# Units of other physical quantities

Having defined the four fundamental dimensions and their units, it is possible to establish units of all other physical quantities (see Table 1.1). Speed, for example, is defined as the distance travelled in unit time. It therefore has the dimension LT-1 and is measured in metres per second (ms-1). It is sometimes desirable and permissible to use kilometres per hour or knots (nautical miles per hour, see Appendix 4) as units of speed, and care must then be exercised to avoid errors of inconsistency.

To find the dimensions and units of more complex quantities, appeal is made to the principle of dimensional homogeneity. This means simply that, in any valid physical equation, the dimensions of both sides must be the same. Thus if, for example, (mass)" appears on the left-hand side of the equation, (mass)" must also appear on the right-hand side, and similarly this applies to length, time and temperature.

Thus, to find the dimensions of force, use is made of Newton’s second law of motion

Force = mass x acceleration

while acceleration is speed -=- time. Expressed dimensionally, this is

Writing in the appropriate units, it is seen that a force is measured in units of kgms-2. Since, however, the unit of force is given the name Newton (abbreviated usually to N), it follows that

1N = 1 kgms 2

It should be noted that there could be confusion between the use of m for milli and its use for metre. This is avoided by use of spacing. Thus ms denotes millisecond while m s denotes the product of metre and second.

The concept of the dimension forms the basis of dimensional analysis. This is used to develop important and fundamental physical laws. Its treatment is postponed to Section 1.4 later in the current chapter.

Table 1.1 Units and dimensions

 Quantity Dimension Unit (name and abbreviation) Length L Metre (m) Mass M Kilogram (kg) Time T Second (s) Temperature в Degree Celsius (°С), Kelvin (K) Area L2 Square metre (m2) Volume L3 Cubic metre (m3) Speed LT-1 Metres per second (ms-1) Acceleration LT-2 Metres per second per second (ms-2) Angle 1 Radian or degree (°) (The radian is expressed as a ratio and is therefore dimensionless) Angular velocity T-1 Radians per second (s-1) Angular acceleration T-2 Radians per second per second (s-2) Frequency j-l Cycles per second, Hertz (s-1 Hz) Density ML -[2] Kilograms per cubic metre (kgm-3) Force MLT-2 Newton (N) Stress ML-‘T2 Newtons per square metre or Pascal (N m-2 or Pa) Strain 1 None (expressed as %) Pressure ML-IT-2 Newtons per square metre or Pascal (N m-2 or Pa) Energy work ml2t-2 Joule (J) Power ml2t-3 Watt (W) Moment ML2T-2 Newton metre (Nm) Absolute viscosity ML’r1 Kilogram per metre second or Poiseuille (kgm-1 s-1 or PI) Kinematic viscosity L2T-1 Metre squared per second (m2 s-1) Bulk elasticity ML"‘T^2 Newtons per square metre or Pascal (N m-2 or Pa)

1.1.1 Imperial units*

Until about 1968, aeronautical engineers in some parts of the world, the United Kingdom in particular, used a set of units based on the Imperial set of units. In this system, the fundamental units were:

mass – the slug length – the foot time – the second

temperature – the degree Centigrade or Kelvin.