- Uniformly Accelerated Motion Calculator
- Electron Gain Calculator
- Gravitational Force Calculator
- Energy In A Lc Circuit Calculator
- Power Factor In A Rlc Circuit Calculator
- Dot (Scalar) Product of Two Vectors Video Tutorial
- Aberration in Lenses
- Conservation Of Momentum In 2 D Calculator
- Image Position And Magnification In Curved Mirrors And Lenses Calculator
- Friction on Inclined Plane Calculator
- Gravitational Field Strength Calculator

Welcome to our Physics lesson on **Dimensional Analysis**, this is the fourth lesson of our suite of physics lessons covering the topic of **Length, Mass and Time**, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.

Dimensional analysis is a method of analysis in which physical quantities are expressed in terms of their fundamental dimensions that is often used when there is not enough information to set up precise equations.

For example, let's suppose you don't remember the equation of velocity v in a uniform motion (the correct equation is v = Δx / t where Δx is the displacement and t is the moving time). You are somehow puzzled whether the correct formula is the abovementioned one, or it is v = Δx × t or v = t /Δx. In this case, you can use the dimensional analysis to remove the doubts if you know the unit of velocity. Thus, giving that the unit of velocity is metre per second [m / s] where metre is the unit of displacement Δx and second is the unit of time t, it is obvious the equation of velocity is v = Δx / t.

However, Dimensional Analysis has an important drawback. Through this method, we are not able to identify dimensionless constants in a formula. For example, if we know the unit of energy is joule [J], we are not able to find the correct formula of the kinetic energy (KE) as

[J] = [N × m] = [*kg × m* */**s*^{2} × m ]= [*kg × m*^{2} */**s*^{2} ] = [*kg × (**m* */**s*)^{2} ]

(Remember that 1 N = 1 kg × ** m/s**2). Giving that [kg] is the unit of mass m and [

KE=m × v^{2}

However, the correct formula of kinetic energy (see the article: Kinetic Energy), is

KE=1/2 m × v^{2}

Hence, we missed the constant dimensionless constant 1/2 when tried to find the equation based on the dimensional analysis. Therefore, to be correct in our response, we must write

KE ~ m × v^{2}

(the symbol ̴ means "is proportional to"). Thus, the correct way to write the equation of KE found by using the dimensional analysis method is

KE=C × m × v^{2}

where C is a dimensionless constant. In the specific case, C = 1/2.

A student believes the speed of sound v in a gas depends on three factors: pressure p density of the medium ρ and the volume V of the medium. He also has determined the approximate equation

v=C × p^{x} × ρ^{y} × V^{z}

where C is a dimensionless constant.

- Calculate the values of x, y and z.
- Does the speed of sound depend on all the above 3 factors?
- Write the correct formula for the speed of sound in a gas.

Pressure = *Force**/**Area*

Density = *Mass**/**Volume*

Force = Mass × Acceleration

Acceleration = *Change in velocity**/**Time*

Speed = *Distance**/**Time*

First, let's write all units in terms of metre, kilogram and second. We have

Unit of speed = *Unit of distance**/**Unit of time*

In symbols,

Unit of speed=[*m**/**s*]

Also,

Unit of pressure = *Unit of force**/**Unit of area* = *Unit of mass × Unit of acceleration**/**Unit of area*

= *Unit of mass × **Unit of velocity**/**(Unit of time))**/**(Unit of area)* = *Unit of mass × Unit of velocity**/**Unit of area × Unit of time*

Giving that speed and velocity have the same unit [** m/s**], we obtain after substituting the symbols of the respective units

Unit of pressure = [*kg × **m**/**s*)*/**(m*^{2} × s ] = [*kg**/**m × s*^{2}]

The same method is used for the other quantities as well. Thus,

Unit of density = *Unit of mass**/**Unit of volume*

In symbols, we have

Unit of density = [*kg**/**m*^{3}]

where [m^{3}] is obviously the unit of volume. Hence, combining all these units, we obtain

[*m**/**s*]=[*kg**/**m × s*^{2}]^{x} × [*kg**/**m*^{3}]^{y} × [*m*^{3}*/**1*]^{z}

[*m*^{1}*/**s*^{1} ] = [*kg*^{x} × kg^{y} × m^{3z}*/**m*^{x} × s^{2x} × m^{3y}]

[*m*^{1}*/**s*^{1} ] = [*kg*^{x+y} × m^{3z})*/**m*^{x+3y} × s^{2x} ]

[*m*^{1}*/**s*^{1} ] = [*kg*^{x+y} × m^{3z-x-3y}*/**s*^{2x} ]

Hence, we obtain three different equations:

From the third equation, we can find the value of x (x = 1/2). Also, since in the first equation we have x + y = 0=>y = - x = - 1/2.

Therefore, substituting the above two values in the second equation, we obtain

3z - *1**/**2* - 3 × - *1**/**2* = 1

3z - *1**/**2* + *3**/**2* = 1

3z + *2**/**2* = 1

3z + 1 = 1

3z = 0

z = 0

Since the volume's exponent is zero (z = 0), the speed of sound does not depend on the volume. This means the volume does not appear in the formula of the sound speed in a gas.

The correct formula is

v = C × p^{1/2} × ρ^{ - 1/2} = C × *p*^{1/2}*/**ρ*^{-1/2} = C × (*p**/**p*)^{1/2}

Giving that m^{1/2} = √x, we obtain

v = C × √(*p**/**p*)

where C is a dimensionless constant.

Enjoy the "Dimensional Analysis" physics lesson? People who liked the "Length, Mass and Time lesson found the following resources useful:

- Dimensional Analysis Feedback. Helps other - Leave a rating for this dimensional analysis (see below)
- Units and Measurements Physics tutorial: Length, Mass and Time. Read the Length, Mass and Time physics tutorial and build your physics knowledge of Units and Measurements
- Units and Measurements Video tutorial: Length, Mass and Time. Watch or listen to the Length, Mass and Time video tutorial, a useful way to help you revise when travelling to and from school/college
- Units and Measurements Revision Notes: Length, Mass and Time. Print the notes so you can revise the key points covered in the physics tutorial for Length, Mass and Time
- Units and Measurements Practice Questions: Length, Mass and Time. Test and improve your knowledge of Length, Mass and Time with example questins and answers
- Check your calculations for Units and Measurements questions with our excellent Units and Measurements calculators which contain full equations and calculations clearly displayed line by line. See the Units and Measurements Calculators by iCalculator™ below.
- Continuing learning units and measurements - read our next physics tutorial: Significant Figures and their Importance

We hope you found this Physics lesson "Length, Mass and Time" useful. If you did it would be great if you could spare the time to rate this physics lesson (simply click on the number of stars that match your assessment of this physics learning aide) and/or share on social media, this helps us identify popular tutorials and calculators and expand our free learning resources to support our users around the world have free access to expand their knowledge of physics and other disciplines.