Dimensions of Physics quantities

What are you referring to when you say “Dimension of Physical Quantities”?

A physical quantity’s dimension is the magnitude to which its fundamental units must be raised to represent it. The fundamental quantities are mass, length, time, temperature, electric current, luminous intensity, and amount of substance. These fundamental quantities can be used to express physical quantities. These seven quantities correspond to the physical world’s seven dimensions. The mass dimension is denoted by [M], the length dimension by [L], the time dimension by [T], the temperature dimension by [K], the electric current measurement by [I], the luminous intensity dimension by [Cd], and the amount of substance dimension by [mol].

Dimensional Analysis

Dimensional analysis is the process of determining the dimensions of physical quantities to verify their relationships. These dimensions are unrelated to numerical multiples and constants, and any quantity in the world can be expressed as a function of these 7 fundamental dimensions. Dimensional analysis studies the relationship between physical quantities using dimensions and units of measurement. Dimensional analysis is critical because it maintains the same units, enabling us to perform mathematical calculations efficiently.

Unit Conversion and Dimensional Analysis

Dimensional analysis is also called the Factor Label Method or Unit Factor Method because it employs conversion factors to obtain the same units. To illustrate the statement, suppose you want to know how many meters make up 3 kilometres.We know that 1000 meters equals 1 km; thus, 3 km equals 3 × 1000 meters equals 3000 meters. The conversion factor is 1000 meters in this case. Dimensional Analysis as a Tool for Verifying the Correctness of Physical Equations

• Assume you’re unsure whether time equals speed/distance or distance/speed
• This can be verified by comparing the dimensions on both sides of the equations
• We obtain this by reducing both equations to their fundamental units on each side of the  equation
• However, it is worth noting that dimensional analysis cannot be used to determine any dimensionless constants in the equation

Advantages of Dimensional Analysis

Dimensional analysis is a critical measurement component and is used extensively in real-world physics. Three primary reasons motivate our use of dimensional analysis:

1. To ensure that a two-dimensional equation is consistent.
2. To establish a relationship between physical quantities present in physical phenomena.
3. To convert units between systems.

Dimensional Analysis’s Limitations

Several limitations of dimensional analysis include the following:

• It provides no information regarding the dimensional constant
• It is impossible to derive formulas containing trigonometric functions, exponential functions, logarithmic functions, etc
• It contains no information about the nature of a physical quantity, such as whether it is a scalar or a vector

Dimensional Formula

The expression indicating the powers to which fundamental units must be raised to obtain one unit of a derived quantity is the quantity’s dimensional formula. Suppose Q is the unit of a derived quantity represented by the expression Q = M^aL^bT^c. In that case,M^aL^bT^c is the dimensional formula, and the exponents a, b, and c are referred to as the dimensions.

What are Dimensional Constants?

Dimensional constants are physical quantities that have dimensions and a fixed value. Examples include the gravitational constant (G), Planck’s constant (h), the universal gas constant (R), and the speed of light in vacuum (c).

What are the Dimensionless quantities?

The term “dimensionless quantities” refers to those lacking dimensions but having a fixed value. Quantities without dimensions: Pure numbers, ln, e, sin, cos, tan, and so on. Quantities without dimensions and units: Radian – angular displacement, relative density etc.

What are dimensional variables?

Dimensional variables are physical quantities that do not have a fixed value or dimensions—for example, velocity, acceleration, force, work, and power.

How are dimensionless variables defined?

Dimensionless variables are physical quantities that lack dimensions and have an undefined value—for instance, specific gravity, refractive index, friction coefficient, and Poisson’s ratio.

Law of Dimensional Homogeneity

1. In any correct equation expressing the relationship between physical quantities, all terms must have the exact dimensions on both sides. Words denoted by a plus sign or a minus sign must have the exact dimensions.
2. When a physical quantity Q has dimensions a, b, and c in length (L), mass (M), and time (T), respectively, and n¹ is its numerical value in a system with fundamental units L¹, M¹, and T¹, and n₂ is its numerical value in another system with fundamental units L², M², and T², respectively, then, n₁ [L¹a M¹b T¹ c ]= n₂ [L²a M²b T² c].

Dimensional Analysis’s Limitations

1. This method cannot be used to determine dimensionless quantities. This method cannot be used to determine the constant of proportionality. They can be discovered either experimentally or theoretically.
2. This procedure does not work with trigonometric, logarithmic, or exponential functions.
3. This method will be difficult to apply to physical quantities dependent on more than three physical quantities.
4. In some instances, the constant of proportionality has dimensions as well. In such cases, we are unable to use this system.
5. We cannot use this method to derive an expression of one side of the equation containing the addition or subtraction of physical quantities.

Quantities Possessing the Identical Dimensional Formula

• Momentum and impulse
• Work, torque, the moment of force, and energy are all terms having the same dimensions
• Rotational impulse, angular momentum, Planck’s constant
• Stress, pressure, elasticity modulus, and energy density
• Surface energy, surface tension, and force constant
• Angular velocity, frequency, and gradient of velocity
• Gravitational potential energy, latent heat
• Thermal capacity, entropy, universal gas, and Boltzmann’s constant are all constants
• The force, the thrust

Applications of Dimensional Analysis

When dealing with physical quantities, dimension analysis is critical. This section will discuss some of the applications of dimensional analysis.Fourier established the principles of dimensional analysis. Dimensional formulas are used to calculate the following:

1. Validate the accuracy of a physical equation.
2. Calculate the relationship between two physical quantities.
3. Converting the units of a physical quantity between systems.

Conclusion

Dimensional analysis is a technique that scientists and engineers use to determine whether their equations and calculations make sense. It cannot prove that calculations are correct, but it can demonstrate that something is incorrect and needs to be corrected. The dimensional analysis makes use of units of measurement: any unit of length, for example, is always expressed in meters, feet, fathoms, or any other unit of distance. You can use these units to ensure that the numbers you’re combining produce the desired result.