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Dimensional Formula of Power and its Derivations

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Last updated date: 28th Feb 2024
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Introduction to 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.

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


Power is  written as:

Power=work/time

I.e  P=w/t .

Power is totally time based, which allows us to know  about the rate of doing work .

The dimensional formula of power is denoted as :


M1 L2 T-3, where

M stands for Mass

L stands for Length

T stands for Time.


The Derivation can be explained as :

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

Since, Work (J) = Force (M x a) × displacement = M1 L1 T-2 × [L]

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

On substituting 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, we can say that power is dimensionally represented as M1 L2 T-3.


Dimension basically refers to the length, height, width or depth of things or objects. The total amount of any measurable space is called Dimension.


The formulas which tell us about how much, and which fundamental qualities are present in any physical quantity are basically known as the Dimensional Formula. A basic aspect of units and measurements.


To check whether a formula is correct dimensionally or not, we need the help of the Dimensional Formula. This formula is required to convert systems from one unit to another. Explanation of how physical quantities can be expressed in fundamental units can be done using the Dimensional Formula.


This Dimensional Formula method can be used only if the multiplication type of dependency is there.  It is not possible to derive formulae containing exponential, trigonometric and logarithmic functions using this method. The formula which has more than one term that is added or subtracted like s = ut+ ½ at2 also cannot be derived using this Dimensional Formula.


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

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

  • 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


  • 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. 


  • 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:

  • 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.

FAQs on Dimensional Formula of Power and its Derivations

1. What  is Dimension?

The powers through which fundamental units can be raised in order to obtain one unit of the particular quantity, can be defined as Dimension. With the help of Dimension we can know how and  which way different physical quantities are related.


The relation between derived quantities and fundamental quantities, dimensions are not related to Magnitude of quantities.

2. Name a few quantities which have the same Dimensional Formula?

The list is quite long as there are a lot of quantities with the same Dimensional Formula.


However, a few important ones are : 

Impulse and momentum,  work, and the moment of force and energy,  Stress, pressure, the  modulus of elasticity and energy density, the Force constant, surface tension and surface energy,  Frequency and  velocity gradient as also the thermal capacity, entropy and the universal gas constant. The  gravitational potential along with latent heat, Power and flux etc. are some quantities with similar Dimensional Formula.

3. What is Power?

The rate of doing some work in unit time is Power. Watt(W) is the standard unit of Power and is also written as Joules per second( J/s). The power that is required to do one joule of work in one second is what we understand as Watts. The rate of doing work is Power and it is calculated by dividing  the work done by time, ie,

 

P =w/t. Here, ‘P’ is the power, and ‘W’ is the work done and  the time taken is ‘T’ .

Not having any direction, Power is said to be a Scalar quantity.

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