S.I. Units, Dimensional Analysis

It is not necessary to use a capital letter when writing the full names of the units, even if they are named after famous scientists. For example, newton, watt, ampere, and metre are all units of measurement. The unit should be written either in full or only using the symbols that have been agreed upon. Units are not expressed in the plural form. For example, 10 kg but not 10 kgs and so on. No full stop or punctuation mark should be used within or at the end of symbols for units unless absolutely necessary.

Dimensions

To obtain one unit of a physical quantity, the dimensions of a physical quantity are the powers to which the fundamental units are raised in order to obtain that quantity.

Dimensional Analysis is a type of statistical analysis that examines the relationship between two or more dimensions. It is the practice of examining relationships between physical quantities by determining the dimensions of the physical quantities in question that is known as dimensional analysis. These dimensions are independent of the numerical multiples and constants, and any quantity in the world can be expressed as a function of the fundamental dimensions, regardless of the numerical multiples or constants used.

Formula for Dimensions

In mathematics, the dimensional formula of a derived quantity is the expression that shows the powers to which the fundamental units must be raised in order to obtain one unit of the derived quantity in question. For example, if Q is the unit of a derived quantity represented by the equation Q = MaLbTc, it is referred to as the dimensional formula and the exponents a,b and c as the dimensions.

Dimensional Constants

Dimensional constants are physical quantities that have dimensions and have a fixed value. Dimensional constants are physical quantities that have dimensions and have a fixed value. There are a number of examples, such as the gravitational constant(G=6.67408 × 10-11 m3 kg-1s-2), Planck’s constant ( h=6.62607004 × 10-34 m2 kg / s   ), the universal gas constant ( R=8.31446261815324 J⋅K-1⋅mol-1 ), and the velocity of light in a vacuum ( c=3108m/s ).

Dimensionless Quantities

Dimensionless quantities are those that have no dimensions but have a fixed value despite the fact that they do not have any dimensions.

Quantities that have no units but are dimensionless: Pure numbers, such as a, e, sin, cos, tan, and so on.

Quantities that have no dimensions and are expressed in units: Angular displacement is measured in radians, Joule’s constant is measured in joules/calories, and so on.

Dimensional variables

Dimensional variables are physical quantities that have dimensions but do not have a fixed value, such as length, width, or height. For example, velocity, acceleration, force, work, and power are all terms that can be used.

Dimensionless variables

Dimensionless variables are physical quantities that do not have dimensions and do not have a fixed value. Dimensionless variables are also known as dimensionally indeterminate variables. Specific gravity, refractive index, coefficient of friction, Poisson’s ratio, and so on are all examples of physical properties.

Limitations of Dimensional Analysis

1. This method cannot be used to determine quantities that have no dimensions. This method does not allow for the determination of the constant of proportionality. They can be discovered either through experiment or through theoretical reasoning.
2. In the case of trigonometric, logarithmic, and exponential functions, this method is not applicable.
3. This method will be difficult to use in the case of physical quantities that are dependent on more than three other physical quantities at the same time.
4. In some cases, the constant of proportionality has both dimensions and a numerical value. In such cases, we are unable to make use of this system.
5. It is not possible to derive an expression using this method if one side of the equation contains the addition or subtraction of physical quantities on the other side of the equation.

Conversions That Are Significant

1 bar = 10⁶ dyne/cm²=10⁵ Nm-2=105Pa 

76 cm of Hg = 1.013106 dyne/cm²= 1.013105 Pa = 1.013 bar= 1.013 cm of Hg

1 torricelli or torr = 1 mm of Hg = 1.333103 dyne/cm² = 1.333 millibar

1 kmph is equal to 5/18 ms-1.

1 dyne is equal to 10-5N, and 1 Horse Power is equals to 746 watts.

1 kilowatt hour is equal to 36105 joules.

1 kg weight is equal to 1 newton.

1 calorie is equal to 4.2 joules.

1 electron volt is equal to 1.60210-19 joules.

1 erg is equal to 10-7 joule.

Some Physical Constants That Are Important

The speed of light in vacuum (c) is equal to 3108 ms-1.

At STP, the sound velocity in air is 331 ms-1, and the acceleration 

due to gravity (g) is 9.81 ms-2.

The avogadro number (N) is equal to 6.023 x 1023/mol.

At 4°C, the density of water is 1000 kgm-3, or one gramme per cubic centimetre.

Absolute zero is equal to -273.15 degrees Celsius or 0 degrees Kelvin.

The atomic mass unit is equal to1.66 10-27 kg.

The quantum of charge (e) is equal to 1.602 10-19 C.

Stefan’s constant is 5.67 x 10–8 Wm-2K-4and is defined as

In the case of Boltzmann’s constant (K), the value is 1.38110-23 JK-1.

One atmosphere is equal to 76 cm Hg ( 1.013105 Pa).

The mechanical equivalent of heat (J) is equal to 4.186 joules per calorie.

h=6.62607004 × 10-34 m2 kg / s is Planck’s constant (hertz).

The universal gas constant (R) is 8.314 joules per mole of water.

S.T.P. = 1.293 kg m-3, which is the density of air at that pressure.

The gravitational constant of the universe is G=6.67408 × 10-11 m3 kg-1s-2.

Conclusion

The Law of Homogeneity of Dimensions is a mathematical principle that states that all dimensions are equal. The dimensions of all terms on both sides of any correct equation representing the relationship between physical quantities must be the same in order for the equation to be correct. Terms that are separated by a ‘+’ or a ‘–’ must have the same height and width.