Dimensional Formula of Angular Momentum

Dimensions of the physical quantity are the power to which the base quantities are raised to represent that quantity. Dimensions of any given quantity tell us about how and which way different physical quantities are related. Finding dimensions of different physical quantities has many real-life applications and is helpful in finding units and measurements. Imagine a physical quantity X which depends mainly on base mass(m), length(L), and time(T) with their respective powers, then we can represent dimensional formula as [MaLbTc]

Dimensional Formula: 

The dimensional formula of any physical quantity is that expression which represents how and which of the base quantities are included in that quantity. 

It is written by enclosing the symbols for base quantities with appropriate power in square brackets i.e ( ).

E.g: Dimension formula of mass is: (M)

Dimensional Equation: 

The equation obtained by equating a physical quantity with its dimensional formula is called a dimensional equation. 

Application of Dimensional Analysis: 

1. To convert a physical quantity from one system of the unit to the other:

It is based on a fact that magnitude of a physical quantity remain same whatever system is used for measurement i.e magnitude = numeric value(n) multiplied by unit (u) = constant

n₁u₁ = n₁u₂

2. To check dimensional correctness of a given physical relation: 

If in a given relation, the terms of both sides have the same dimensions, then the equation is dimensionally correct. This concept is best known as the principle of homogeneity of dimensions. 

3. It determines the relationship between various physical quantities. 

Using the principle of homogeneity of dimension, the new relation among physical quantities can be derived if the dependent quantities are known. 

Limitation of This Method

1. This method can be used only if dependency is of multiplication type. The formula containing exponential, trigonometric, and logarithmic functions can not be derived using this method. The formula containing more than one term which is added or subtracted likes s = ut+ ½ at² also cannot be derived.

2. The relation derived from this method gives no information about the dimensionless constants. 

Angular Momentum:

Angular momentum signifies the product of mass and the velocity of the object. If any object is moving with mass, then they possess momentum. The major difference present in angular momentum is that they deal with bodies who have to rotate and spin objects. So they are almost similar to linear momentum. It is a vector quantity, which implies that they have both magnitudes as well as direction. 

Angular momentum can be defined as the vector quantity which clearly represents the product of the body's rotational inertia and rotational velocity about a particular axis in radians per second.

Difference between linear momentum and the angular momentum

The main difference between the linear momentum and the angular momentum is that the linear momentum is applicable only for the object moving in a direct path whereas the angular momentum is applied for the objects only in the angular motions.

The Formula of Angular Momentum:

If an object is accelerating around a fixed point, then it also possesses angular momentum. Hence it can be given as:

L= r × p

L=l × ω

Where L is the angular momentum, I is rotational inertia and ω is the angular velocity. 

Dimensional Formula of Angular Momentum:

We known that dimensional formula of angular momentum is written as, M¹ L² T⁻¹

Where, mass M, length L, Time T.

Derivation of the Dimensional Formula of Angular Momentum:

We know that angular momentum can be written as:

We know that angular momentum = Angular Velocity × Moment of Inertia . . . . (1)

As, Angular Velocity = Angular displacement × TimeTime⁻¹ = M⁰L⁰T⁰M⁰L⁰T⁰ TT⁻¹

Therefore the dimensional formula of Angular Velocity = M⁰ L⁰ T⁻¹ . . . . . . (2)

And, the Moment of Inertia is written as, M.O.I = Mass × (Radius of Gyration)

Therefore the dimensional formula of M.O.I = M¹ L² T⁰ . . . . . (3)

Putting the  value of equation (2) and (3) in equation (1) we obtain,

Angular Momentum = Angular Velocity × Moment of Inertia

Or, M = M⁰L⁰T⁻¹M⁰L⁰T⁻¹ × M¹L²T⁰M¹L²T⁰⁻¹ = M¹ L² T⁻¹.

Therefore, the angular momentum is dimensionally represented as M¹ L² T⁻¹.

A Particle's Angular Momentum

The moment of force is the rotational equivalent of force. At the same time, angular momentum is the rotational equivalent of linear momentum. We'll now determine the angular momentum of a single moving particle. The notion of angular will subsequently be extended to particle systems, including rigid bodies. Angular momentum, like the moment of force, is a vector product. 

Consider a particle with mass m and linear momentum P, which is located at a distance r from the origin O. The particle's angular momentum L with respect to the origin o will then be \[L = R \times P\]

An Angular Momentum Example

Before touching the water's surface, a diver jumps from the springboard and exhibits in the air. He curls his hands inwards after stepping off the springboard. As a result, his angular velocity rises (since the moment of inertia decreases). As a result, the total angular momentum does not change.

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Conclusion

This is how angular momentum is calculated by deriving a formula specific to the requirement. Focus on how different physical quantities are used to derive the formula and used to calculate it.