Dimensional Formula of Angular Momentum

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Dimensions:

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. 


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 × [Time]⁻¹ = [M⁰L⁰T⁰] [T]⁻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 = [M⁰ L⁰ T⁻¹] × [M¹ L² T⁰]⁻¹ = M¹ L² T⁻¹.

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

FAQ (Frequently Asked Questions)

1. Write a Few Limitations of Dimensional Formula?

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.

2. Explain the Term 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. 

L=l×ω

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

3. Gives an Example Based on Angular Momentum?

One of the famous examples of angular momentum in an  ice skater executing a spin, as shown in. Net torque is almost near zero in them as there is very little friction is present between them and friction which is exerted by them is very close to the pivot point.