How to Calculate Dimensional Formula?

The dimensional formula expresses a physical quantity in terms of the fundamental dimensions of mass (M), length (L), and time (T), raised to certain powers.

Understanding Dimensional Formula

According to the provided reference, the dimensional formula is expressed as:

Q = M^aL^bT^c

Where:

• Q represents the derived physical quantity.
• M, L, and T are the base dimensions for mass, length, and time, respectively.
• a, b, and c are the exponents that denote the power to which each base dimension is raised.

This formula demonstrates how a derived quantity can be expressed in terms of the fundamental units.

Steps to Calculate Dimensional Formula

Here's a breakdown of the steps to determine a dimensional formula:

1. Identify the physical quantity: First, clearly identify the physical quantity whose dimensional formula you want to calculate (e.g., velocity, force, energy).

2. Express the quantity in terms of fundamental quantities: Express the chosen quantity using its definition in terms of fundamental quantities like mass, length, and time. For example:

   • Velocity = Displacement / Time
   • Force = Mass × Acceleration

3. Substitute the dimensional symbols: Replace each fundamental quantity with its respective dimensional symbol:

   • Mass (M)
   • Length (L)
   • Time (T)

4. Determine the exponents: Determine the powers to which each of the fundamental dimensions (M, L, and T) must be raised.

Examples

Let's look at a couple of examples to illustrate how to calculate the dimensional formula:

1. Velocity

• Definition: Velocity = Displacement / Time
• Dimensional Representation:
  • Displacement has the dimension of length (L).
  • Time has the dimension of time (T).
• Dimensional Formula: Velocity = L/T = M⁰L¹T^-1

2. Force

• Definition: Force = Mass × Acceleration
• Dimensional Representation:
  • Mass is represented by M.
  • Acceleration is defined as velocity change over time which has dimension L/T².
• Dimensional Formula: Force = M * L/T² = M¹L¹T^-2

3. Energy

• Definition: Energy = Force × Displacement
• Dimensional Representation:
• Force has the dimension M¹L¹T^-2.
• Displacement has the dimension L.
• Dimensional Formula: Energy = M¹L¹T^-2 * L = M¹L²T^-2

Key Takeaways

• The dimensional formula shows how a physical quantity is related to the fundamental dimensions.
• It is crucial for checking the correctness of physical equations.
• It helps in understanding the nature of physical quantities.
• The formula is typically expressed as Q = M^aL^bT^c, as stated in the reference.