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Physics - Dimension Analysis

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Published in: Physics
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'Dimension Analysis' is the topic of this sample note.

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  1. Dimension Analysis The fundamental quantities, irrespective of the system of units used for their measurement and their corresponding quantitative values, characterize different independent groups or classes of basic physical quantities. In other words, a fundamental quantity belongs to a class of physical quantities of its own kind only, each of them having no dependence with other fundamental quantities. The attribute that is common to a class of physical quantities is what is called their dimensionality. 'OOpx 200 soo 4 3 10 210 80 oo Most physical quantities can be expressed in terms of combinations of five basic dimensions. 72pt 2 f 4-4 20 30 40 50 oo 70 These are mass (M), length (L), time (T), electrical current (l), and temperature, represented by the Greek letter theta (9). These five dimensions have been chosen as being basic because they are easy to measure in experiments. Dimensions aren't the same as units. For example, the physical quantity, speed, may be measured in units of meters per second, miles per hour etc.; but regardless of the units used, speed is always a length divided a time, so we say that the dimensions of speed are length divided by time, or simply LIT. Similarly, the dimensions of area are L2 since area can always be calculated as a length times a length. For example, although the area of a circle is conventionally written as -rrr, we could write it as TTr (which is a length) x r (another length). Dimensions of a physical quantity are the powers to which the fundamental units be raised in order to represent that quantity. A dimension deal with qualitative part of measurement. By international agreement a small number of physical quantities such as length, time etc. are chosen and assigned standards. These quantities are called 'base quantities' and their units as 'base units'. All other physical quantities are expressed in terms of these 'base quantities'. The units of these dependent quantities are called 'derived units'.
  2. The standard for a unit should have the following characteristics. (a) It should be well defined. (b) It should be invariable (should not change with time) (c) It should be convenient to use (d) It should be easily accessible The 14th general conference on weights and measures (in France) picked seven quantities as base quantities, thereby forming the International System of Units abbreviated as Sl (System de International) system. Dimensional Formula and Dimensional Equation: — Dimensional formula of a physical quantity is the formula which tells us how and which of the fundamental units have been used for the measurement of that quantity. An equation written in the following manner is called dimensional equation. How to Write Dimensions of Physical Quantities : — Dimensions of a physical quantity can be determined as follows: [MOL2T0] (a) Write the formula for that quantity, with the quantity on L.H.S. of the equation. (b) Convert all the quantities on R.H.S. into the fundamental quantities mass, length and time. (c) Substitute M,L, and T for mass, length and time respectively. (d) Collect terms of M,L and T and find their resultant powers (a,b,c) which give the dimensions of the quantity in mass, length and time respectively. Base quanti ties and their units : — The seven base quantities and their units are, Base quantity Length Mass Time Electric current Temperature Luminous intensity Amount of substance Derived uni ts : Unit Meter Kilogram Second Ampere Kelvin Candela Mole Symbol Kg Sec Cd Mole We can define all the derived units in terms of base units. For example, speed is defined to be the ratio of distance to time. Unit of Speed = (unit of distance (length))/(unit of time)
  3. = m/s = ms-I (Read as meter per sec.)

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