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Accommodation of Standard Technical Measures It is Company policy to accommodate, and not interfere with, standard technical measures it determines are reasonable under the circumstances, i. Click here - to use the wp menu builder Click here - to use the wp menu builder. Where Are They Now? StudySpace w w w. Smartwork online homework systema subscription service. In a quite literal sense, physics is the greatest of all natural sciences: it encompasses the small- est particles, such as electrons and quarks; and it also encom- Note that the power of 10, or the exponent on the 10, simply tells us how many zeros follow the 1 in the number if the power of 10 is positive or how many zeros follow the passes the largest bodies, such as galaxies and the entire Universe.
In In scientific notation, a number that does not coincide the pictures on the following pages we will survey the world of with one of the powers of 10 is written as a product of a dec- physics and attempt to develop some rough feeling for the sizes imal number and a power of For example, in this nota- of things in this world. This preliminary survey sets the stage tion, is written as 1.
Alternatively, this for our explanations of the mechanisms that make things behave number could be written as 15 or as 0. Such explanations are at the heart of physics, in scientific notation it is customary to place the decimal point and they are the concern of the later chapters of this book. The same rule applies Since the numbers we will be dealing with in this prelude to numbers smaller than 1; thus, 0.
In this notation, numbers are written with powers of In the first sequence we zoom out: we begin with a picture of 10; thus, hundred is written as , thousand is written as , a womans face and proceed step by step to pictures of the ten thousand is written as , and so on. The following table lists some powers of ten: decreasing in steps of factors of Many of these have 0.
For some of our pictures no 0. Erin belongs to the phylum Chordata, class Mammalia, order Primates, family Hominidae, genus Homo, species sapiens. She is made of 5. The matter in Erins body and the matter in her immediate environ- ment occur in three states of aggregation: solid, liquid, and gas.
All these forms of matter are made of atoms and molecules, but solid, liquid, and gas are qualitatively different because the arrangements of the atomic and molecular building blocks are different. In a solid, each building block occupies a definite place.
When a solid is assembled out of molecular or atomic building blocks, these blocks are locked in place once and for all, and they cannot move or drift about except with great difficulty. This rigidity of the arrangement is what makes the aggregate hardit makes the solid solid. In a liquid, the molecular or atomic building blocks are not rigidly connected. They are thrown together at random and they move about fairly freely, but there is enough adhesion 0 0. Finally, in a gas, the molecules or atoms are almost completely independent of one another.
They are distributed at random over the volume of the gas and are separated by appreciable distances, coming in touch only occasionally during collisions. A gas will disperse spontaneously if it is not held in con- finement by a container or by some restraining force.
This library holds 1. The streets in this part of the city are laid out in a regular rec- tangular pattern. The library is the building in the park in the middle of the picture.
The photograph was taken early in the morning, and the high buildings typical of New York cast long shadows. The photograph was taken from an airplane flying at an altitude of a few thousand meters.
North is at the top of the photograph. We can barely recognize the library and its park as a small rectangular patch slightly above the center of the picture. The central mass of land is the island of Manhattan, with the Hudson River on the left and the East River on the right. This photograph and the next two were taken by satellites orbiting the Earth at an altitude of about kilometers.
On this scale, we can no longer distinguish the pattern of streets in the city. The vast expanse of water in the lower right of the picture is part of the Atlantic Ocean. The mass of land in the upper right is Long Island.
Parallel to the south shore of Long Island we can see a string of very narrow islands; they almost look man-made. These are barrier islands; they are heaps of sand piled up by ocean waves in the course of thousands of years.
Cape Cod is the hook near the northern end of the coastline, and Cape Fear is the promontory near the southern end of the coastine. Note that on this scale no signs of human habitation are visible. However, at night the lights of large cities would stand out clearly. This photograph was taken in the fall, when leaves had brilliant colors.
Streaks of orange trace out the spine of the Appalachian moun- tains. Through the gap in the clouds in the lower middle of the pic- ture, we can see the coast of California and Mexico. We can recognize the peninsula of Baja California and the Gulf of California. Erins location, the East Coast of the United States, is covered by a big system of swirling clouds on the right of the photograph.
Note that a large part of the area visible in this photograph is ocean. The atmosphere covering this surface is about kilometers thick; on the scale of this photograph, its thickness is about 0. Seen from a large distance, the predominant colors of the planet Earth are blue oceans and white clouds. Sunlight is striking the Earth from the top of the picture. As is obvious from this and from the preceding photograph, the Earth is a sphere.
Its radius is 6. As in the preceding picture, the Sun is far below the apogee bottom of the picture. The position of the Moon is that of January 1, Jan. The solid red curve in the drawing is the orbit of the Moon, and the dashed green curve is a circle; by comparing these two curves we can see how little the ellipse deviates from a circle centered on the Earth. The point on the ellipse closest to perigee the Earth is called the perigee, and the point farthest from the Earth.
The distance between the Moon and the Earth is 2. The Moon takes On this scale, both the Earth and the Moon look like small dots. Again, the Sun is far below the bottom of the picture. In the middle, we see the Earth and the Moon in their positions for January 1, On the right and on the left we see, respectively, their positions for 1 day before and 1 day after this date.
Note that the net motion of the Moon consists of the Dec. However, Venus itself is beyond the edge of the picture.
The small circle is the orbit of the Moon. The dot repre- senting the Earth is much larger than what it should be, although the artist has drawn it as minuscule as possible.
On this scale, even the Sun is quite small; if it were included in this pic- ture, it would be only 1 millimeter across. The positions of the planets are those of January 1, The orbits of all these planets are ellipses, but they are close to Earth circles. The point of the orbit nearest to the Sun is called the per- Venus ihelion and the point farthest from the Sun is called the aphe- lion. The Earth reaches perihelion about January 3 and aphelion Mars about July 6 of each year.
All the planets travel around their orbits in the same direction: counterclockwise in our picture. The marks along the Mercury orbit of the Earth indicate the successive positions at intervals of 10 days. Furthermore, a large number of comets orbit around the 0 0.
Most of these have pronounced elliptical orbits. The comet Halley has been included in our drawing. On the scale of the picture, the Sun looks like a very small dot, even smaller than the dot drawn here. The mass of the Sun is 1. The matter in the Sun is in the plasma state, sometimes called the fourth state of matter.
Plasma is a very hot gas in which violent collisions between the atoms in their random thermal motion have fragmented the atoms, ripping electrons off them. An atom that has lost one or more electrons is called an ion. Thus, plasma con- sists of a mixture of electrons and ions engaging in frequent colli- sions. These collisions are accompanied by the emission of light, making the plasma luminous.
SCALE On this scale, the orbits of the inner planets Ha are barely visible. As in our other pictures, the positions of the lley planets are those of January 1, Sat The outer planets move slowly and their orbits are very large; Ju ur n thus they take a long time to go once around their orbit.
The pit er extreme case is that of Pluto, which takes years to complete one orbit. Uranus, Neptune, and Pluto are so far away and so faint that us their discovery became possible only through the use of tele- an scopes.
Uranus was discovered in , Neptune in , and Ur the tiny Pluto in Pluto is now known as one of several ne dwarf planets. Although this space is shown empty in the picture, the Solar System is encircled by a large cloud of millions of comets whose orbits crisscross the sky in all directions. Furthermore, the interstellar space in this pic- ture and in the succeeding pictures contains traces of gas and of dust. The interstellar gas is mainly hydrogen; its density is typi- cally 1 atom per cubic centimeter.
The small circle is the orbit of Pluto. On this scale, the Solar System looks like a minuscule dot, 0. All three are in the constellation Centaurus, in the southern sky.
The star closest to the Sun is Proxima Centauri. This is a Centauri A and B Sun very faint, reddish star a red dwarf , at a distance of 4.
Astronomers like to express stellar distances in light-years: Proxima Centauri is 4. There are many more stars in this Castor Capella Caph Denebola Arcturus box besides those shownthe total number of stars in this box Procyon Pollux Vega is close to Sirius is the brightest of all the stars in the night sky.
If it Altair Fomalhaut Sun were at the same distance from the Earth as the Sun, it would be 28 times brighter than the Sun. Menkent Sirius Cent. Alnair 0 0. The total number of Mirfak stars within this box is about 2 million. In this diagram, Starbursts signify single stars, cir- Sun Antares Spica cles with starbursts indicate star clusters, and a circle with a Acrux single star indicate a star cluster with its brightest star.
Now there are so many stars in our field of view that they appear to form clouds of stars. There are about a million stars in this photograph, and there are many more stars too faint to show up distinctly.
Although this photograph is not centered on the Sun, it is similar to what we would see if we could look toward the Solar System from very far away. Its clouds of stars are arranged in spiral arms wound around a central bulge. The bright central bulge is the nucleus of the galaxy; it has a more or less spherical shape. The surrounding region, with the spiral arms, is the disk of the galaxy. The stars making up the disk circle around the galactic center in a clockwise direction.
Our Sun is in a spiral galaxy of roughly similar shape and size: the Milky Way Galaxy. The total number of stars in this galaxy is about The Sun is in one of the spiral arms, roughly one-third inward from the edge of the disk toward the center. Some of these clusters consist of just a few galaxies, others of hundreds or even thousands. The photograph shows a cluster, or group, of galaxies beyond the constellation Fornax. The group contains an elliptical galaxy like a luminous yellow egg center , three large spiral galaxies left , and a spiral with a bar bottom left.
According to recent investigations, the dark, apparently empty, space near galaxies contains some form of distributed matter, with a total mass 20 or 30 times as large as the mass in the luminous, visible galaxies. But the composition of this invis- ible, extragalactic dark matter is not known.
This is a cluster of clusters of galaxies. At the center of the Local Supercluster is the Virgo Cluster with several thousand galaxies. Seen from a large distance, our super- cluster would present a view comparable with this photograph, which shows a multitude of galaxies beyond the constellation Fornax, all at a very large distance from us. The photograph was taken with the Hubble Space Telescope coupled to two very sensitive cameras using an exposure time of almost hours.
All these distant galaxies are moving away from us and away from each other. The very distant galaxies in the photo are moving away from us at speeds almost equal to the speed of light. This motion of recession of the galaxies is analogous to the outward motion of, say, the fragments of a grenade after its explosion.
The motion of the galaxies suggests that the 0 0. Thus, the galaxies are too small to show up clearly on a photograph. Instead we must rely on a plot of the positions of the galaxies. The plot shows the positions of about galaxies. The dense cluster of galaxies in the lower half of the plot is the Virgo Cluster. Since we are looking into a volume of space, some of the galaxies are in the foreground, some are in the background; but our plot takes no account of perspective.
The luminous stars in the galaxies constitute only a small fraction of the total mass of the Universe. The space around the galaxies and the clusters of galaxies contains dark matter, and the space between the clusters contains dark energy, a strange form of matter that causes an acceleration of the expansion of the Universe.
The false color in this image indicates the distancered for shorter distances, blue for larger distances. This is the last of our pictures in the ascending series.
We have reached the limits of our zoom out. If we wanted to draw another picture, 10 times larger than this, we would need to know the shape and size of the entire Universe. We do not yet know that. The surface of her skin appears smooth and firm. But this is an illusion. Matter appears continuous because the number of atoms in each cubic centimeter is extremely large. In a cubic centimeter of human tissue there are about atoms. This large number creates the illusion that matter is continu- ously distributedwe see only the forest and not the individual trees.
The solidity of matter is also an illusion. The atoms in our bodies are mostly vacuum. As we will discover in the following pictures, within each atom the volume actually occupied by sub- atomic particles is only a very small fraction of the total volume.
The photograph shows the pupil and the iris of Erins eye. Annular muscles in the iris change the size of the pupil and thereby control the amount of light that enters the eye.
In strong light the pupil automatically shrinks to about 2 millimeters; in very weak light it expands to as much as 7 millimeters. The rear surface of the retina is densely packed with two kinds of cells that sense light: cone cells and rod cells. In a human retina there are about 6 million cone cells and million rod cells. The cone cells distinguish colors; the rod cells distinguish only brightness and darkness, but they are more sensitive than the cone cells and therefore give us vision in faint light night vision.
This and the following photographs were made with various kinds of electron microscopes. An ordinary micro- scope uses a beam of light to illuminate the object; an electron microscope uses a beam of electrons. Electron microscopes can achieve much sharper contrast and much higher magnification than ordinary microscopes. For this photograph, the retina was cut apart and the microscope was aimed at the edge of the cut.
In the top half of the picture we see tightly packed rods. Each rod is connected to the main body of a cell containing the nucleus. In the bottom part of the picture we can distinguish tightly packed cell bodies of the cell. The upper portions of the rods contain a special pig- mentvisual purplewhich is very sensitive to light.
The absorption of light by this pigment initiates a chain of chemi- cal reactions that finally trigger nerve pulses from the eye to the brain. DNA is found in the nuclei of cells. It is a long molecule made by stringing together a large number of nitrogenous base molecules on a backbone of sugar and phosphate molecules.
The base mole- cules are of four kinds, the same in all living organisms. But the sequence in which they are strung together varies from one organism to another. This sequence spells out a messagethe base molecules are the letters in this message. The message contains all the genetic instructions governing the metabolism, growth, and reproduction of the cell. The strands of DNA in the photograph are encrusted with a variety of small protein molecules.
At intervals, the strands of DNA are wrapped around larger protein molecules that form lumps looking like the beads of a necklace. This picture was prepared with such a microscope. The picture shows strands of DNA deposited on a substrate of graphite. In contrast to the strands of the preceding picture, these strands are uncoated; that is, they are without protein encrustations.
The strand consists of a pair of helical coils wrapped around each other. This picture was generated by a computer from data obtained by illuminating DNA samples with X rays X-ray scattering. Here we have visual evidence of the atomic structure of matter. The palladium atoms are arranged in a symmetric, repetitive hexagonal pattern. Materials with such regular arrangements of atoms are called crystals. Each of the palladium atoms is approximately a sphere, about 3 meter across.
However, the atom does not have a sharply defined boundary; its surface is somewhat fuzzy. Atoms of other elements are also approximately spheres, with sizes that range from 2 to 4 meter across. At present we know of more than kinds of atoms or chemical elements.
The lightest atom is hydrogen, with a mass of 1. This atom consists of 10 electrons orbiting around a nucleus. In the drawing, the electrons have been indi- cated by small dots, and the nucleus by a slightly larger dot at the center of the picture.
These dots have been drawn as small as possible, but even so the size of these dots does not give a correct impression of the actual sizes of the electrons and of the nucleus.
The electron is smaller than any other particle we know; maybe the electron is truly pointlike and has no size at all. The nucleus has a finite size, but this size is much too small to show up on the drawing.
Note that the electrons tend to cluster near the center of the atom. However, the overall size of the atom depends on the distance to the outermost electron; this electron defines the outer edge of the atom. The electrons move around the nucleus in a very compli- cated motion, and so the resulting electron distribution resem- bles a fuzzy cloud, similar to the STM image of the previous picture. This drawing, however, shows the electrons as they 0 0. The mass of each electron is 9.
We are seeing the central part of the atom. Only two electrons are in our field of view; the others are beyond the margin of the drawing. The size of the nucleus is still much smaller than the size of the dot at the center of the drawing. At this magnification, the nucleus of the neon atom looks like a small dot, 0.
Since the nucleus is extremely small and yet contains most of the mass of the atom, the density of the nuclear mate- rial is enormous. If we could assemble a drop of pure nuclear material of a volume of 1 cubic centimeter, it would have a mass of 2.
Our drawings show clearly that most of the volume within the atom is empty space. The nucleus occupies only a very small fraction of this volume. The nucleus has a nearly spherical shape, but its surface is slightly fuzzy. The nucleus of the neon atom is made up of 10 protons white balls and 10 neutrons red balls. Each proton and each neutron is a sphere with a diameter of about 2 meter, and a mass of 1. In the nucleus, these protons and neutrons are tightly packed together, so tightly that they almost touch.
The protons and neutrons move around the volume of the nucleus at high speed in a complicated motion. These pointlike bodies are quarkseach proton and each neutron is made of three quarks.
Recent experiments have told us that the quarks are much smaller than protons, but we do not yet know their precise size. Hence the dots in the drawing probably do not give a fair description of the size of the quarks. The quarks within protons and neutrons are of two kinds, called up and down.
The proton consists of two up quarks and one down quark joined together; the neutron consists of one up quark and two down quarks joined together. This final picture takes us to the limits of our knowledge of the subatomic world. As a next step we would like to zoom in on the quarks and show what they are made of.
According to a speculative theory, they are made of small snippets or loops of strings, m long. But we do not yet have any evidence for this theory. The thrust of the powerful rocket engines, including the shuttle orbiters main engines shown here, accelerates the entire launch vehicle to the speed of sound in just 45 seconds. It emits a pulse of laser light toward a mirror placed at some unknown Reference Frames distance and measures the time taken by the pulse to travel to the mirror 1.
From this round-trip time and the known speed of light, it then calculates the distance. Since the speed of light is very large, the round-trip 1.
How far does light travel in a small fraction of a second? Example 1, Consistency of Units and page 10 Conversion of Units?
How is the precision of the distance determination limited by the precision of the time measurement? Example 3, page 15 2. Phenomena happen at points in space and at points in time. A complicated phenom- enon, such as a collision between two ships, is spread out over many points of space and time.
But no matter how complicated, any phenomenon can be fully described by stat- ing what happened at diverse points of space at successive instants of time. Measurements of positions and times require the use of coordinate grids and reference frames, which we will discuss in the first section of this chapter.
Ships and other macroscopic bodies are made of atoms. Since the sizes of the atoms are extremely small compared with the sizes of macroscopic bodies, we can regard atoms as almost pointlike masses for most practical purposes. A pointlike mass of no discernible size or internal structure is called an ideal particle.
At any given instant of time, the ideal particle occupies a single point of space. Furthermore, the particle has a mass. And that is all: if we know the position of an ideal particle at each instant of time and we know its mass, then we know everything that can be known about the particle.
Position, time, and mass give a complete description of the behavior and the attrib- utes of an ideal particle. Thus, measurements of position, time, and mass are of fundamental significance in physics.
We will discuss the units for these measurements in later sec- tions of this chapter. In giving these instructions, the attendant is taking the service station as origin, and he is specifying the position of Moose Jaw relative to this origin. To achieve a precise, quantitative description of the position of a particle, physicists use much the same pro- cedure. They first take some convenient point of space as origin and then specify the position of the particle relative to this origin.
For this purpose, they imagine a grid of lines around the origin and give the location of the particle within this grid; that is, they 70 km imagine that the ground is covered with graph paper, and they specify the position of Moose Jaw the particle by means of coordinates read off this graph paper.
The most common coordinates are rectangular coordinates x and y, which rely Destination on a rectangular grid. Figure 1. The mutually per- Position is specified 90 km pendicular lines through the origin O are called the x and y axes.
The coordinates of by distances north- the grid point P, where the particle is located, simply tell us how far we must move ward and westward. For example, the point P shown in Fig. Origin If we move from the origin in a direction opposite to that indicated by the arrow on the axis, then the coordinate is negative; thus, the point P shown in Fig. Electricity is the the automobile has to travel 90 km north and subject of Chapters However if we want to describe the three-dimensional eastwest, northsouth, and updown motion of an aircraft flying through the air or a submarine diving through the ocean, then we need a three-dimensional grid, with x, y, and z axes.
When we determine the position of a particle by means of a coor- A coordinate in the direction and away from of the axis arrow is positive it is negative. For instance, a harbormaster might use a coordinate grid with the origin at the harbor; but a municipal engineer might prefer a displaced coordinate grid with its origin at the center of town see Fig. The nav- igator of a ship might find it convenient to place the origin at the midpoint of her ship and to use a coordinate grid erected around this origin; the grid then moves with the ship see Fig.
If the navigator plots the track of a second ship on this grid, she can tell at a glance what the distance of closest approach will be, and whether the other ship is on a collision course whether it will cross the origin.
For the description of the motion of a particle, we must specify both its position and the time at which it has this position. To determine the time, we use a set of syn- chronized clocks which we imagine arranged at regular intervals along the coordinate grid. When a particle passes through a grid point P, the coordinates give us the position of the particle in space, and the time registered by the nearby clock gives us the time t.
Such a coordinate grid with an array of synchronized clocks is called a reference frame. Like the choice of origin and the choice of coordinate grid, the choice of reference frame is a matter of convenience. For instance, Fig.
Thus, we speak of the reference frame of the harbor, the reference frame of the to x y. This coordinate grid has This coordinate grid is This coordinate grid a origin at center of town. Consider an extended body, such as a bowling ball. What quantities can you measure about the bowling ball, besides position and mass? Do you know the units of any of these quantities?
Mark a point P on this grid. Is the x value for the given O x point P larger or smaller than the x value? What about the y and y values? In this book we will use the metric system of units, which is based on the meter as the unit of length, the second as the unit of time, and the kilogram as the unit of mass.
These units of length, time, This reference frame and mass, in conjunction with the unit of temperature and the unit of electric charge moves with the ship.
Scientists and engineers refer to this set of units as the International of a coordinate grid and a set of synchronized System of Units, or SI units from the French, Systme International. This is a bar made of platinumiridium alloy with a fine scratch mark near each end see Fig. By definition, the distance between these scratch marks was taken to be exactly one meter. The length of the meter was originally chosen so as to make the polar circumference of the Earth exactly 40 million meters see Fig.
The height or thickness of each block serves as a length standard. Copies of the prototype standard meter were manufactured in France and dis- tributed to other countries to serve as secondary standards.
The length standards used in industry and engineering have been derived from these secondary standards. The precision of the standard meter is limited by the coarseness of the scratch marks at the ends. For the sake of higher precision, physicists developed improved definitions of the standard of length.
The most recent improvement emerged from the development of stabilized lasers see Fig. These lasers emit light waves of extreme uniformity which make it possible to determine the speed of light with extreme precision.
Table 1. Many of the quantities listed in the table have already been mentioned in the Prelude. Quantities indicated with an approximately equals sign are not precisely defined; these quan- tities are rough approximations. The prefixes used in Table 1. Prefixes representing powers of ten that differ by factors of are often used with any unit, for convenience or conciseness.
These standard prefixes and their abbreviations are listed in Table 1. In the British system of unitsabandoned by Britain and almost all other coun- tries, but regrettably still in use in the United Statesthe unit of length is the foot ft , which is exactly 0. For quick mental conversion from feet to meters,. Sporadic efforts to adopt metric units in the United States have failed, although most American automobile manufacturers now use metric units, and so does the U.
Army note that in U. Army slang, a kilometer is called a klick, a usage com- mendable for its brevity. Checkup 1. How many mil- limeters in a kilometer? The solar day is the time interval required for the Earth to complete one rotation relative to the Sun. The length of the solar day depends on the rate of rotation of the Earth, which is subject to a host of minor variations, both seasonal and long-term, which make the rotation of the Earth an imperfect timekeeper. To avoid any variation in the unit of time, we now use an atomic standard of time.
This standard is the period of one vibration of microwaves emitted by an atom of cesium. The second 1 s is defined as the time needed for 9 vibrations of a cesium atom. In this clock, the feeble vibrations of the cesium atoms are amplified to a level that permits them to control the dial of the clock. Good cesium clocks are very, very goodthey lose or gain no more than 1 second in 20 million years.
These time signals can be picked up worldwide on shortwave receivers tuned to 2. Precise time signals are also announced continuously by telephone [in the United States, the telephone number is ; the precise time is also available online at www. National Institute of Standards and Table 1. How far does light travel in one billionth of a second a nanosecond? The ruler drawn diagonally across this page shows the distance light travels in 1 nanosecond. How many picoseconds in a microsecond? The standard of mass is a cylinder of platinum iridium alloy kept at the International Bureau of Weights and Measures see Fig.
The kilogram 1 kg is defined as exactly equal to the mass of this cylinder. Mass is the only fundamental unit for which we do not, as yet, have an atomic standard. Mass is measured with a balance, an instrument that compares the weight of an unknown mass with a known force, such as the weight of the standard mass or the pull of a calibrated spring. Weight is directly proportional to mass, and hence equal weights imply equal masses the precise distinction between mass and weight will be spelled out in Chapter 5.
The watt balance is effectively a spring balance, but instead of a mechanical spring suspension it uses a magnetic suspension with calibrated magnetic forces. To relate the mass of an atom to the kilogram mass we need to know Avogadros number NA, or the number of atoms per mole.
One mole of any chemical element or any chemical compound is that amount of matter containing as many atoms or molecules as FIGURE 1. The atomic mass of a chemical element kilogram. Thus, according to the table of atomic masses in Appendix 8, one mole of carbon atoms C has a mass of The available experimental data yield the following value for NA: NA 6.
Thus, the carbon atom has a mass of 12 u, the hydrogen atom has a mass of 1. Atomic masses appear in Appendix 8. According to the periodic table of chemical elements in Appendix 8, the atomic mass of nickel is Thus, the mass of one nickel atom is The number of atoms in our 5. In the British system of units, the unit of mass is the pound, which is exactly 0.
The mass of a person is typically 60 to 75 kg. The distance from the centerline of the body to the end of the outstretched arm is about 1 m. How many metric tons in one milligram? Any other physical quantity can be measured by introducing a derived unit constructed by some combination of the base units.
For example, area can be measured with a derived unit that is the square of the unit of length; thus, in the metric system, the unit of area is the square meter 1 m 1 m 1 m2 , which is the area of a square, one meter on a side Fig. And volume can be measured with a derived unit that is the cube of the unit of length; in the metric system, the unit of volume is the cubic meter 1 m 1 m 1 m 1 m3 , which is the volume of a cube, one meter on FIGURE 1.
Tables 1. Similarly, density, or mass per unit volume, can be measured with a derived a b unit that is the ratio of the unit of mass and the unit of volume. We will see in later chapters that other physical quantities, such as speed, acceleration, force, etc.
It is based on the meter, the Sailboat Vector Simple Experiment second, and the kilogram, plus a special unit for temperature and a special unit for electric current. How many cubic centimeters are there in a cubic meter?
A B C D Online 1. This not only has the advantage that very large or very small numbers can be written compactly, but it also serves to indicate the precision of the numbers.
For instance, a scientist observing the Berlin marathon at which Ronaldo da Costa set the world record of 2 h 6 min 5. The first of these watches permits measurements to within s, the second to within about 1 s, the third to within 10 or 20 s, and the fourth to within 1 or 2 minutes if the scientist is good at guessing the position of the hand on the blank face.
We will adopt the rule that only as many digits, or significant figures, are to be written down as are known to be fairly reliable. In accordance with this rule, the number 7. Thus, the scientific notation gives us an immediate indication of the pre- cision to within which the number has been measured.
When numbers in scientific notation are multiplied or divided in calculations, the final result should always be rounded off so that it has no more significant figures than the original numbers, because the final result can be no more accurate than the original numbers on which it is based.
Thus, the result of multiplying 7. When numbers are added or subtracted, the result should be rounded to the largest decimal place among the last digits of the original numbers. Thus, What is the round-trip distance, expressed with the correct number of significant figures? This fraction-of-a-meter accuracy agrees with the distance calculated for a nanosecond time interval in Example 1. Sometimes even a number known to many significant figures is rounded off to fewer significant figures for the sake of convenience, when high accuracy is not required.
For instance, the exact value of the speed of light is 2. This consistency is illustrated by the calculations in Example 3, where we see that on the right side of Eq. It is a general rule that in any calculation with the equations of physics, the units can be multiplied and divided as though they were algebraic quantities, and this automatically yields the cor- rect units for the final result.
This requirement of consistency of units in the equa- tions of physics can be reformulated in a more general way as a requirement of consistency of dimensions. In this context, the dimensions of a physical quantity are said to be length, time, mass, or some product or ratio of these if the units of this physical quantity are those of length, time, mass, or some product or ratio of these. In any equation of physics, the dimensions of the two sides of the equation must be the same.
For instance, we can test the consistency of Eq. A test of the consistency of dimensions tells us no more than a test of the consistency of units, but has the advan- tage that we need not commit ourselves to a particular choice of units, and we need not worry about conversions among multiples and submultiples of the units.
Bear in mind that if an equation fails this consistency test, it is proved wrong; but if it passes, it is not proved right. Dimensions are sometimes used to find relationships between physical quantities.
Such a determination of the appropriate proportionality between powers of relevant quantities is called dimensional analysis. Such analysis is performed by requiring the consistency of dimensions of units on each side of an equation. Dimensional analysis will prove useful when we have become familiar with more physical quantities and their dimensions.
Engineers at Lockheed Martin provided spacecraft operating data needed for navigation in British units rather than metric units. Flight controllers assumed the data were in metric units, and thus the probe did not behave as intended when the relevant thrusters were fired near Mars.
Such conversions involve no more Orbiter spacecraft. For example, the density of water is 1. Thus, starting with 1. To change the units of a quantity, simply multiply the quantity by one or several conversion factors that will bring about the desired cancel- lation of the old units.
The ratio of two quantities with identical dimensions or units will have no dimen- sions at all. For example, the slope of a path relative to the horizontal direction is defined as the ratio of the increment of height to the increment of horizontal distance. Since this is the ratio of two lengths, it is a dimensionless quantity.
Likewise, the sine of an angle is defined as the ratio of two lengths; in a right triangle, the sine of one of the acute angles is equal to the length of the opposite side divided by the length of the hypotenuse see Math Help: Trigonometry of the Right Triangle for a review.
Thus the slope and the sine, cosine, and tangent of an angle are examples of dimensionless quantities. What is the slope of the path of ascent of the airliner? What altitude does it reach at a horizontal distance of m, or 2. Therefore [slope] tan With our calculator, we find that the tangent of 12 is 0. Hence, [slope] 0. Slopes are often quoted as ratios; thus a slope of 0.
By proportions, the height reached for m of horizontal advance must be times as large as the height for 1 m of horizontal advance; that is, [height] m 0. For instance, in though they were algebraic quantities. This will automatically Example 3, we rounded off the final result to four significant yield the correct units for the final result. If it does not, you figures, since four significant figures were specified in the have made some mistake in the calculation.
Thus, it is always time interval measured by the device. Any additional signif- worthwhile to keep track of the units in calculations, because icant figures in the final result would be unreliable and mis- this provides some extra protection against costly mistakes.
A leading. In fact, even the fourth significant figure in the answer failure of the expected cancellations is a sure sign of trouble! It is always wise to else multiply the old units by whatever conversion factors will doubt the accuracy of the last significant figure, in the final bring about the cancellation of the old units.
Take a droplet of oil and let it spread out on a smooth surface of water. When the oil slick attains its maximum area, it consists of a monomolecular layer; that is, it consists of a single layer of oil molecules which stand on the water surface side by side.
Given that an oil droplet of mass 8. This latter volume can be expressed in terms of the thickness and the area of the oil slick: [volume] [thickness] [area] Consequently, [volume] [thickness] [area] 9. This theorem implies that 1 sin2 cos2.
In principle, the numerical value of the sine, cosine, or The figure shows a right triangle with an angle , its oppo- tangent of any angle can be found by laying out a right triangle site and adjacent sides, and the hypotenuse. The sine, cosine, with this angle, measuring its sides, and evaluating the ratios and tangent of are defined as follows: given in the definitions. In practice, numerical values of tan- gents, cosines, and sines are obtained from handheld elec- [opposite side] tronic calculators.
Accordingly, to how many significant figures can you calculate the volume? How many significant figures are there in the result? What is the result? When multiplying or dividing two or more quantities, the result has the same number of significant figures as the least number in the original quantities.
When adding or subtracting two or more quantities, the number of significant figures in the result is determined by the largest decimal place among the last digits in the original quantities. Try to estimate by eye the lengths, in centimeters or meters, of and observation of the position of the Sun in the sky to find a few objects in your immediate environment. Then measure longitude.
How good were your estimates? How close is your watch to standard time right now? Roughly how many minutes does your watch gain or lose per month? What is meant by the phrase a point in time? Mechanical clocks with pendulums were not invented until the tenth century A.
What clocks were used by the ancient Greeks and Romans? By counting aloud One Mississippi, two Mississippi, three Mississippi, etc. Try to measure 30 seconds in this way. How good a timekeeper are you? In an accurate chronometer built by John Harrison see 8. During this voyage, the chronometer accu- mends that each ship carry three chronometers for accurate mulated an error of less than 2 minutes.
For this achievement, timekeeping. What can the navigator do with three Harrison was ultimately awarded a prize of that the chronometers that cannot be done with two? British government had offered for the discovery of an accu- 9. Suppose that by an act of God or by the act of a thief the rate method for the determination of geographical longitude standard kilogram at Svres were destroyed.
Would this at sea. Explain how the navigator of a ship uses a chronometer destroy the metric system? Estimate the masses, in grams or kilograms, of a few bodies in longest list you can and give the units.
Are all these units your environment. Check the masses with a balance if you derived from the meter, second, and kilogram? Could we take length, time, and density as the three funda- Consider the piece of paper on which this sentence is printed. What could we use as a standard of density?
If you had available suitable instruments, what physical quan- Could we take length, mass, and density as the three funda- tities could you measure about this piece of paper?
Make the mental units? Length, mass, and speed? What is your height in feet? In meters? With a ruler, measure the thickness of this book, excluding the fiber-optic interferometer can measure a distance times cover. Deduce the thickness of each of the sheets of paper the size of a wavelength. How does such precision compare making up the book. A football field measures yd yd.
The thread of a screw is often described either in terms of the these lengths in meters. If each step you take is 0. For to cover 1. The pica is a unit of length used by printers and book thread of 80 turns per inch or a thread of 0. A nautical mile nmi equals 1. Show that dard sheet of paper, 11 in.
Express the last four entries in Table 1. A physicist plants a vertical pole at the waterline on the shore 1 1 1 1 of a calm lake. When she stands next to the pole, its top is at 8 , 16 , 32 , and 64 in. Express one mil one thousandth of an inch in micrometers eye level, cm above the waterline.
She then rows across microns. Express one millimeter in mils. Analogies can often help us to imagine the very large or is blocked by the curvature of the surface of the lake; that is, very small distances that occur in astronomy or in atomic the entire pole is below the horizon Fig. She finds that physics. From a If the Sun were the size of a grapefruit, how large would this information, deduce the radius of the Earth. How far away would the nearest star be?
How large would a red blood cell be? One of the most distant objects observed by astronomers is 5m the quasar Q, at a distance of If you wanted to plot the position of this R quasar on the same scale as the diagram at the top of page xliii of the Prelude, how far from the center of the diagram would you have to place this quasar?
An interferometer uses the pattern created by mixing laser light waves in order to measure distances extremely accurately. What is your age in days? In seconds? What is your mass in pounds?
In kilograms? In atomic mass The age of the Earth is 4. Express this in units? What percentage of the mass of the Solar System is in the A computer can perform a single calculational step each planets?
What percentage is in the Sun? Use the data given in nanosecond s. How many steps can be performed in the table printed inside the cover of this book. What is the ratio of the largest to the smallest length listed An Olympic marathon record of 2 h 9 min 21 s was set by in Table 1. The ratio of the longest to the shortest time in Carlos Lopes of Portugal in Express this time in sec- Table 1. The ratio of the largest to the smallest mass in onds. Do you see any coincidences or near-coincidences between these numbers?
Some physicists have proposed that Joan Benoit of the United States set the womens Olympic coincidences between these large numbers must be explained marathon record in with a time of 2 h 24 min 52 s.
Express this time in seconds. The atom of uranium consists of 92 electrons, each of mass The solar day is the interval for the Earth to complete one 9.
What percentage of the total rotation in relation to the Sun, and the sidereal day is the mass is in the electrons and what percentage is in the nucleus interval for the Earth to complete one rotation in relation to of the atom? The solar day has exactly 24 hours. How many hours and minutes are there in one sidereal day? Hint: 1 year A laboratory microbalance can measure a mass of one-tenth of is How many atoms are there in such a speck of gold, which has grams in one A mechanical wristwatch ticks 4 times per second.
Suppose mole? How often does it tick in this time interval? English units use the ordinary pound, also called the avoirdu- pois pound, to specify the mass of most types of things. How many days is a million seconds? However, the troy pound is often used to measure precious How many hours are there in a week? How many seconds? Your heart beats 71 times per minute.
How often does it beat 0. If we adopt these different in a year? Each day at noon a mechanical wristwatch was compared with an avoirdupois pound of feathers? WWV time signals. The watch was not reset.
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