Standards and Units

As we learned in Section 1.1, physics is an experimental science. Experiments require measurements, and we generally use numbers to describe the results of measurements. Any number that is used to describe a physical phenomenon quantitatively is called a physical quantity. For example, two physical quantities that describe you are your weight and your height. Some physical quantities are so fundamental that we can define them only by describing how to measure them. Such a definition is called an operational definition. Two examples are measuring a distance by using a ruler and measuring a time interval by using a stopwatch. In other cases we define a physical quantity by describing how to calculate it from other quantities that we can measure. Thus we might define the average speed of a moving object as the distance traveled (measured with a ruler) divided by the time of travel (measured with a stopwatch).

   When we measure a quantity, we always compare it with some reference standard. When we say that a Ferrari 458 Italia is 4.53 meters long, we mean that it is 4.53 times as long as a meter stick, which we define to be 1 meter long. Sucha standard defines a unit of the quantity. The meter is a unit of distance, and the second is a unit of time. When we use a number to describe a physical quantity, we must always specify the unit that we are using; to describe a distance as simply “4.53” wouldn’t mean anything.

   To make accurate, reliable measurements, we need units of measurement that do not change and that can be duplicated by observers in various locations. The system of units used by scientists and engineers around the world is commonly called “the metric system,” but since 1960 it has been known officially as the International System, or SI (the abbreviation for its French name, Système International). Appendix A gives a list of all SI units as well as definitions of the most fundamental units.


From 1889 until 1967, the unit of time was defined as a certain fraction of the mean solar day, the average time between successive arrivals of the sun at its highest point in the sky. The present standard, adopted in 1967, is much more precise. It is based on an atomic clock, which uses the energy difference between the two lowest energy states of the cesium atom (133Cs). When bombarded by microwaves of precisely the proper frequency, cesium atoms undergo a transition from one of these states to the other. One second (abbreviated s) is defined as the time required for 9,192,631,770 cycles of this microwave radiation (Fig. 1.3a).


In 1960 an atomic standard for the meter was also established, using the wavelength of the orange-red light emitted by excited atoms of krypton 186Kr2. From this length standard, the speed of light in vacuum was measured to be 299,792,458 m>s. In November 1983, the length standard was changed again so that the speed of light in vacuum was defined to be precisely 299,792,458 m>s.Hence the new definition of the meter (abbreviated m) is the distance that light travels in vacuum in 1>299,792,458 second (Fig. 1.3b). This modern definition provides a much more precise standard of length than the one based on a wavelength of light.

The standard of mass, the kilogram (abbreviated kg), is defined to be the mass of a particular cylinder of platinum–iridium alloy kept at the International Bureau of Weights and Measures at Sèvres, near Paris (Fig. 1.4). An atomic standard of mass would be more fundamental, but at present we cannot measure masses on an atomic scale with as much accuracy as on a macroscopic scale. The gram (which is not a fundamental unit) is 0.001 kilogram.

   Other derived units can be formed from the fundamental units. For example, the units of speed are meters per second, or m>s; these are the units of length (m) divided by the units of time (s).

Unit Prefixes
Once we have defined the fundamental units, it is easy to introduce larger and smaller units for the same physical quantities. In the metric system these other units are related to the fundamental units (or, in the case of mass, to the gram) by
multiples of 10 or 1 10 Thus one kilometer 11 km2 is 1000 meters, and one centimeter 11 cm2 is 1
100 meter. We usually express multiples of 10 or 1 10 in exponential notation: 1000 = 103, 1
1000 = 103, and so on. With this notation, 1 km = 10m3 and 1 cm = 10m2.
   The names of the additional units are derived by adding a prefix to the name of the fundamental unit. For example, the prefix “kilo-,” abbreviated k, always means a unit larger by a factor of 1000; thus

   1 kilometer = 1 km = 103meters = 103 m
1 kilogram = 1 kg = 103 grams = 103 g
1 kilowatt = 1 kW = 103
 watts = 10W

A table in Appendix A lists the standard SI units, with their meanings and abbreviations.

   Table 1.1 gives some examples of the use of multiples of 10 and their prefixes with the units of length, mass, and time. Figure 1.5 shows how these prefixes are used to describe both large and small distances.

Standards and Units

The British System

Finally, we mention the British system of units. These units are used in only the United States and a few other countries, and in most of these they are being replaced by SI units. British units are now officially defined in terms of SI units, as follows:

Length: 1 inch = 2.54 cm (exactly)
                              Force: 1 pound = 4.448221615260 newtons (exactly)

The newton, abbreviated N, is the SI unit of force. The British unit of time is the second, defined the same way as in SI. In physics, British units are used in mechanics and thermodynamics only; there is no British system of electrical units. In this book we use SI units for all examples and problems, but we occasionally give approximate equivalents in British units. As you do problems using SI units, you may also wish to convert to the approximate British equivalents if they are more familiar to you (Fig. 1.6). But you should try to think in SI units as much as you can.

Standards and Units


Frequently Asked Questions

Ans: Different techniques are useful for solving different kinds of physics problems, which is why this book offers dozens of Problem-Solving Strategies view more..
Ans: Introduce the systems of units used to describe physical quantities and discuss ways to describe the accuracy of a number. view more..
Ans: Experiments require measurements, and we generally use numbers to describe the results of measurements. Any number that is used to describe a physical phenomenon quantitatively is called a physical quantity. view more..
Ans: An equation must always be dimensional consistent. You can’t add apples and automobiles; two terms may be added or equated only if they have the same units. view more..
Ans: Measurements always have uncertainties. If you measure the thickness of the cover of a hardbound version of this book using an ordinary ruler, your measurement is reliable to only the nearest millimeter, and your result will be 3 mm. It would be wrong to state this result as 3.00 mm; given the limitations of the measuring device, you can’t tell whether the actual thickness is 3.00 mm, 2.85 mm, or 3.11 mm. view more..
Ans: We have stressed the importance of knowing the accuracy of numbers that represent physical quantities. But even a very crude estimate of a quantity often gives us useful information. Sometimes we know how to calculate a certain quantity, but we have to guess at the data we need for the calculation. Or the calculation might be too complicated to carry out exactly, so we make rough approximations. view more..
Ans: Some physical quantities, such as time, temperature, mass, and density, can be described completely by a single number with a unit. But many other important quantities in physics have a direction associated with them and cannot be described by a single number. view more..
Ans: A body that can be modeled as a particle is in equilibrium whenever the vector sum of the forces acting on it is zero. But for the situations we’ve just described, that condition isn’t enough. If forces act at different points on an extended body, an additional requirement must be satisfied to ensure that the body has no tendency to rotate: The sum of the torques about any point must be zero. This requirement is based on the principles of rotational dynamics view more..
Ans: In this chapter we’ll apply the first and second conditions for equilibrium to situations in which a rigid body is at rest (no translation or rotation). Such a body is said to be in static equilibrium view more..
Ans: In most equilibrium problems, one of the forces acting on the body is its weight. We need to be able to calculate the torque of this force. The weight doesn’t act at a single point; it is distributed over the entire body. But we can always calculate the torque due to the body’s weight by assuming that the entire force of gravity (weight) is concentrated at a point called the center of gravity view more..
Ans: We can often use symmetry considerations to locate the center of gravity of a body, just as we did for the center of mass. The center of gravity of a homoge-neous sphere, cube, or rectangular plate is at its geometric center. The center of gravity of a right circular cylinder or cone is on its axis of symmetry. view more..
Ans: There are just two key conditions for rigid-body equilibrium: The vector sum of the forces on the body must be zero, and the sum of the torques about any point must be zero. To keep things simple, we’ll restrict our attention to situations in which we can treat all forces as acting in a single plane, which we’ll call the xy-plane view more..
Ans: Here are some solved examples to help your concepts to be more clear. view more..
Ans: The rigid body is a useful idealized model, but the stretching, squeezing, and twisting of real bodies when forces are applied are often too important to ignore. view more..
Ans: The simplest elastic behavior to understand is the stretching of a bar, rod, or wire when its ends are pulled (Fig. 11.12a). Figure 11.14 shows an object that initially has uniform cross-sectional area A and length l0. We then apply forces of equal magnitude F# but opposite directions at the ends (this ensures that the object has no tendency to move left or right). We say that the object is in tension. view more..
Ans: When a scuba diver plunges deep into the ocean, the water exerts nearly uniform pressure everywhere on his surface and squeezes him to a slightly smaller volume. This is a different situation from the tensile and compressive stresses and strains we have discussed. view more..
Ans: The third kind of stress-strain situation is called shear. The ribbon in Fig. 11.12c is under shear stress: One part of the ribbon is being pushed up while an adjacent part is being pushed down, producing a deformation of the ribbon. view more..

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