What Are the Dimensions of Volume?

Volume, as a physical quantity, represents the space occupied by a body in three-dimensional space. Its dimensional formula is crucial for dimensional analysis in physics and is used to verify equations as well as derive relationships between various physical quantities.

Definition and Concept of Volume

Volume is defined as the measure of the amount of space enclosed within a three-dimensional boundary. It is applicable to solids, liquids, and gases occupying physical space in all three dimensions: length, width, and height.

Mathematical Expression of Volume

The general mathematical expression for the volume of a rectangular solid is given by the product of its three linear dimensions: length ($l$), breadth ($b$), and height ($h$).

This relationship is written as $V = l \times b \times h$, where $V$ denotes volume.

Dimensional Formula of Volume

Each linear dimension such as length, breadth, or height has the dimensional formula $[M^0L^1T^0]$. Therefore, multiplying three linear quantities results in:

$V = [M^0L^1T^0] \times [M^0L^1T^0] \times [M^0L^1T^0] = [M^0L^3T^0]$

Thus, the dimensional formula of volume is $[M^0L^3T^0]$. The exponent 3 for length denotes three-dimensional space occupied by an object.

SI and Other Units of Volume

The SI unit of volume is the cubic metre ($\mathrm{m}^3$). Other commonly used units include cubic centimetre ($\mathrm{cm}^3$), litre (L), and millilitre (mL), where $1~\mathrm{L} = 1000~\mathrm{cm}^3$.

Measurements are always expressed in cubic units since volume is a three-dimensional quantity.

Physical Significance of Volume Dimensions

The dimension of volume reflects that it is independent of mass and time, depending only on the cube of length. This principle is used for checking the consistency of physical equations and for converting between physical quantities.

Volume is fundamental for understanding related physical concepts such as density, pressure, and work, which rely on the volume calculation and its dimensional formula. For more information, refer to Dimensions Of Density.

Derivation of Volume for Different Shapes

For a cube with side $a$, the volume is $V = a^3$, maintaining the same dimensional formula $[M^0L^3T^0]$. For a cylinder of radius $r$ and height $h$, the volume formula is $V = \pi r^2 h$, where both $r$ and $h$ are lengths.

Similarly, for a sphere of radius $r$, the formula $V = \dfrac{4}{3} \pi r^3$ confirms the dimensional formula, as $r^3$ implies the cube of length.

Table: Conversion of Volume Units

Solved Examples on Dimensions of Volume

For a cube having side 5 cm, the volume $V = 5^3 = 125~\mathrm{cm}^3$. For a cuboid with length 8 m, breadth 3 m, and height 2 m, the volume is $V = 8 \times 3 \times 2 = 48~\mathrm{m}^3$.

If a cylindrical tank has radius 2 m and height 7 m, then $V = \pi \times 2^2 \times 7 = 28\pi~\mathrm{m}^3$. The dimensional formula used in each calculation is $[M^0L^3T^0]$.

Uses of Volume Dimensions in Physics

The dimensional formula of volume is essential for checking the homogeneity of physical equations, analysing derived quantities, and converting between units. It is also crucial in fluid mechanics, thermodynamics, and material science.

To understand dimensions of related physical quantities, see Dimensions Of Pressure.

Dimensional Analysis and Related Quantities

Dimensional analysis using the volume formula helps in deriving units of physical quantities like volumetric flow rate and volumetric strain. For example, the volumetric flow rate has the dimensions $[M^0L^3T^{-1}]$.

Other related concepts include volumetric density, work, and force, which utilize volume dimensions in their formulations. Visit Dimensions Of Mass and Dimensions Of Force for further reference.

Important Properties of Volume Dimensions

• Independent of mass and time dimensions
• Depends only on the cube of length
• Used for verifying physics equations
• SI unit is always in cubic metres
• Can be applied to regular and irregular shapes
• Commonly used in determination of density

Summary of Volume Dimensions in Physics

The dimensions of volume serve as a fundamental concept in physics, particularly for dimensional analysis, unit conversion, and understanding related physical properties. Mastery of volume dimensions is necessary for solving problems in JEE and other competitive exams.

Detailed explanations of dimensions for energy and work can be found at Dimensions Of Energy and Dimensions Of Work.