Dimensional Formula of Power and its Derivation

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Power:

Power can be defined as the rate of doing work or it is the work done in unit time. Standard unit of power is watt(W) which is also written as joules per second(J/s). We use terms horsepower (hp) for power of motor vehicles and other machines. One horsepower is equal to 745.7 watts. Whereas the average power is defined as total energy consumed divided by total time taken i.e total work done per unit time. 

In mathematical term power can be written as:

Power=work/time

I.e P=w/t

It is totally time based, which tells us about the rate of doing work or rate of doing work.


Dimensional Formula of Power:

The dimensional formula of power is written as, M1 L2 T-3

Where M represents mass, L represents length and T represents time. 


Derivation of Dimensional Formula of Power:

We know that power is written as:

Power (P) = Work × time-1 = Joule × second-1 . . . . . (1)

As, Work (J) = N × m = M1 L1 T-2 × [L]

Therefore, the dimensional formula of work = M1 L2 T-2 . . . . (2)

On putting equation (2) in equation (1) we get,

Power (P) = Work × time-1

Or, P = [M1 L2 T-2] × [T-1] = M1 L2 T-3.

Therefore, power is dimensionally written as M1 L2 T-3.


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 measurement. 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 we get by equating a different physical quantity with its dimensional formula is called a dimensional equation and is written in standard form.

Following are the applications of the dimension formula


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= n2u2


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. To Derive a Relationship Between Given Different 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.

FAQ (Frequently Asked Questions)

1. Explain Term Dimension?

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 measurement. 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]

2. Explain Term Power?

Power can be defined as the rate of doing work or it is the work done in unit time. Standard unit of power is watt(W) which is also written as joules per second(J/s). We use terms horsepower (hp) for power of motor vehicles and other machines. One horsepower is equal to 745.7 watts. Whereas the average power is defined as total energy consumed divided by total time taken i.e total work done per unit time. 

In mathematical term power can be written as:

Power=work/time

I.e P=w/t

It is totally time based, which tells us about the rate of doing work or rate of doing work.

3. Write a Few Limitations of Dimensional Formula?

  • 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.

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

4.  Write a Few Sets that Have the Same Dimension Formula.

Some of the sets having the same dimension formula that are discussed below:

  • Strain, refractive index, relative density, distance gradient, relative permeability, angle of content.

  • Mass and inertia.

  • Momentum and impulse.

  • Thrust, force, weight, tension, energy gradient.

  • Angular momentum and Planck’s constant.

  • Surface tension, surface area, force gradient, spring constant.

  • Latent heat and gravitational potential.

  • Thermal capacity, Boltzman constant, entropy.