How Is the Dimension of Density Derived and Used?
Density is an important physical quantity in physics, representing the amount of mass contained within a unit volume of a substance. Understanding the dimensions of density is essential in dimensional analysis and solving related physics problems at the JEE level.
Definition and Expression of Density
Density is mathematically defined as the mass per unit volume of a substance. It is given by the formula $ \text{Density} = \dfrac{\text{Mass}}{\text{Volume}} $. This relationship provides the basis for determining the dimensional formula of density.
Fundamental Dimensions in Physics
Physical quantities can be expressed in terms of fundamental quantities such as mass (M), length (L), and time (T). The dimensions of a derived quantity indicate how these base quantities are involved in its definition. This concept is critical in dimensional analysis.
Dimensional Formula of Density
The mass (M) has the dimensional formula $[M^1]$. The volume (V) is a measure of space and is expressed as $[L^3]$ in terms of dimension. Substituting these into the formula for density gives the dimensional formula of density.
$ \text{Density} = \dfrac{\text{Mass}}{\text{Volume}} = \dfrac{[M^1]}{[L^3]} = [M^1L^{-3}] $
Stepwise Derivation of Density Dimensions
To derive the dimensions of density, consider that mass is fundamentally $[M]$ and volume is $[L^3]$. Using the division rule for exponents, the dimensional formula results in $[M^1L^{-3}]$. This shows that density depends on mass directly and on the inverse cube of length.
Dimensional Formula Table
| Quantity | Dimensional Formula |
|---|---|
| Mass | $[M^1]$ |
| Volume | $[L^3]$ |
| Density | $[M^1L^{-3}]$ |
Dimensional Analysis and Applications
Knowledge of density's dimensions is useful for verifying equations, converting units, and establishing relationships with other physical quantities. Dimensional analysis provides a systematic method for checking the consistency of physical equations in mechanics.
Examples of Density Dimensions
For the density of water, the dimensional formula remains $[M^1L^{-3}]$. For liquids or gases, irrespective of the substance, the dimensions of density do not change, as they depend only on mass and volume.
Relationship with Other Physical Quantities
The dimensional formula $[M^1L^{-3}]$ can be compared with other quantities such as pressure. For detailed study on related dimensions, refer to Dimensions Of Density And Specific Gravity.
Important Points about Dimensions of Density
- Density has one dimension in mass
- Density has minus three dimensions in length
- Density has zero dimension in time
- The dimensional formula applies to all substances
Comparison with Volume and Related Quantities
The volume of an object has the dimensional formula $[L^3]$. Density’s dimensions show its dependence on both mass and volume. For a detailed understanding of volume, refer to Dimensions Of Volume.
Further Applications in Physics
Density’s dimensional analysis helps in solving problems involving force, electric flux, and stress. The use of dimensional formulas is essential for checking the validity of equations and deriving relationships. Explore more at Dimensions Of Electric Flux.
Summary Table: Dimensions of Related Quantities
| Quantity | Dimensional Formula |
|---|---|
| Force | $[M^1L^1T^{-2}]$ |
| Electric Flux | $[M^1L^3T^{-3}A^{-1}]$ |
| Stress | $[M^1L^{-1}T^{-2}]$ |
| Magnetic Flux | $[M^1L^2T^{-2}A^{-1}]$ |
Related Physics Dimensions
The study of density’s dimensions is linked to other physics quantities like force and magnetic flux. Learn more about these by visiting Dimensions Of Force and Dimensions Of Magnetic Flux.
Conclusion on Dimensions of Density
The dimensional formula of density is $[M^1L^{-3}]$, indicating its dependence on mass and inverse volume. A proper understanding of density’s dimensions enables accurate analysis and solution of a wide range of physics problems, which is vital in JEE Main preparation.
FAQs on Understanding the Dimensions of Density
1. What are the dimensions of density?
Density is defined as mass per unit volume, and its dimensions are [M L-3].
• Mass (M) is expressed in kilograms (kg)
• Length (L) cubed represents volume (m3), so the exponent is -3
• Density = Mass/Volume, so [M]/[L3] = [M L-3]
2. What is the SI unit and dimensional formula of density?
The SI unit of density is kilogram per cubic metre (kg/m3), and its dimensional formula is [M L-3].
• SI unit: kg/m3
• Dimensional formula: [M L-3]
• Density combines mass (M) and volume (L3) in its calculation.
3. How do you calculate density?
Density is found by dividing mass by volume.
• Formula: Density = Mass/Volume
• Mass is measured in kg or g
• Volume is measured in m3 or cm3
• The answer gives you the density in kg/m3 or g/cm3
4. What is the physical meaning of the dimensional formula of density?
The dimensional formula of density, [M L-3], shows how density depends on mass and volume.
• M (mass): Indicates density directly depends on mass
• L (length): The negative sign (-3) shows it is inversely proportional to the cube of length (volume)
• This means: Increasing mass increases density, but increasing volume decreases density.
5. Which quantities have the same dimensional formula as density?
Any physical quantity defined as mass per unit volume will have the same dimensional formula as density ([M L-3]).
• Examples include: mass concentration, specific gravity (relative density without units), mass density
• All express a relationship of mass distributed within a certain volume.
6. Why is density considered a derived quantity?
Density is considered a derived quantity because it is calculated from two fundamental quantities: mass and volume.
• It is not a base physical quantity
• It uses the formula: Density = Mass/Volume
• The dimensions are derived as [M L-3]
7. Can you derive the dimensional formula of density step by step?
Yes, the dimensional formula of density can be derived in steps:
1. Density = Mass / Volume
2. Mass has the dimension [M]
3. Volume has the dimension [L3]
4. So, Density = [M] / [L3] = [M L-3]
8. What is the difference between mass and density?
Mass is the total quantity of matter in a body, while density describes how much mass exists in a certain volume.
• Mass is measured in kg (fundamental quantity).
• Density is measured in kg/m3 (derived quantity).
• Density = Mass divided by Volume.
9. What unit is commonly used for density in laboratories?
In laboratories, density is often measured in grams per cubic centimetre (g/cm3) or kilograms per cubic metre (kg/m3).
• g/cm3 is practical for small objects and chemicals
• kg/m3 is the SI unit used in most scientific contexts
10. Is density a scalar or vector quantity?
Density is a scalar quantity because it is described completely by its magnitude alone, without direction.
• Only has magnitude (value)
• No direction involved
• Example: 1000 kg/m3 for water (just a value, no direction)
