Dimensions of Physical Quantities

Dimensions of a physical quantity are the powers to which you must raise the basic units to describe that quantity. We use square brackets ‘[ ]’ to show these dimensions, which help us understand what the quantity is about.

There are seven basic units: mass, length, time, temperature, electric current, luminous intensity, and amount of substance. Every physical quantity can be described using these basic units and their dimensions. The dimensions of a physical quantity match the dimensions of its unit.

A unit is a measure of stable quantities like mass and length. Physical quantities are split into two types: fundamental quantities and derived quantities. Measurement is the process of comparing a physical quantity to a standard to figure out how big or small it is.

The dimensional formula is a way to express the combination of basic units needed to derive a physical quantity. Physical quantities are shown using dimension equations, which connect a physical quantity with its dimensional formula.

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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.

Fundamental Quantites and Units

1. Basic quantities are those that stand alone and do not depend on other quantities.
2. The units used to measure these basic quantities are known as fundamental units.
3. These units are unique and cannot be formed by combining other units. In the world of physics, there are seven basic quantities. All other physical quantities can be determined using these seven.
4. To measure a basic quantity, systems like CGS, MKS, FPS, and SI are employed.

The following table lists the seven basic quantities, along with their units and symbols.

Dimensions of Physical Quantity

The concept of dimensional analysis involves expressing a physical quantity in terms of the basic quantities from which it is derived, such as mass, length, and time. These fundamental quantities are represented by powers, or exponents, in the dimensional formula.

For instance, if a physical quantity A can be described by the dimensional formula

MaLbTc, then the exponents

a,

b, and

c indicate the dimensions of A in terms of mass, length, and time respectively. Here,

[M] represents the dimension of mass,

[L] represents the dimension of length, and

[T] represents the dimension of time.

Consider the example of an area, which is calculated by multiplying length by width. Since both length and width have the dimension of length

[L], the area has a dimensional formula of

[L2]. This indicates that the area has two dimensions in length and zero in both mass and time.

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Similarly, volume is calculated by multiplying length, width, and height—each of which is a dimension of length. Thus, the volume’s dimensional formula is

[L3], suggesting that volume has three dimensions in length and zero in mass and time.

It is important to note that these dimensional formulas do not consider the actual magnitude of the quantities. Therefore, various quantities like velocity, average velocity, initial velocity, and final velocity, though they may have different magnitudes, share the same dimensional representation as

[L]/[T] or

[LT−1].

How do you write Dimensions of physical quantity units?

1. 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.
2. The unit should be written in its entirety or using only the symbols that have been agreed upon.
3. 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.
4. 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.

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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.