Dimensions and units
Any physical quantity can be characterized by dimensions. The arbitrary magnitudes assigned to the dimensions are called units. Some basic dimensions such as mass m, length L, time t, and temperature T are selected as primary or fundamental dimensions, while others such as velocity Ѵ, energy E, and volume V are expressed in terms of the primary dimensions and are called secondary dimensions or derived dimensions.
Two sets of units are still in common use today: the English system, which is also known as the United States Customary System (USCS), and the metric SI (from Le System International d’ Unites), which is also known as the International System. The SI is a simple and logical system based on a decimal relationship between the various units, and it is being used for scientific and engineering work in most of the industrialized nations, including England. The English system, however, has no numerical base, and various units in this system are related to each other rather arbitrarily which makes it confusing and difficult to learn. The United States is the only industrialized country that has not yet fully converted to the metric system.
Some SI and English Unis
In SI, the units of mass, length and time are the kilogram (kg), meter (m), and second (s), respectively. The respective units in the English system are the pound-mass (lbm), foot (ft), and the second (s). The pound symbol lb is actually the abbreviation of libra, which was the ancient Roman unit of weight. The English retained this symbol even after the end of the Roman occupation of Britain in 410. The mass and length units in the two systems are related to each other by
1 lbm = 0.45359 kg
1ft = 0.3048m
In the English system, force is usually considered to be one of the primary dimensions and is assigned a nonderived unit. This is a source of confusion and error that necessitates the use of a conversion factor in many formulas. To avoid this nuisance, we consider force to be a secondary dimension whose unit is derived from Newton’s second law, i.e.,
Force = (Mass) (Acceleration)
F = ma
In SI, the force unit is the Newton (N), and it is defined as the force required to accelerate a mass of 1kg at a rate 1m/s². In the English system, the force unit is the pound-force (lbf) and is defined as the force required to accelerate a mass of 32.174 lbm (1 slug) at a rate of 1ft/s². That is,
1 N = 1kg.m/s²
1 lbf = 32.174 lbm.ft/s²
A force of 1 newton is roughly equivalent to the weight of a small apple (m = 102g), whereas a force of 1 pound-force is roughly equivalent to the weight of 4 medium apples (m total = 454g), Another force unit in common use in many European countries is the kilogram-force (kgf), which is the weight of 1kg mass at sea level (1kgf = 9.807N)
The term weight is often incorrectly used to express mass, particularly by the “weight watchers”. Unlike mass, weight W is a force, It is the gravitational force applied to a body, and its magnitude is determined from Newton’s second law,
W = mg (N)
where m is the mass of the body, and g is the local gravitational acceleration (g is 9.807 m/s² or 32.174 ft/s² at sea level and 45° latitude). The ordinary bathroom scale measures the gravitational force acting on a body. The weight of a unit volume of a substance is called the specific weight w and is determined from ѡ=𝜌g, where 𝜌 is density.