# Dimensional Analysis Calculator + Online Solver With Free Steps

Dimensional Analysis Calculator is an online tool that helps analyze the dimensions of physical quantities belonging to the same class. The **calculator **takes the details of two physical quantities as input.

**Dimensional analysis** is a technique in which physical quantities are expressed in the form of basic dimensions. It determines the relation between quantities using their units and dimensions in real-life problems where they are related to each other.

The calculator is capable of doing unit conversions, unit comparisons, and calculating the total of two physical quantities.

## What Is a Dimensional Analysis Calculator?

**A Dimensional Analysis Calculator is an online tool that is used to perform dimensional analysis of mathematical problems by bringing the involved physical quantities to the same scale.**

**Dimensional analysis** means equalizing the **units** of all those quantities in a problem that represent the same thing but have different units. For instance, two quantities represent weight in different units, so it will convert both quantities into one identical unit.

Due to this reason, it is widely used by researchers in the fields like **physics**, **chemistry**, and **mathematics** as it helps them to manipulate and reduce the complexity of the problem.

It seems to be an easy process but you must have a prior vast knowledge about all the units, the relation between the units, and what is the process of converting one unit to the other.

You don’t have to go through the above hectic process if you use the **Dimensional Analysis Calculator**. This calculator will do dimensional analysis for your problem quickly and will give you the perfect results.

This online **calculator** is readily available in the browser, you can get it by searching just like you search for anything else on the internet. Therefore it frees you from doing any downloading and installation.

Moreover, the functionality of the **calculator** is very simple. You don’t need any skill to use this calculator because the interface is super friendly and easy to understand. Just enter the fields required and the rest of the task will be handled by the calculator.

## How To Use the Dimensional Analysis Calculator?

You can use the **Dimensional Analysis Calculator** by inserting various physical quantities in the respective boxes. The calculator is reliable and efficient as it provides you with the most accurate and precise solutions.

The calculator can take at most **two** physical quantities at one time and both quantities should represent the same dimension. Once you fulfill these requirements then you are **ready** to use the calculator.

Now to achieve optimal performance of the calculator, you may follow the given step-by-step guidelines:

### Step 1

Enter the first quantity in the **Physical Quantity 1 **box. It should have a numerical value and a valid unit.

### Step 2

Now insert the second quantity in the **Physical Quantity 2 **field with a value and unit.

### Step 3

Finally, click the **Submit **button for getting the results.

### Result

First of all, the calculator gives the interpretation of the insert quantities, then the unit of both quantities is made equivalent in the **Unit Conversion** tab. It can convert the second quantity’s unit equal to the unit of the first quantity or the other way around. Both scenarios are shown in the solution.

Also, the calculator compares the first quantity with the second one and describes the relationship between the two quantities in the **Comparisons **tab.

It explains how many **times** the first quantity is either smaller or greater than the second quantity and how much the first quantity is less or more than the second quantity in terms of **unit**.

Last, the **Total **section displays the sum of the quantities in both units. The calculator can perform unit conversions for any kind of quantity like length, mass, time, angle, volume, electric current, etc.

## How Does the Dimensional Analysis Calculator Work?

The Dimensional Analysis calculator works by finding the **comparison** and** relationship** between different physical quantities and by identifying base quantities and units of measure. It determines the dimensional consistency of physical quantities.

It **converts** the units and simplifies the ratio of given physical quantities. This calculator converts the lowest unit of measurement to a higher unit of measurement and a higher unit of measurement to the lowest unit.

To better understand the working of the calculator we should know what is the dimensional analysis and what are its applications.

### What Is Dimensional Analysis?

Dimensional analysis is the study of the **relationship **between different physical quantities based on their **dimensions** and** units**. This analysis helps to determine the relationship between two physical quantities.

The need for this analysis is because only those quantities can be added or subtracted which have the **same **units therefore the units and dimensions should be the same while solving mathematical and numerical problems.

### Base and Derived Units

There are two types of physical quantities:** base **quantities and** derived **quantities. Base quantities are those which have **base** units and they are not derived from any other quantity, whereas derived quantities are obtained from combining two or more base quantities and they have **derived** units.

There are **seven** base quantities and their corresponding units are called base units. These quantities are length, mass, time, electric current, temperature, amount of substance, and luminous intensity.

Their corresponding base units are meter (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mole), and candela (cd). Other than these seven base units, all units are derived.

### Conversion Factor

A **conversion factor** is a number that is used to change the set of units of one quantity to another by **multiplying **or** dividing**. This conversion factor is important because when the conversion of units becomes mandatory, then a suitable factor must be used.

The dimensional analysis is also called the **Factor Label Method** or** Unit Factor Method** because to find the dimensions or units, the conversion factor is used.

The conversion factor is used for the conversion within imperial units, within the System International units (SI). It can also be used for the conversion between SI units and imperial units.

However, the conversion of units must take place within the **same **physical quantities as it is impossible to convert units of different quantities. To change the time measurement from minutes to hours, the conversion factor of 1hr=60mins will be used.

**Time in hours = time in minutes*(1 hr/60 mins)**

Here (1hr/ 60mins) is the conversion factor.

### Principle of Homogeneity of Dimension

The principle of homogeneity of dimensions states that “For an equation to be dimensionally correct, the dimension of each term on the left side of the equation must be **equa**l to the dimension of each term on the right side.”

It means that the equation can not represent the physical units if the dimensions on** both sides** are not the same. For instance, the equation X+Y=Z is dimensionally correct if and only if the dimensions of X, Y, Z are the same.

The basis of this principle is the rule that two physical quantities can be added, subtracted, or compared if they have the exact dimensions. To check if the equation P.E= mgh is dimensionally correct, compare the dimension on both sides.

Dimensions of P.E (LHS)= $[ML^2T^-2]$

Dimensions of mgh (RHS)= $[M][LT^-2][L]= [ML^2T^-2]$

Since the dimensions on both sides are the same, this equation is dimensionally correct.

### Methods of Dimensional Analysis

There are different methods of dimensional analysis, which are explained below.

#### Simple Conversion Factors

This method allows algebraic simplification while analysis because the conversion factor is placed in the form of a **fraction** so that the desired unit is in the numerator and the converting unit is in the denominator.

This arrangement is done to algebraically cancel the converting units and obtain the desired unit. For example, to convert km to m%, the conversion factor should be in the form of m/km.

#### Multi-Dimensional Conversion

The multi-dimensional conversion is mostly of derived physical quantities. If the unit conversion includes multi-dimensional quantity then the conversion factor is also applied corresponding** multiple times**.

For instance, the volume of a cube is Length x Width x Height. The volume is a derived quantity, and its derived units are cubic meters ($m^3$), cubic centimeters ($cm^3$), cubic decimeters ($dm^3$), and cubic feet ($ft^3$)

Now in the conversion of cubic meters to cubic feet, the conversion factor is 3.28ft/1m. This factor will be multiplied by three** times** to convert the cubic meters to cubic feet.

#### Fractional Unit Conversion

Fractional units are those which are in** fraction **form. When these units are needed to be converted into some other fractional unit, then the conversion factor must be applied to both the** numerator** and** denominator** of the given fractional unit.

To illustrate this type of conversion, suppose the conversion of km/h to m/s is required. Since the given unit is in fractional form, the conversion factor is applied to the numerator and denominator.

As we know, 1km=1000m and 1h=3600s, therefore the conversion factor is **1000m/3600s**. This factor will be multiplied by a given fractional unit to obtain the desired unit in m/s.

## Applications of Dimensional Analysis

Dimensional Analysis is the main feature of the measurement. It has many applications in physics and mathematics which are listed below.

- It is used to determine the consistency of a dimensional equation through the principle of homogeneity. The equation will be consistent if the dimension on the
**left-hand side**is equal to the**right-hand side**. - This analysis is useful in determining the nature of physical quantity.
- Dimensional analysis is applied when there is a need to convert the value of a physical quantity from one system of units to another system of units.
- It is easy to find the dimensions of any quantity because the dimension expressions can be operated as algebraic quantities.
- This analysis is convenient in deriving the relation between physical quantities in physical phenomena.
- It is used to derive formulas.

## Limitations of Dimensional Analysis

Dimensional analysis is useful but there are also some limitations to this analysis. These limitations are given below:

- The dimensional analysis
**does not**give knowledge about the dimensional constant. The dimensional constant is a physical quantity that has dimensions but has a fixed value such as Planck’s constant and gravitational constant. - This analysis can not derive exponential, logarithmic, and trigonometric functions.
- It does not provide information about the scalar or vector identity of a physical quantity.
- Dimensional analysis can not derive any formula of that physical quantity that depends on
**more than three**factors having the dimensions. - This method can not be used to derive relations other than the product of power functions.

## History of Dimensional Analysis

Dimensional Analysis has an interesting history and many researchers made their contributions to its development. For the first time, an article by Francois Daviet has been cited as the written application of dimensional analysis.

As a result, it was determined that the equations of all the fundamental laws must be homogeneous in terms of the units used to measure the quantities involved. This concept was then observed in the Buckingham theorem.

In 1822, a theory was developed by Joseph Fourier that the physical principle such as $F=ma$ should be independent of the quantifying units for their physical variables. Later in 1833, the term dimension was established by Simeon Poisson.

The concept of dimensional analysis was further modified when James Clerk Maxwell declared mass, time, and length as the basic units. The quantities other than these were considered as derived. The Mass, Length and Time was represented by the units M, T, and L respectively.

Therefore using these fundamental units he derived units for other quantities as well. He determined the dimension of Gravitational Mass as $M = T^{-2} L^{3}$. Then the unit for the electrostatic charge was defined as $Q = T^{-2} L^{3/2} M^{1/2}$.

If the dimensions derived for Mass above are entered in the formula for Q, then its new dimension would be equal to $Q=T^{-2} L^{3}$ which is the same as that of the original mass.

Afterward, Lord Rayleigh published the dimensional analysis method in one of his works in 1877. The actual meaning of the word dimension is the value of exponents of base units that was presented in Fourier’s Theorie de la Chaleur.

But Maxwell proposed that dimensions will be the unit with the exponents in their power. For instance, the dimension for speed is 1 and -1 with respect to length and time respectively. But according to the Maxwell theory, it is represented as $T^{-1} L^{1}$.

But nowadays in physics, there are seven quantities that are considered the base. The rest of the physical quantities are derived using these bases.

## Solved Examples

The best way to check the performance of the **Dimensional Analysis Calculator** is to observe the examples solved by the calculator. Here are some examples for your better understanding:

### Example 1

Consider the two given physical quantities:

**P1 = 10 mi**

** P2 = 1 km **

Find the **relationship** between two quantities.

### Solution

The calculator shows the following results:

#### Input Interpretation

The interpretation of the calculator is shown as the ratio of two quantities with their units:

**10 miles | 1 meter **

#### Unit Conversions

The units of the quantities are made the same in this section. There are two ways for unit conversions. Let’s have a look at each of them.

One way is to represent two quantities in the bigger unit.

**10 mi : 0.6214 mi **

The other way is to convert both quantities into smaller units.

**16.09 km : 1 km **

#### Unit Comparison

The relationship between quantities is determined by comparing them. The first method is to show how much the quantities are different from each other.

**10 mi is 16.09 times larger than 1 km**

The second method describes the relation in terms of units.

**10 mi is 9.379 mi more than 1 km **

#### Total

In this section, it adds the two quantities and the resultant quantity is represented in both units.

**10.62 mi**

**17.09 km **

### Example 2

Let’s take below physical quantities that represent mass.

**P1 = 500 g **

**P2 = 20 lb **

Compare them using** Dimensional Analysis Calculator**.

### Solution

#### Input Interpretation

The interpretation of the calculator is shown as the ratio of two quantities with their units:

**500 grams : 20 lb (pounds) **

#### Unit Conversions

Both ways of unit conversion for the problem are shown below:

** 500 g : 9072 g **

** 1.102 lb : 20 lb **

#### Unit Comparison

The quantities are compared with each other. It describes how much 500 grams differs from the 20 pounds both in terms of ratio and units.

**500 g is 0.05512 times smaller than 20 lb **

**500 g is 8572 less than 20 lb **

#### Total

The sum of the input quantities is:

**9572 g **

**21.1 lb **

### Example 3

A maths student is given two quantities that represent angles.

**P1 = 2 radians **

**P2 = 6 degrees **

The student is asked to perform a **dimensional analysis** for this problem.

### Solution

The solution can be quickly obtained using **Dimensional Analysis Calculator**.

#### Input Interpretation

The calculator’s interpretation:

**2 radians : $6^{\circ}$ (degrees) **

#### Unit Conversions

The quantities are converted to one single unit.

**2 rad : 0.1047 rad **

**$114.6^{\circ}$ : $6^{\circ}$**

#### Unit Comparison

The comparison of the units clears the relation between the two quantities which is given as:

**2 rad is 19.1 times larger than $6^{\circ}$ **

**2 rad is 1.895 rad more than $6^{\circ}$ **

#### Total

The two quantities are first added and then demonstrated in both dimensions.

**2.105 rad **

**$126.6^{\circ}$**