Units And Measurement

To determine the dimensional formula for angular momentum, we'll analyze its fundamental components and derive the formula step-by-step.

1. Understanding Angular Momentum:
Angular momentum (L) is defined as the rotational equivalent of linear momentum, given by: L = r × p where: - r is the position vector (distance) - p is the linear momentum (p = mv)

2. Breaking Down the Components:
a) Linear Momentum (p): - p = mass × velocity - Dimensional formula: [M][L][T]^-1 b) Position Vector (r): - Represents distance - Dimensional formula: [L]

3. Deriving the Dimensional Formula:
Since angular momentum is the cross product of r and p: L = r × p The dimensional formula becomes: [L] = [r] × [p] = [L] × [M][L][T]^-1 = [M][L]²[T]^-1

4. Verification:
We can verify this using another expression for angular momentum: L = Iω (moment of inertia × angular velocity) - I (moment of inertia) = [M][L]² - ω (angular velocity) = [T]^-1 Thus: [M][L]² × [T]^-1 = [M][L]²[T]^-1

5. Physical Interpretation:
The dimensions show: - [M]: Depends on mass - [L]²: Depends on both distance and moment arm - [T]^-1: Includes rotational time dependence

Final Answer:
The dimensional formula for angular momentum is [M][L]²[T]^-1.