Calculating the dimensional formula of a physical quantity involves expressing it in terms of the fundamental physical quantities. The dimensional formula is the statement of a physical quantity in terms of its fundamental unit with suitable dimensions.
Understanding Dimensional Formulas
Every physical quantity, whether fundamental or derived, can be expressed using a unique combination of fundamental dimensions. The internationally accepted fundamental dimensions typically include:
- Mass [M]
- Length [L]
- Time [T]
- Electric Current [A]
- Thermodynamic Temperature [K]
- Amount of Substance [mol]
- Luminous Intensity [cd]
The dimensional formula of a quantity shows how it depends on these fundamental dimensions. For example, Velocity is displacement (Length) divided by time, so its dimensions are [L]/[T] or [L¹ T⁻¹].
Steps to Calculate a Dimensional Formula
Follow these steps to determine the dimensional formula for a physical quantity:
- Identify the Physical Quantity: Determine the quantity for which you want to find the dimensional formula (e.g., Force, Work, Power, Pressure).
- Recall the Formula or Unit: Find the definition or formula that relates the quantity to other simpler quantities, or identify its standard unit in the SI system.
- Substitute with Dimensions: Replace each quantity or unit in the formula with its corresponding dimensional formula or the dimensions of its fundamental units. For example, replace mass (kg) with [M], length (m) with [L], and time (s) with [T].
- Simplify the Expression: Combine the dimensions using the rules of algebra (adding/subtracting exponents when multiplying/dividing the same base).
- Write the Final Dimensional Formula: Express the result in the standard format [Mᵃ Lᵇ Tᶜ ...], where a, b, c are the exponents of the fundamental dimensions.
Examples of Calculating Dimensional Formulas
Let's look at a couple of examples, including those mentioned in the reference.
Example 1: Force (F)
- Quantity: Force
- Formula/Unit: Force can be defined by Newton's second law: F = mass × acceleration. The unit of Force is Newton (N), which is equivalent to kg⋅m/s².
- Substitute with Dimensions:
- Mass (kg) corresponds to [M¹]
- Length (m) corresponds to [L¹]
- Time (s) corresponds to [T¹]
- Therefore, acceleration (m/s²) corresponds to [L¹] / [T²] = [L¹ T⁻²]
- Force = Mass × Acceleration
- Force dimensions = [M¹] × [L¹ T⁻²]
- Simplify: The dimensions combine directly.
- Final Dimensional Formula: [M¹ L¹ T⁻²] or simply [M L T⁻²]. As stated in the reference: "Dimensional force is an example. [M L T-2] F = [M L T-2] The reason for this is that the unit of Force is Newton, or kg*m/s2."
Example 2: Pressure (P)
- Quantity: Pressure
- Formula/Unit: Pressure is defined as Force per unit Area.
- Substitute with Dimensions:
- From Example 1, the dimensional formula for Force is [M¹ L¹ T⁻²].
- Area is Length × Length, so its dimensional formula is [L¹] × [L¹] = [L²].
- Pressure = Force / Area
- Pressure dimensions = [M¹ L¹ T⁻²] / [L²]
- Simplify: Using exponent rules (L¹ / L² = L¹⁻² = L⁻¹), we get:
- [M¹ L¹⁻² T⁻²] = [M¹ L⁻¹ T⁻²]
- Final Dimensional Formula: [M¹ L⁻¹ T⁻²]. This matches the calculation shown in the reference: "Answer: P = [M1 L1 T–2] × [L2]–1 = M1 L-1 T –2." Here, multiplying by [L2]–1 is equivalent to dividing by [L2].
Why Use Dimensional Analysis?
Understanding and calculating dimensional formulas is crucial in physics and engineering for several reasons:
- Checking Formula Consistency: You can use dimensional analysis to check if a physical equation is dimensionally correct. Both sides of an equation must have the same dimensions.
- Deriving Relationships: In some cases, dimensional analysis can help suggest the relationship between different physical quantities in a physical phenomenon.
- Unit Conversion: Dimensional formulas provide a systematic way to convert units from one system to another.
Fundamental Dimensions
Here's a summary of the basic fundamental dimensions commonly used:
| Quantity | Symbol in Formula | Unit Symbol (SI) |
|---|---|---|
| Mass | [M] | kg |
| Length | [L] | m |
| Time | [T] | s |
| Electric Current | [A] | A |
| Thermodynamic Temperature | [K] | K |
| Amount of Substance | [mol] | mol |
| Luminous Intensity | [cd] | cd |
By expressing any physical quantity in terms of these fundamental dimensions with appropriate exponents, you obtain its dimensional formula.
