Dimensions of Physical Quantities: Understanding Measurement Units

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Dimensions of physical quantities

Physical Quantity Dimensions and Dimensional Analysis: What are the dimensions of physical quantities? How do you generate dimensional formulae for various physical quantities?

The dimensions of any physical quantity determine its nature. We have seven fundamental units from which the rest of the units are formed. These are called derived units and can be stated using any combination of the seven fundamental units. As a result, these seven numbers are referred to as dimensions. The dimensions of any physical quantities are indicated by square ‘[]′ brackets.

From this article, You will learn how to generate dimensional formulae for various physical quantities from this article; next, we will look at a list of different dimensional formulae and, ultimately, what they are used for and how to use them.

Overview

A standard of measurement is required to measure physical quantities such as length, mass, and time. The unit of that physical quantity is the measuring standard. The unit of length, for example, is the meter, and a standard length of one meter has a specific definition. To discover how many times this standard length meter is included in the length of a room, measure the length of an object.

Comparing a physical quantity to a standard quantity is known as measurement. The number of times the fundamental units of mass, length, and time appear in the physical quantity can be characterized as the dimension of the units of a derived physical quantity. The dimensional formula of a derived quantity is an equation that shows the powers to which the fundamental units must be increased to acquire one unit of that quantity.

Dimensions, Physical Quantities and Units

There are two parts to every measurement. The first is a number (n), followed by a unit (u). Q is equal to nu. An object’s length, for example, is 40 cm. A physical quantity’s magnitude is inversely proportional to the unit employed to express it.

n₁u1 = n₂u₂ if n₁ and n₂are the numerical values of physical quantities corresponding to the units u₁ and u₂. For example, 2.8 m = 280 cm; 6.2 kg = 6200 g.

How do you write Dimensions of physical quantity units?

Full names of units should not begin with a capital letter, even if they are named after a scientist. For example, newton, watt, ampere, and meter.
The unit should be written in its entirety or using only the symbols that have been agreed upon.
There is no plural form for units. For example, 10 kg is not the same as 10 kgs, and 20 w is not the same as 20 was.
Within or at the end of symbols representing units, no full stop or punctuation mark should be used—for example, 10 W but not 10 W.

What are Dimensions of physical quantity?

The powers to which the fundamental units are elevated to obtain one unit of a physical quantity are called the dimensions of that quantity.

Dimensional Analysis

Dimensional analysis is the process of determining the dimensions of physical quantities in order to check their relationships. All quantities in the world can be stated as a function of the fundamental dimensions independent of numerical multiples and constants.

Dimensional Formula

The dimensional formula of a derived quantity is an equation that shows the powers to which the fundamental units must be increased to acquire one unit of that quantity.

If Q is the unit of a derived quantity, M^aL^bT^c is the dimensional formula, and the exponents a, b, and c are the dimensions.

What are Dimensional Constants?

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

What are the Dimensionless quantities?

Quantities with no dimensions but a set value are known as dimensionless quantities. Quantities with no units and no dimensions: e, sin, cos, tan, and other pure numbers Quantities without dimensions and units: Joule’s constant – joule/calorie, etc. Angular displacement – radian

What are Dimensional variables?

Physical quantities with dimensions but no set value are referred to as dimensional variables—for instance, velocity, acceleration, force, work, power, and so on.

The Law of Dimensional Homogeneity

The size of all terms on both sides must be the same in any accurate equation describing the relationship between physical quantities. The dimensions of terms separated by a ‘+’ or a ‘–’ must be the same.

When a physical quantity Q has dimensions a, b, and c in length (L), mass (M), and time (T), and n1 is its numerical value in a system where the fundamental units are L1, M1, and T1, and n2 is its numerical value in a system where the fundamental units are L2, M2, and T2, respectively, then.

n₂=n₁[L₁/L₂]^a[M₁/M₂]^b[T₁/T₂]^c

Dimensional Analysis’ Limitations

This approach is incapable of determining dimensionless quantities. This approach cannot calculate the proportionality constant. They can be discovered by experiment (or) theory.
Trigonometric, logarithmic, and exponential functions are not relevant to this approach.
This strategy will be difficult to use when physical quantities are reliant on more than three physical qualities.
In some circumstances, the proportionality constant also has dimensions. We are unable to use this system in such circumstances.
We cannot use this method to get the expression if one side of the equation comprises the addition or subtraction of physical quantities.

Quantities Having the Same Dimensional Formula

Impulse and momentum are two terms that are used interchangeably.
Work, torque, the moment of force, and energy are all terms used to describe how much work is done.
Rotational impulse, angular momentum, Planck’s constant
Stress, pressure, modulus of elasticity, and energy density are all terms used to describe stress and pressure.
Surface energy, surface tension, and force constant
Angular velocity, frequency, and velocity gradient are all terms used to describe how fast something moves.
Thermal capacity, entropy, the universal gas constant, and Boltzmann’s constant are all terms used to describe the properties of a gas.

Dimensional Analysis in Practice

When working with physical quantities, dimensional analysis is critical. We’ll learn about several dimensional analysis applications in this part.

The foundations of dimensional analysis were laid by Fourier. The dimensional formulas are used to do the following:

Check if a physical equation is correct.
Determine a physical quantity’s relationship.
The conversion of a physical quantity’s units from one system to another.

Dimensional Analysis is a technique for determining the size and shape of objects. It assists us in mathematically studying the nature of items. It includes geometrical features like flatness and straightness, as well as lengths and angles. The essential principle of dimension is that only quantities with the same dimensions can be added or subtracted. Similarly, they are equal if two physical quantities have the same dimensions. Physical Quantity Dimensions and Dimensional Analysis: The dimensions of any physical quantity determine its nature. We have seven fundamental units from which the rest of the units are formed. These are called derived units and can be stated using any combination of the seven fundamental units. As a result, these seven numbers are referred to as dimensions. The dimensions of any physical quantities are indicated by square ‘[]′ brackets.

FAQs

Question: What is the force dimension?

Answer: We all know that,

F = ma — (1)

Acceleration is a derived quantity that may be expressed in terms of fundamental quantities, whereas mass is a fundamental quantity.

Using (1) and (2),

F = [MLT⁻²]

This is the dimension of force.

What is the meaning of dimension in physics?

It is a formula that connects derived and fundamental quantities. However, the magnitude of the generated quantity has nothing to do with it.

What is dimensional analysis?

Dimensional analysis is founded on the idea that two quantities with the same dimensions can only be compared. I can compare kinetic and potential energy and assert that they are equal or that one is larger than the other because they have the same dimension. I can compare kinetic and potential energy and assert that they are equal or that one is larger than the other because they have the same dimension.

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