dimensions and units

Introduction to dimensions and units

All the physical quantities are given by a few fundamental quantities or their combinations. The units of such fundamental quantities are called base units, combinations of them being called derived units. The system in which length, mass and time is adopted as the basic quantities, and from which the units of other quantities are derived, is called the absolute system of units.

Dimension:

A dimension is the measure by which a physical variable is expressed quantitatively.

Unit:

A unit is a particular way of attaching a number to the quantitative dimension.

Measured physical properties have a basic dimension in which they are measured. There may be many units that are used to measure this dimension. This is best shown by example. The thickness of an object has the dimension of length. Length can be measured in a wide range of units including inches, feet, yards, meters, kilometers, micrometers, Angstrom units, furlongs, fathoms, light-years and many more. The thickness of an object cannot be measured in kilograms, however. That is because the kilogram is a unit used to measure quantities that have the fundamental dimension of mass.

The choice of the units to use in a particular measurement is a matter of convention or convenience. Whatever units you use, however, must correspond to the correct dimension for the physical quantity.

Systems of units usually start by making arbitrary definitions of a unit for fundamental dimensions. Typically these fundamental dimensions are mass, length, time, electric charge and temperature. Once these units are selected for the fundamental dimensions the units for other physical quantities can be determined from the physical relations among quantities having the fundamental units. For example velocity is found as distance divided by time. Thus the dimensions of velocity must be length/time. Similarly the dimensions of acceleration, found as velocity divided by time, must be length/time2, and the dimensions of force can be found from Newton’s second law: force equals mass times acceleration. This gives the dimensions of force as the dimensions of mass times the dimensions of acceleration or (mass) times (length) divided by (time). The symbols M, L, and T are usually used to represent dimensions of mass, length, and time, respectively.

We can continue in this fashion. Pressure is force per unit area; this must have dimensions of (force) / (length)2 = (mass) (length) / [(time)2(length)2] = (mass) / [(time)2(length)]. Work is the product of force time’s distance. Thus the dimensions of work must be the dimensions of force times the dimensions of distance. This means that the dimensions of work are (mass)(length)2 /(time)2. Because work is a form of energy we should expect that any kind of energy should have the same dimensions. We can check this by recalling the formula for kinetic energy: mV2/2. This has the dimensions of mass times the dimensions of velocity squared. But, this is just (mass) times (length/time)2, which is the result that we just found for work. Similarly, potential energy, mgz, has dimensions of mass time’s dimensions of acceleration time’s dimensions of length. This gives (mass) times [(length)/(time)2] times (length). Again the same result: the dimensions of energy are (mass) (length)2 /(time)2. Finally, power which is energy divided by time must have dimensions of (mass) (length)2 /(time)3.

SI units

In the SI system of units the first three fundamental dimensions are mass, length and time. The units used to measure these dimensions (and their abbreviations) are defined to be the kilogram (kg), the meter (m) and the second (s). With these definitions the units for velocity and acceleration become meters per second (m/s) and meters per second per second (or meters per second2, m/s2), respectively.

The kilogram is defined as the mass of a reference material located at the International Bureau of Weights and Measures. The meter is defined as the length of a path traveled by light in a vacuum during a time interval of 1/299,792,458 of a second. (This is equivalent to defining the speed of light as 299,792,458 m/s.) The second is defined in terms of the wavelength of a certain emission from a cesium atom and the kelvin (temperature) is defined as 1/273.16 of the triple point temperature of water. The ampere is defined as the basic unit of electrical current.

As mentioned above the force units must be the same as the mass unit’s times the acceleration units. Thus the SI units of force are kg•m/s2. These units are given a special name, the newton (N). Note that we use a capital N when we spell the last name of Sir Isaac Newton, but a lower case n when we spell the name of the SI force unit, the newton. However, we use an upper case N when we abbreviate one newton as 1 N. This is the common practice in SI unit naming conventions.

This definition of the newton can be written as a unit-conversion factor equation:

1 N = 1 kg•m•s-2

Similarly the units for pressure, energy and power, which can be written in terms of fundamental units, are also given separate names for convenience. Table 1 gives a summary of the various units in the SI system of units.

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Prefixes

The wide scale of units in engineering applications is handled by using prefixes that account for multiplicative factors on the SI units. Thus a kilopascal (abbreviation kPa) means 1000 Pascals. The following prefixes, and their associated factors and abbreviations are commonly used with SI units.

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