# Dimensional Analysis Calculator

The Dimensional Analysis Calculator is a free online tool that analyses the dimensions for two given physical quantities. STUDYQUERIES’S online dimensional calculator tool makes the calculation faster, and it analyses the two physical quantities in a fraction of seconds.

## How to Use the Dimensional Analysis Calculator?

To use the Dimensional Analysis calculator, follow these steps:

• Step 1: Enter two physical quantities in their respective input fields
• Step 2: To receive the analysis, click “Submit”
• Step 3: In the new window, you will see the dimensional analysis

## A Step by Step Guide to Dimensional Analysis

The dimension of any physical quantity expresses its dependence on the base quantities as a product of symbols (or powers of symbols) representing the base quantities. The table lists the base quantities and the symbols used for their dimension.

For example, a measurement of length is said to have dimension $$[L]\ or\ [L^1]$$, a measurement of mass has dimension $$[M]\ or\ [M^1]$$, and a measurement of time has dimension $$[T]\ or\ [T^1]$$

Like units, dimensions obey the rules of the algebra. Thus, the area is the product of two lengths and so has dimension $$[L^2]$$, or length squared. Similarly, volume is the product of three lengths and has dimension $$[L^3]$$, or length cubed. Speed has dimension length over time, $$\left[\frac{L}{T} \right]\ or\ [L^1T^{-1}]$$ Volumetric mass density has dimension $$\left[\frac{M}{L^3} \right]\ or\ [M^1L^{–3}]$$, or mass over length cubed. In general, the dimension of any physical quantity can be written as

$$\left[{M^a}{L^b}{T^c}{I^d}{\Theta^e}{N^f}{J^g} \right]$$

for some powers a, b, c, d, e, f, and g. We can write the dimensions of a length in this form with a = 1 and the remaining six powers all set equal to zero:

$$\left[M^1\right] = \left[{M^1}{L^0}{T^0}{I^0}{\Theta^0}{N^0}{J^0}\right]$$

Any quantity with a dimension that can be written so that all seven powers are zero $$(that\ is\, its\ dimension\ is\ \left[{M^1}{L^0}{T^0}{I^0}{\Theta^0}{N^0}{J^0}\right] )$$ is called dimensionless (or sometimes “of dimension 1,” because anything raised to the zero power is one). Physicists often call dimensionless quantities pure numbers.

$$Base\ Quantity\longrightarrow Symbol\ for\ Dimension$$
$$Length\longrightarrow [L]$$
$$Mass\longrightarrow [M]$$
$$Time\longrightarrow [T]$$
$$Electric\ Current\longrightarrow [I]$$
$$Thermodynamic\ Temperature\longrightarrow [\Theta]$$
$$Amount\ of\ Substance\longrightarrow [N]$$
$$Luminous\ Intensity\longrightarrow [J]$$

Physicists often use square brackets around the symbol for a physical quantity to represent the dimensions of that quantity.

For example, if r is the radius of a cylinder and h is its height, then we write [r] = L and [h] = L to indicate the dimensions of the radius and height are both those of length, or L. Similarly if we use the symbol A for the surface area of a cylinder and V for its volume, then [A] = L² and [V] = L³. If we use the symbol m for the mass of the cylinder and ρ for the density of the material from which the cylinder is made, then [m] = M and [ ρ ] = ML−3.

Dimension plays a vital role in mathematics since any equation relating to physical quantities must be dimensionally consistent, which means that it must obey the following rules:

• The terms of an expression must all have the same dimensions; one cannot add or subtract quantities of different dimensions (think of the old saying, “You can’t add apples and oranges”). The expressions on both sides of equality in an equation must have the same dimensions.
• The arguments of any standard mathematical functions such as trigonometric functions (such as sine and cosine), logarithms, or exponential functions must be dimensionless. The functions take pure numbers as inputs and return pure numbers as outputs.

How to Write a Literary Analysis Essay

An equation that violates either of these rules cannot possibly be a true statement of physical law because it cannot be dimensionally consistent. You can use this simple fact to check for typos or algebraic mistakes, to help remember the various laws of physics, and even to propose the form that new laws of physics might take. You will undoubtedly learn about this last use of dimensions later in your academic career since it is beyond the scope of this text.

### Using Dimensions to Remember an Equation

Suppose we need the formula for the area of a circle for some computation. Like many people who learned geometry too long ago to recall with any certainty, two expressions may pop into our minds when we think of circles: $$\pi r^2\ and\ 2\pi r$$ One expression is the circumference of a circle of radius r and the other is its area. But which is which?

Strategy

Searching for information from a reputable source could take some time, but this would be a natural strategy. Furthermore, even if we trust the source, we shouldn’t believe everything we read. I like that I can double-check something just by thinking about it.

We might also be in a situation where we cannot look things up (such as during a test). By using the fact that dimensions follow algebraic rules, the strategy is to find the dimensions of both expressions. Either expression cannot possibly be the correct equation for the area of a circle if it does not have the same dimensions as the area.

Solution

We know the dimension of the area is $$[L^2]$$ Now, the dimension of the expression πr² is

$$[\pi r^2]=[π]⋅[r]^2=1⋅L^2=L^2$$

since the constant π is a pure number and the radius r is a length. Therefore, πr² has the dimension of the area. Similarly, the dimension of the expression 2πr is

$$[2\pi r]=[2]⋅[\pi]⋅[r]=1⋅1⋅L=L$$

since the constants 2 and π are both dimensionless and the radius r is a length. We see that 2πr has the dimension of length, which means it cannot possibly be an area.

We rule out 2πr because it is not dimensionally consistent with being an area. We see that πr² is dimensionally consistent with being an area, so if we have to choose between these two expressions, πr² is the one to choose.

Significance

It may seem like a silly example, but these ideas are very general. When we know the dimensions of the physical quantities in an equation, we can check whether it is dimensionally consistent. Alternatively, since true equations are dimensionally consistent, we can match expressions from imperfect memories to quantities for which they might be expressions.

While this will not help us remember dimensionless factors that appear in the equations (for example, if you accidentally conflated the two expressions from the example into 2πr²), it will help us remember the correct basic form of the equation.

### Conversion Ratios

It is possible to make simple unit conversions mentally or with a simple one-step multiplication or division. There are, however, more complex conversions that require several conversions between numerous units.

A conversion ratio (or unit factor) is a ratio equal to one. This ratio carries the names of the units to be used in the conversion. It can be used for conversions within the English and Metric Systems, as well as for conversions between the systems. The conversion ratio is based upon the concept of equivalent values. In the example below, one foot is substituted for its equivalent measure of 12 inches.

$$\frac{12\ Inches}{12\ Inches}=\frac{1\ Foot}{12\ Inches}$$

The main idea in Dimensional Analysis is to create a conversion ratio (unit factor) that has the units you want in the numerator and the units you already have in the denominator. It may be necessary to multiply by more than one conversion ratio in more difficult problems. Remember that you are setting up for one, or more, of the units to cancel until only the desired units remain. ALL conversion ratios (unit factors) must equal one.

$$Units\ You\ Have\times \frac{Units\ You\ Want}{Units\ You\ Have}=Units\ You\ Want$$

### Unit Conversion

We use conversion factors to get the same units when performing dimensional analysis, which is also called Factor Label Method or Unit Factor Method. How many meters does it take to make 3 kilometers? Let’s use an example to help you better understand the statement.

We know that 1000 meters make 1 km, Therefore, 3 km = 3 × 1000 meters = 3000 meters

Here, the conversion factor is 1000 meters.

### Using Dimensional Analysis to Check the Correctness of Physical Equation

Let’s say that you don’t remember whether

time = speed/distance, or
time = distance/speed

We can check this by making sure the dimensions on each side of the equations match.

Reducing both the equations to their fundamental units on each side of the equation, we get

$$[T]=\left[\frac{LT^{-1}}{L}\right]=[T^{-1}]\ (Wrong)$$

$$[T]=\left[\frac{L}{LT^{-1}}\right]=[T^{1}]\ (Right)$$

However, it should be kept in mind that dimensional analysis cannot help you determine any dimensionless constants in the equation.

### Homogeneity Principle of Dimensional Analysis

According to Homogeneity, each term in a dimensional equation should have the same dimensions on both sides. Our ability to convert units from one form to another is made possible by this principle. Let’s see an example to better understand the concept:

Check the correctness of the physical equation $$s=ut+\frac{1}{2}at^2$$ In the equation, s is the displacement, u is the initial velocity, v is the final velocity, a is the acceleration and t is the time in which change occurs.

We know that $$L.H.S = s\ and\ R.H.S = ut+\frac{1}{2}at^2$$

The dimensional formula for the L.H.S can be written as $$s = [L^1M^0T^0]\longrightarrow(1)$$

We know that R.H.S is ut + ½ at² , simplifying we can write R.H.S as [u][t] + [a] [t]²

$$R.H.S=[L^0 M^0 T^{-1}][L^0 M^0 T^1] +[L^1 M^0 T^{-2}][L^0 M^0 T^2]$$

$$=[L^1M^0T^0]\longrightarrow(2)$$

From (1) and (2), we have [L.H.S] = [R.H.S]

Hence, by the principle of homogeneity, the given equation is dimensionally correct.

### Applications of Dimensional Analysis

Real-life physics relies on dimension analysis as a fundamental aspect of measurement. There are three major reasons for using dimensional analysis:

• Checking the consistency of a dimensional equation
• To derive the relationship between physical quantities in physical phenomena
• Changing units from one system to another

### Limitations of Dimensional Analysis

Dimensional analysis has the following limitations:

• It does not provide information about the dimensional constant.
• Formulas containing trigonometric functions, exponential functions, logarithmic functions, etc. cannot be derived.
• The function doesn’t indicate whether a physical quantity is a scalar or a vector.

## Dimensional Analysis Calculator With Steps:

A dimensional analysis calculator with steps is a tool that helps users perform dimensional analysis calculations by providing step-by-step instructions and guidance. It allows users to input values with specific units and obtain the desired result with the correct units. This type of calculator is particularly useful in scientific and engineering fields where units and dimensions play a crucial role.

Example: Suppose you need to convert a length measurement from meters (m) to feet (ft). Using a dimensional analysis calculator with steps, you would input the value in meters and follow the provided steps, which might include multiplying the value by a conversion factor of 3.28084 to obtain the equivalent length in feet.

Solution: By using a dimensional analysis calculator with steps, you can follow a systematic approach to perform conversions or calculations involving units. The calculator guides you through the necessary steps, ensuring accuracy and helping you understand the process.

## Dimensional Analysis Converter Calculator:

A dimensional analysis converter calculator is a tool that allows users to convert measurements between different units using dimensional analysis principles. It provides a user-friendly interface where you can input the value and select the initial and desired units for conversion. The calculator then performs the necessary calculations to convert the measurement accurately.

Example: Let’s say you have a temperature measurement in Celsius (°C) and want to convert it to Fahrenheit (°F). Using a dimensional analysis converter calculator, you would input the temperature value in Celsius and select Celsius as the initial unit and Fahrenheit as the desired unit. The calculator would then apply the appropriate conversion formula (e.g., °F = °C × 9/5 + 32) to obtain the converted temperature in Fahrenheit.

Solution: A dimensional analysis converter calculator simplifies the process of converting measurements between different units. It eliminates the need for manual calculations and reduces the chances of errors, providing quick and accurate results.

## Dimensional Analysis Chemistry Calculator:

A dimensional analysis chemistry calculator is a specialized tool used in chemistry to perform calculations involving units and dimensions. It helps chemists convert between different units, determine molar ratios, calculate concentrations, and solve other chemistry-related problems using dimensional analysis principles.

Example: Suppose you need to calculate the molar mass of a compound given its chemical formula. Using a dimensional analysis chemistry calculator, you would input the formula and follow the appropriate steps to determine the molar mass. The calculator might provide a breakdown of the formula, allowing you to calculate the molar mass by summing the atomic masses of the individual elements.

Solution: A dimensional analysis chemistry calculator streamlines the process of performing calculations in chemistry. It ensures that the correct units are considered, simplifying complex calculations and reducing the likelihood of errors.

## Dimensional Analysis Conversion Calculator:

A dimensional analysis conversion calculator is a versatile tool that allows users to convert measurements between different units by applying dimensional analysis principles. It can handle various types of conversions, such as length, volume, time, temperature, and more, based on the specific units and dimensions involved.

Example: Let’s say you want to convert a volume measurement from milliliters (mL) to liters (L). Using a dimensional analysis conversion calculator, you would input the volume value in milliliters and select milliliters as the initial unit and liters as the desired unit. The calculator would then perform the necessary calculations, such as dividing the value by 1000, to obtain the volume in liters.

Solution: A dimensional analysis conversion calculator offers a convenient way to convert measurements between different units. It eliminates the need for manual calculations and provides accurate results, making it a valuable tool in various fields, including science, engineering, and everyday life.

## Dimensional Analysis Calculator With Work Shown:

A dimensional analysis calculator with work shown is a tool that not only provides the final result of a dimensional analysis

calculation but also displays the step-by-step work or calculations involved in obtaining that result. It allows users to understand the process and ensures transparency in the calculation.

Example: Suppose you need to convert a speed measurement from kilometers per hour (km/h) to meters per second (m/s). Using a dimensional analysis calculator with work shown, you would input the speed value in kilometers per hour and follow the displayed calculations. The calculator might show the conversion factor used and the steps involved in multiplying the value by the conversion factor to obtain the speed in meters per second.

Solution: A dimensional analysis calculator with work shown helps users comprehend the calculation process and verify the accuracy of the results. It serves as an educational tool, especially for students and those learning dimensional analysis concepts.

## Dimensional Analysis Calculator With Steps Chemistry:

A dimensional analysis calculator with steps specifically tailored for chemistry provides a guided approach to perform chemistry-related calculations involving units and dimensions. It assists in solving problems related to chemical reactions, stoichiometry, concentrations, and other chemistry concepts using the principles of dimensional analysis.

Example: Let’s say you need to calculate the concentration of a solution in moles per liter (mol/L) given the mass of the solute and the volume of the solution. Using a dimensional analysis calculator with steps for chemistry, you would input the given values and follow the provided instructions. The calculator might guide you through the steps of converting the mass to moles using the molar mass and then dividing the result by the volume to obtain the concentration.

Solution: A dimensional analysis calculator with steps for chemistry enables users to apply dimensional analysis techniques to solve chemistry problems accurately. It ensures that the correct units are considered and provides a structured approach to tackling complex calculations in chemistry.

## Dimensional Analysis Calculator Moles:

A dimensional analysis calculator for moles is a tool specifically designed to perform calculations involving the unit of moles (mol). It allows users to convert between moles and other units, calculate molar ratios, determine the number of particles, or perform other calculations related to moles.

Example: Suppose you want to calculate the number of moles in a given mass of a substance. Using a dimensional analysis calculator for moles, you would input the mass value and follow the appropriate steps. The calculator might provide the molar mass of the substance and guide you through the process of dividing the given mass by the molar mass to obtain the number of moles.

Solution: A dimensional analysis calculator for moles simplifies calculations involving moles, enabling users to convert between different units or perform mole-related calculations accurately. It is particularly useful in stoichiometry problems and other calculations in chemistry.

## Dimensional Analysis Formula:

The dimensional analysis formula is a general framework or set of rules used to perform calculations involving units and dimensions. It helps ensure that the dimensions of the input values match the desired dimensions of the output and provides a systematic approach to solving dimensional analysis problems.

Example: The formula for converting Celsius (°C) to Fahrenheit (°F) using dimensional analysis is °F = °C × 9/5 + 32. This formula takes the input temperature in Celsius, multiplies it by 9/5, and then adds 32 to obtain the temperature in Fahrenheit.

Solution: The dimensional analysis formula acts as a guide to perform accurate calculations involving units and dimensions. By understanding and applying the appropriate formulas, users can convert between units, solve problems, and ensure consistency in their calculations.

## Density Dimensional Analysis Calculator:

A density dimensional analysis calculator is a tool specifically used to calculate the density of a substance or material using dimensional analysis principles. It allows users to input the mass and volume values and obtain the density with the correct units.

Example: Let’s say you have the mass of a substance in grams (g) and the volume in cubic centimeters (cm³). Using a density dimensional analysis calculator, you would input the given values and follow the necessary steps. The calculator might guide you through the process of dividing the mass by the volume to obtain the density in grams per cubic centimeter (g/cm³).

Solution: A density dimensional analysis calculator simplifies the calculation of density by considering the units of mass and volume. It ensures that the resulting density has the appropriate units and assists in solving problems related to density in various fields, including physics, chemistry, and materials science.

## Dimensional Analysis Chemistry:

Dimensional analysis in chemistry is a method used to perform calculations and conversions involving units and dimensions in chemical systems. It allows chemists to convert between different units, determine molar ratios, calculate concentrations, and solve other chemistry-related problems while maintaining the integrity of the units.

Example: Suppose you need to calculate the number of moles of a substance involved in a chemical reaction. Using dimensional analysis chemistry, you would use the balanced equation for the reaction to establish the molar ratios between the reactants and products. By applying dimensional analysis principles, you can convert between the given units (such as mass or volume) and moles, ensuring the units cancel out appropriately.

Solution: Dimensional analysis in chemistry provides a structured and consistent approach to solving problems and performing calculations in the field of chemistry. It helps maintain the correct units and dimensions throughout the calculations, ensuring accurate and meaningful results.

## Buckingham Pi Theorem Calculator:

The Buckingham Pi theorem calculator is a tool used in dimensional analysis to determine the dimensionless combinations (π terms) that describe a physical system. It helps identify the relevant variables and their relationships, allowing for the reduction of complex physical problems to simpler, dimensionless equations.

Example: Let’s consider the problem of determining the drag force on a sphere moving through a fluid. By using the Buckingham Pi theorem calculator, you would input the relevant variables such as the velocity of the fluid, the density of the fluid, the diameter of the sphere, and the viscosity of the fluid. The calculator would then determine the dimensionless π terms that govern the drag force, such as the Reynolds number (Re) and the drag coefficient (Cd).

Solution: The Buckingham Pi theorem calculator assists in simplifying complex physical problems by identifying dimensionless combinations of variables. It allows engineers and scientists to express relationships and dependencies in a more concise and general form, facilitating analysis, modeling, and design in various fields, including fluid mechanics, heat transfer, and structural engineering.

## FAQs

What is dimensional analysis?

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

What is the principle of Homogeneity of dimension?

The principle of homogeneity states that the dimensions of each of the terms of a dimensional equation on both sides are the same.

How many kilograms go into a gram?

1 gram (g) is equal to 0.001 kilograms (kg).

What is the purpose of dimensional analysis?

dimensional analysis, a technique used in the physical sciences and engineering to reduce physical properties, such as acceleration, viscosity, energy, and others, to their fundamental dimensions of length (L), mass (M), and time (T).

How To Solve Dimensional Analysis On A Calculator?

To solve dimensional analysis using a calculator, follow these steps:
1. Identify the given quantity and the desired quantity you want to convert or calculate.
2. Determine the conversion factors or formulas needed to convert between the units.
3. Input the given value into the calculator and multiply it by the appropriate conversion factors, ensuring that units cancel out correctly.
4. Perform the necessary arithmetic operations, such as multiplication or division, to obtain the final result.
5. Verify that the units of the final result are correct and make sense.

Example: Suppose you want to convert a distance of 5 kilometers (km) to miles (mi) using a calculator. You would input the value 5 into the calculator and multiply it by the conversion factor of 0.62137119 mi/km. The calculator would perform the calculation and provide the result of approximately 3.10685596 miles.

How To Do Dimensional Analysis On A Graphing Calculator?

Performing dimensional analysis on a graphing calculator follows a similar process to using a regular calculator. Here are the steps:
1. Identify the given quantity and the desired quantity you want to convert or calculate.
2. Determine the conversion factors or formulas needed to convert between the units.
3. Input the given value into the calculator and use the appropriate functions or operations to apply the conversion factors.
4. Follow the prompts or steps provided by the calculator to ensure the correct units cancel out.
5. Perform any additional calculations or operations to obtain the final result.
6. Verify that the units of the final result are correct.

Graphing calculators often have built-in functions and features that make dimensional analysis calculations more convenient, such as unit conversion functions or the ability to define custom formulas.

How To Do Dimensional Analysis Without Calculator?

While a calculator can be helpful, it is possible to perform dimensional analysis without one. Here’s how:
1. Identify the given quantity and the desired quantity you want to convert or calculate.
2. Determine the conversion factors or formulas needed to convert between the units.
3. Write down the given value and apply the conversion factors or formulas manually, ensuring that units cancel out correctly.
4. Perform any necessary arithmetic operations, such as multiplication or division, to obtain the final result.
5. Verify that the units of the final result are correct and make sense.

Doing dimensional analysis without a calculator may require more manual calculations and attention to detail, but it is still an effective method for solving dimensional analysis problems.

How To Convert 210 KPa To Psi Dimensional Analysis Calculator?

To convert 210 kilopascals (kPa) to pounds per square inch (psi) using a dimensional analysis calculator, follow these steps:
1. Identify the given value of 210 kPa.
2. Determine the conversion factor between kPa and psi. The conversion factor is approximately 0.145037738 psi/kPa.
3. Input the value 210 into the calculator and multiply it by the conversion factor.
4. The calculator will perform the calculation, resulting in approximately 30.23846418 psi.

Using the dimensional analysis calculator ensures that the units cancel out correctly and provides an accurate conversion from kPa to psi.

What Are The 3 Steps For Dimensional Analysis?

The three steps for dimensional analysis are as follows:
1. Identify the given quantity and the desired quantity you want to convert or calculate.
2. Determine the conversion factors or formulas needed to convert between the units or relate the given and desired quantities.
3. Apply the conversion factors or formulas by multiplying or dividing the given value appropriately, ensuring that the units cancel out correctly. Perform any necessary arithmetic operations to obtain the final result.

By following these three steps, dimensional analysis allows for systematic

and accurate conversion or calculation of quantities while maintaining the integrity of units.

What Is An Example Of A Dimensional Analysis?

An example of dimensional analysis is converting a distance from meters (m) to centimeters (cm). Using dimensional analysis, you would follow these steps:
1. Identify the given quantity: Let’s say the given distance is 5 meters.
2. Determine the conversion factor: The conversion factor between meters and centimeters is 100 cm/m.
3. Apply dimensional analysis: Multiply the given value of 5 meters by the conversion factor of 100 cm/m.
4. The result is 500 centimeters.

In this example, dimensional analysis ensures that the units of meters cancel out, leaving the desired unit of centimeters.

What Is Dimensional Analysis To Convert Quantity?

Dimensional analysis is a method used to convert or relate quantities by considering their units and dimensions. It involves using conversion factors, formulas, or ratios to perform calculations that maintain the integrity of units. By analyzing the dimensions of the given and desired quantities, dimensional analysis ensures that the units are consistent throughout the calculation, leading to accurate conversions.

For example, to convert a speed from kilometers per hour (km/h) to meters per second (m/s), dimensional analysis considers the relationship between the units of distance and time. By multiplying the given value by an appropriate conversion factor (e.g., 1 km = 1000 m and 1 hour = 3600 seconds), dimensional analysis ensures that the units cancel out correctly, resulting in the desired unit of meters per second.