1. What is Dimensional Formula?

Definition: Dimensional formula expresses the physical quantity in terms of the fundamental dimensions of mass (M), length (L), and time (T).

Purpose: It helps in checking the consistency of equations, converting units, and deriving relationships between physical quantities.

2. Dimensional Formula of Force

The dimensional formula for force is:

Derivation:

• From Newton's second law: Force = mass × acceleration
• Mass has dimension [M]
• Acceleration has dimension [L T^{-2}]
• Therefore, Force = [M] × [L T^{-2}] = [M L T^{-2}]

3. Dimensional Formula of Power

The dimensional formula for power is:

Derivation:

• Power = Work / Time
• Work = Force × Distance
• Force has dimension [M L T^{-2}]
• Distance has dimension [L]
• Therefore, Work = [M L T^{-2}] × [L] = [M L^2 T^{-2}]
• Power = [M L^2 T^{-2}] / [T] = [M L^2 T^{-3}]

4. Importance of Dimensional Analysis

Applications:

• Verifying the correctness of physical equations
• Deriving relationships between physical quantities
• Converting units from one system to another
• Predicting how one quantity depends on others

5. Frequently Asked Questions (FAQ)

Q1: What do M, L, and T represent?
A: M stands for mass (kg), L for length (m), and T for time (s) - the fundamental base quantities.

Q2: Why is force [M L T^{-2}]?
A: Because force equals mass times acceleration, and acceleration is length per time squared.

Q3: How is power's dimensional formula different from force?
A: Power has an extra L and T^{-1} because it's work (force × distance) per unit time.

Q4: Can dimensional analysis prove an equation is correct?
A: No, it can only prove an equation is dimensionally consistent, not necessarily physically correct.

Q5: What would be the dimensions of energy?
A: [M L^2 T^{-2}], same as work, since they're equivalent quantities.