What is the dimensional formula of density?
ML-3T0
Understanding the Dimensional Formula of Density
Dimensional analysis is a powerful tool in physics used to check the consistency of equations and to understand the relationship between different physical quantities. The dimensional formula of a physical quantity expresses it in terms of the fundamental dimensions of mass (M), length (L), time (T), temperature (K), electric current (A), luminous intensity (cd), and amount of substance (mol).
Defining Density
Density is a fundamental property of a substance that relates its mass to the volume it occupies. It is defined by the formula:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
Deriving the Dimensional Formula of Density
To find the dimensional formula of density, we need to determine the dimensions of mass and volume.
- The dimension of Mass is simply [M]. In terms of M, L, and T, this is \([M^1 L^0 T^0]\).
- Volume is derived from length. The volume of a cube, for example, is side × side × side, which involves multiplying length three times. Therefore, the dimension of Volume is \([L^3]\). In terms of M, L, and T, this is \([M^0 L^3 T^0]\).
Now, substitute these dimensions into the formula for density:
\[ [\text{Density}] = \frac{[\text{Mass}]}{[\text{Volume}]} \]
\[ [\text{Density}] = \frac{[M^1 L^0 T^0]}{[M^0 L^3 T^0]} \]
Using the rules of exponents (\(\frac{a^m}{a^n} = a^{m-n}\)), we combine the dimensions:
\[ [\text{Density}] = [M^{1-0} L^{0-3} T^{0-0}] \]
\[ [\text{Density}] = [M^1 L^{-3} T^0] \]
Thus, the dimensional formula of density is \([ML^{-3}T^0]\). This indicates that density depends on mass raised to the power of 1, length raised to the power of -3, and is independent of time (time raised to the power of 0).
| Physical Quantity | Common Symbol | SI Unit | Dimensional Formula |
|---|---|---|---|
| Mass | m or M | kilogram (kg) | \([M]\) |
| Length | l or L | meter (m) | \([L]\) |
| Time | t or T | second (s) | \([T]\) |
| Volume | V | cubic meter (\(m^3\)) | \([L^3]\) |
| Density | \(\rho\) | kilogram per cubic meter (\(kg/m^3\)) | \([ML^{-3}]\) or \([ML^{-3}T^0]\) |
Comparing this derived dimensional formula with the given options:
- ML-3T
- ML-3T0
- ML3T
- ML-2T
The dimensional formula \([ML^{-3}T^0]\) matches the second option.
Revision Table: Key Dimensions
| Quantity | Dimensions | Derivation/Formula |
|---|---|---|
| Length (L) | \([L]\) | Base Quantity |
| Mass (M) | \([M]\) | Base Quantity |
| Time (T) | \([T]\) | Base Quantity |
| Area (A) | \([L^2]\) | \(L \times L\) |
| Volume (V) | \([L^3]\) | \(L \times L \times L\) |
| Density (\(\rho\)) | \([ML^{-3}T^0]\) | \(\frac{\text{Mass}}{\text{Volume}} = \frac{[M]}{[L^3]}\) |
Additional Information: Uses of Dimensional Analysis
Dimensional analysis is useful for:
- Checking Consistency: Ensuring that the dimensions on both sides of an equation are the same. If they are different, the equation is incorrect.
- Deriving Relationships: Sometimes, the relationship between physical quantities can be derived or predicted based on their dimensions (Buckingham Pi theorem).
- Unit Conversion: Helps in converting units from one system to another by understanding how the dimensions change.
It's important to remember that while dimensional consistency is necessary for an equation to be correct, it is not sufficient. A dimensionally correct equation might still have incorrect numerical constants.
Important Questions from Physical Quantities
Which of the following is a vector quantity?
A. Time
B. Temperature
C. Distance
D. Velocity
Which of the following is a scalar quantity?
Pressure is measured in terms of
A. Mass & Density
B. Work done
C. Force and Area
D. Force and Distance
1 barrel of oil = ________ litre.
The energy $U$ stored in an inductor carrying a current $I$ is related to the magnetic flux $\Phi_B$ by the expression $U = \frac{1}{2} \Phi_B I$. What is the dimensional formula of magnetic flux ($\Phi_B$)?
