived quantity—density. The density p (Greek letter rho) of any substanceis defined as its mass per unit volume:
p = density
m = mass
V = volume
The numbers of protons and neutrons in the nucleus of an atom of an element are related to the atomic mass of the element.
QUICK QUIZ 1.1
Problem: In a machine shop, two cams are produced, one of aluminium and one of iron. Both cams have the same mass
Question: Which cam is larger?
(a) The aluminium cam
(b) The iron cam
(c) Both cams have the same size
Answer: (a) Because the density of aluminum is smaller than an iron, a larger volume of aluminum is required for a
given mass than iron.
Problem : A solid cube of aluminum has P 2.70 g/cm3 and V0.200 cm3 and contains 6.02 X 1023 atoms
Question : How many aluminum atoms are contained the cube?
Answer :Because density equals mass per unit volume, the mass of the cube is
m = pV = (2.70 g/
cm3) (0.200 cm3) = 0.540 g
To solve this problem, we will set up a ratio based on the fact that the mass of a sample of material is proportional to the number of atoms contained in the sample.
m = kN
m = mass of the sample
N = number of atoms in the Sample
k = unknown constant
msampel = kNsampel
m27.0 g = kN27.0 g
>> msampel / m27.0 g = kNsampel /kN27.0 g
We now substitute the values:
0.540 g / 27.0 g = Nsampel/6.02 x 1023
Nsampel = (0.540 g)(6.02 x 1023) / (27.0 g)
= 1.20 x 1022
1.4 Dimensional Analysis
The word dimension has a special meaning in physics. It denotes the physical nature of a quantity. Whether a distance is measured in units of feet or meters or fathoms, it is still a distance. We say its dimension is length. The symbols we use specify the dimensions of length, mass, and time are L, M, and T. We shall often use brackets [ ] to denote the dimensions of a physical quantity. For example, the symbol we use for speed is v, and in our notation the dimensions of speed are written [v] = L/T. As another example, the dimensions of area A are [A] = L². In many situations, you may have to derive or check a specific equation. A useful and powerful procedure called dimensional analysis can be used to assist in the derivation or to check your final expression. Dimensional analysis makes use of the fact that dimensions can be treated as algebraic quantities. For example, 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 on both sides of the equation are the same.
1.5 Conversion Units
Sometimes it is necessary to convert units from one measurement system to another, or to convert within a system, for example, from kilometers to meters. Equalities between SI and U.S. customary units of length are as follows:
1 mile =1609 m = 1,609 km 1 ft = 0,304 8 m = 30,48 cm
1 m = 39,37 in. = 3.281 ft 1 in. = 0,0254 m = 2,54 cm
Units can be treated as algebraic quantities that can cancel each other. For example, suppose we wish to convert 15.0 in. to centimeters. Because 1 in. is defined as exactly 2.54 cm,
15,0 in.=(15,0 in.)(2,54cm / 1 in. ) = 38,1 cm
we find that where the ratio in parentheses is equal to 1. Notice that we choose to put the unit of an inch in the denominator and it cancels with the unit in the original quantity. The remaining unit is the centimeter, which is our desired result.
1.6 Estimates and Order of Magnitude
The purpose of order of magnitude is to prove something without resorting to involved calculations. It has its power, it prevents us from wasting a lot of time -- but it also has its limitations, sometimes we must use involved calculations. It is often useful to compute an approximate answer to a given physical problem even when little information is available. This answer can then be used to determine whether or not a more precise calculation is necessary. Such an approximation is usually based on certain assumptions, which must be modified if greater precision is needed. We will sometimes refer to an order of magnitude of a certain quantity as the power of ten of the number that describes that quantity. Usually, when an order-of magnitude calculation is made, the results are reliable to within about a factor of 10. If a quantity increases in value by three orders of magnitude, this means that its value increases by a factor of about 10³ = 1 000. We use the symbol ~ for “is on the order of.” Thus,
0,008 6 ~10 ̄² 0,002 1 ~ 10¯³ 720 ~10 ³
1.7 Significant figures
When certain quantities are measured, the measured values are known only to within the limits of the experimental uncertainty. The value of this uncertainty can depend on various factors, such as the quality of the apparatus, the skill of the experimenter, and the number of measurements performed. The number of significant figures in a measurement can be used to express something about the uncertainty. As an example of significant figures, suppose that we are asked in a laboratory experiment to measure the area of a computer disk label using a meter stick as a measuring instrument. Let us 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. Likewise, if the label’s width is measured to be 6,4 cm, the actual value lies between 6,3 cm and 6,5 cm. Thus we could write the measured values as (5,5 ± 0,1) cm and (6,4 ±0,1) cm.
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 cm², 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. A good rule of thumb to use in determining the number of significant figures that can be claimed in a multiplication or a division is as follows:
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 cm², realizing that the value can range between (5,4 cm)(6,3 cm) = 34 cm² 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 0,007 5 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 1 500 g. This value is ambiguous because we do not know whether the last two zeros are being used to locate the decimal po]int 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 1,5x10³ g if there are two significant figures in the measured value, 1,50 x10³ g if there are three significant figures, and 1,500 x10³g if there are four. The same rule holds for numbers less than 1, so that 2.3x10 ̄⁴ has two significant figures (and so could be written 0,000 23) and 2,30x10 ̄⁴ has three significant figures (also written 0,000 230). 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.
Quick Quiz 1.2
True or False: Dimensional analysis can give you the numerical value of constants of proportionality that may appear in an algebraic expression.
answer : False. Dimensional analysis gives the units of the proportionality constant but provides no information about its numerical value. To determine its numerical value requires either experimental data or geometrical reasoning.For example, in the generation of the equation x = 1/2 at²,because the factor is dimensionless, there is no way of determining it using dimensional analysis.
Quick Quiz 1.3
The distance between two cities is 100 mi. The number of kilometers between the two cities is (a) smaller than 100 (b) larger than 100 (c) equal to 100.
answer: (b). Because kilometers are shorter than miles, a larger number of kilometers is required for a given distance than miles.
Quick Quiz 1.4
Suppose you measure the position of a chair with a meter stick and record that the center of the seat is 1,043 860 564 2 m from a wall. What would a reader conclude from this recorded measurement?
answer : Reporting all these digits implies you have determined the location of the center of the chair’s seat to the nearest ±0,000 000 000 1 m. This roughly corresponds to being able to count the atoms in your meter stick because each of them is about that size! It would be better to record the measurement as 1.044 m: this indicates that you know the position to the nearest millimeter, assuming the meter stick has millimeter markings on its scale.
Installing a Carpet
A carpet is to be installed in a room whose length is measured to be 12.71 m and whose width is measured to be 3,46 m.
Question: Find the area of the room.
Solution : If you multiply 12,71m by 3,46m on your calculator, you will see an answer of 43,9766m². How many of these numbers should you claim? Our rule of thumb for multiplication tells us that you can claim only the number of significant figures in your answer as are present in the measured quantity having the lowest number of significant figures. In this example, the lowest number of significant figures is three in 3.46 m, so we should express our final answer as 44,0 m².