dimensional analysis exercising problem solving skills

GKT101: General Knowledge for Teachers – Math

Practice with Dimensional Analysis

Complete these exercises and check your answers.

Practice - Dimensional Analysis

Use dimensional analysis to convert the following:

Use dimensional analysis to solve the following:

19) On a recent trip, Jan traveled 260 miles using 8 gallons of gas. How many miles per 1-gallon did she travel? How many yards per 1-ounce?

20) A chair lift at the Divide ski resort in Cold Springs, WY is 4806 feet long and takes 9 minutes. What is the average speed in miles per hour? How many feet per second does the lift travel?

21) A certain laser printer can print 12 pages per minute. Determine this printer's output in pages per day, and reams per month. (1 ream =5000 pages)

22) An average human heart beats 60 times per minute. If an average person lives to the age of 75, how many times does the average heart beat in a lifetime?

24) You are buying carpet to cover a room that measures 38 ft by 40 ft. The carpet cost $18 per square yard. How much will the carpet cost?

25) A car travels 14 miles in 15 minutes. How fast is it going in miles per hour? in meters per second?

26) A cargo container is 50 ft long, 10 ft wide, and 8 ft tall. Find its volume in cubic yards and cubic meters.

28) Computer memory is measured in units of bytes, where one byte is enough memory to store one character (a letter in the alphabet or a number). How many typical pages of text can be stored on a 700 -megabyte compact disc?

Assume that one typical page of text contains 2000 characters. (1 megabyte = 1,000,000 bytes)

29) In April 1996, the Department of the Interior released a "spike flood" from the Glen Canyon Dam on the Colorado River. Its purpose was to restore the river and the habitants along its bank. The release from the dam lasted a week at a rate of 25,800 cubic feet of water per second. About how much water was released during the 1 -week flood?

30) The largest single rough diamond ever found, the Cullinan diamond, weighed 3106 carats; how much does the diamond weigh in miligrams? in pounds?

(1 carat - 0.2 grams)

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Dimensional Analysis

• October 22, 2023

 Dimensional Analysis: 10 Practice Problems:

Ever found yourself struggling with complex problems that involve multiple units of measurement?

By understanding the importance of using units in calculations, you’ll unlock a powerful tool that simplifies problem-solving.

But don’t worry if you’re new to this concept – we’ve got you covered. Get ready to tackle some practice problems that will sharpen your skills and boost your confidence.

So, are you ready to embark on this journey?

Let’s buckle up and explore the wonders of dimensional analysis together.

Understanding conversion factors in dimensional analysis

Dimensional analysis is a method used in physics and engineering to convert units and solve problems involving different dimensions. Here are some practice problems for you to work on:

In dimensional analysis practice problems, conversion factors play a crucial role. These factors help us convert between different units of measurement and ensure accurate calculations.

Let’s delve deeper into understanding conversion factors and how to use them effectively.

Explore Conversion Factors and Their Role in Dimensional Analysis

Conversion factors are ratios that express the relationship between two different units of measurement. They act as a bridge, allowing us to convert from one unit to another within the same dimension.

For example, the conversion factor for converting inches to centimeters is 2.54 cm/inch.

Learn How to Identify and Use Conversion Factors Effectively

To identify a conversion factor, examine the relationship between the starting unit and the desired unit of measurement. Look for a known ratio or equivalence that can be used as a conversion factor.

Once you have identified the appropriate conversion factor, multiply it by the given value to obtain the desired result.

Understand the Relationship Between Different Units and Conversion Factors

Conversion factors are derived from established relationships between different units of measurement.

These relationships are based on fundamental constants or agreed-upon standards. Understanding these relationships allows us to navigate smoothly between various measurement systems.

Master the Art of Converting Between Different Measurement Systems

Converting between different measurement systems involves using multiple conversion factors in sequence.

By breaking down complex conversions into smaller steps, we can simplify the process and ensure accuracy. Start by converting from one unit to an intermediate unit using a conversion factor, then proceed to convert from that intermediate unit to the final desired unit.

By mastering conversion factors and their application in dimensional analysis, you will gain confidence in solving practice problems efficiently and accurately.

Applying dimensional analysis in chemical calculations

See how dimensional analysis is applied in chemistry for accurate calculations.

Dimensional analysis is a powerful tool used in chemistry to ensure accurate calculations. It allows us to convert between different units, such as grams, moles , and liters, by utilizing conversion factors.

By applying this method, we can confidently solve chemical problems and obtain precise results.

Learn how to convert between grams, moles, and other chemical units using dimensional analysis.

In dimensional analysis, we use conversion factors to move from one unit to another.

These conversion factors are derived from the relationships between different units.

For example, if we want to convert grams of a substance into moles, we can use the molar mass of the substance as a conversion factor. By multiplying the given quantity by the appropriate conversion factor, we can cancel out the unwanted units and obtain the desired unit.

Understand the significance of balancing equations when using dimensional analysis.

Balancing chemical equations is crucial when applying dimensional analysis. This step ensures that the ratios between different substances are accurately represented in the equation. Without a balanced equation, our calculations may yield incorrect results. Balancing equations involves adjusting coefficients so that both sides of the equation have an equal number of atoms for each element present.

Practice solving chemical problems with confidence.

To get better at using dimensional analysis for chemistry problems, you need to practice. Do lots of exercises and problems that involve converting and balancing equations. It’s okay to make mistakes because they can help you learn and become a better chemist!

Solving medical dosage problems using dimensional analysis

Discover how healthcare professionals use dimensional analysis for precise medication dosing..

Healthcare workers use a method called dimensional analysis to make sure patients get the right amount of medicine. They calculate the dosage based on things like weight and concentration.

Learn how to calculate correct dosages based on patient weight, concentration, and other factors.

When figuring out how much medicine to give, doctors and nurses have to think about a few things. They need to know how heavy the patient is and how strong the medicine is. By using dimensional analysis, they can change these things into the right units and figure out the right amount of medicine for each person.

Understand the importance of unit conversions in medical dosage calculations.

Unit conversions are a critical aspect of medical dosage calculations. When dealing with different units of measurement, such as milliliters and grams, it is essential to convert them to a common unit before performing any calculations. This ensures that accurate dosages are administered to patients based on their specific needs.

Gain practical skills for solving medical dosage problems accurately.

By mastering dimensional analysis techniques, healthcare professionals acquire practical skills for solving medical dosage problems accurately. They learn how to navigate complex calculations involving multiple units and measurements confidently. These skills enable them to provide precise medication dosing tailored to each patient’s requirements.

Mastering unit conversion and handling powers of units

Developing a solid foundation in unit conversions is essential for tackling dimensional analysis problems. By practicing comprehensive exercises, you can sharpen your skills and become a master at converting between different units. Learning techniques to handle powers of units will make conversions even easier.

Techniques for Handling Powers of Units

Converting between metric prefixes like kilo-, milli-, and micro- may seem daunting at first, but with dimensional analysis, it becomes a breeze. Here are some tips to help you handle powers of units effectively:

• Identify the prefix :   Determine the metric prefix associated with the given unit. For example, if you have grams (g), the prefix is “none” since it’s the base unit.
• Convert to base unit : If necessary, convert the given value to the base unit by moving the decimal point or adjusting the power of ten accordingly. For instance, if you have 1000 milliliters (ml), convert it to liters (L) by dividing by 1000.
• Apply conversion factors : Use conversion factors based on known relationships between units to perform the conversion accurately. For example, if you need to convert kilometers (km) to miles (mi), use the conversion factor 0.6214 mi/km.
• Adjust for powers of ten : When dealing with powers of ten during conversions, adjust both the numerical value and exponent appropriately while ensuring consistency across all units involved in the calculation.

By following these techniques, you’ll gain confidence in handling powers of units and be able to convert between different metric prefixes seamlessly using dimensional analysis.

Practice Makes Perfect

To truly master unit conversion and handling powers of units, practice is key! Engage in various exercises that involve converting measurements such as mass (grams or kilograms), length (meters or centimeters), volume (liters or milliliters), and speed (kilometers per hour or miles per hour). The more you practice, the more comfortable and proficient you’ll become in dimensional analysis.

So, embrace the challenge of unit conversions, sharpen your skills through practice, and conquer those dimensional analysis problems!

Dimensional analysis practice problems on moles and solutions

Apply knowledge of moles and solutions.

Ready to put your knowledge of moles and solutions to the test?

Get ready for some challenging practice problems that will help you solidify your understanding. By practicing dimensional analysis, you’ll become a pro at converting units and solving problems related to moles and solutions.

Solve Mole-to-Mole Ratios with Stoichiometry Principles

One important skill in chemistry is being able to calculate mole-to-mole ratios using stoichiometry principles.

This involves understanding the balanced chemical equation and using it to determine the ratio of moles between different substances. With practice, you’ll be able to confidently convert between different compounds and understand their relationships in chemical reactions.

Calculate Concentrations, Dilutions, and Molarity with Confidence

In solution chemistry, it’s crucial to be able to calculate concentrations , dilutions, and molarity accurately.

These calculations involve determining the amount of solute dissolved in a given volume of solvent. By mastering dimensional analysis, you’ll be able to confidently perform these calculations without breaking a sweat.

Strengthen Your Understanding of Solution Chemistry Concepts

By working through these dimensional analysis practice problems on moles and solutions, you’ll deepen your understanding of solution chemistry concepts.

You’ll gain more insight into how different substances interact when they dissolve in liquids like water or soda. This will enhance your overall comprehension of solution properties and their applications in various fields.

Get ready to practice stoichiometry, concentrations, dilutions, molarity, and solution chemistry. These problems will help you improve and ace your tests!

Quiz or worksheet to test understanding of dimensional analysis

Assess comprehension and problem-solving skills.

Ready to put your dimensional analysis skills to the test? Take a quiz or complete a worksheet to assess your understanding of this important concept. These interactive exercises will evaluate your ability to apply conversion factors, solve complex problems, and convert units effectively.

Evaluate Unit Conversions and Their Role

You can test your knowledge of unit conversions by doing a quiz or worksheet on dimensional analysis. These activities will make you convert between different units, like grams to moles or meters to kilometers. They will also help you understand how conversion factors work in dimensional analysis.

Immediate Feedback for Improvement

One of the great benefits of quizzes and worksheets is that you receive immediate feedback on your answers.

This feedback allows you to identify areas where you may need improvement, helping you focus on specific topics that require further practice.

Whether it’s converting density from g/cm³ to kg/m³ or determining the number of atoms in a sample using Avogadro’s number, these exercises provide an opportunity for self-assessment.

Sample Problems and Examples

Quizzes and worksheets often include sample problems and examples that illustrate the concepts covered

. These examples serve as valuable learning tools by providing step-by-step solutions and explanations. By working through these practice problems, you’ll gain confidence in applying dimensional analysis techniques to various scenarios.

So why wait? Dive into a quiz or worksheet today to enhance your understanding of dimensional analysis!

Benefits of practicing dimensional analysis

• Helps in solving complex mathematical problems
• Provides a systematic approach to problem-solving
• Improves understanding of units and conversions
• Enhances critical thinking skills
• Increases accuracy in calculations
• Builds confidence in mathematical abilities
• Useful in various scientific and engineering fields
• Facilitates clear communication of measurements and quantities

Conclusion:

Great job on finishing the sections before the conclusion! You’ve learned about dimensional analysis, a really useful tool for solving tough problems and doing accurate calculations. Practicing dimensional analysis has lots of benefits that can help you in many ways..

By learning conversion factors in dimensional analysis, you can easily change units of measurement.

This is super helpful in subjects like chemistry, physics, and engineering. Also, using dimensional analysis in chemical calculations helps you figure out amounts and concentrations in reactions. This is really important for getting the right results in experiments and manufacturing.

Using dimensional analysis to solve medical dosage problems helps make sure that medications are given correctly, which keeps patients safe.

Learning how to convert units and work with different measurements helps avoid mistakes. Doing practice problems on moles and solutions is a good way to get better at using dimensional analysis.

Now that you understand the benefits of practicing dimensional analysis, it’s time to take action! Challenge yourself further by attempting a quiz or worksheet to test your understanding. Keep honing your skills through regular practice, as this will enhance your problem-solving abilities across various disciplines.

Problem 1: Convert 25 miles per hour to meters per second.

Problem 2: The formula for the area of a rectangle is A = l × w , where A is the area, l is the length, and w is the width. If A is measured in square meters, l is measured in meters, and w is measured in centimeters, express the formula in consistent units.

Problem 3: The speed of light in a vacuum is approximately 3×1083×108 meters per second. Convert this speed to miles per hour.

Problem 4: The density of a substance is given by the formula ρ = Vm ​, where ρ is density, m is mass (in kilograms), and V is volume (in cubic meters). Express the density in units of grams per cubic centimeter.

Problem 5: The period T of a pendulum is given by the formula T =2 πgL ​​, where L is the length of the pendulum (in meters) and g is the acceleration due to gravity (in meters per second squared). Determine the dimensions of g in terms of length, mass, and time.

Problem 6: A car travels a distance of 150 miles in 2.5 hours. Calculate its average speed in both miles per hour and meters per second.

Problem 7: The kinetic energy E of an object is given by the formula E =21​ mv 2, where m is mass (in kilograms) and v is velocity (in meters per second). Express kinetic energy in terms of joules.

Problem 8: The pressure P exerted by a force F on an area A is given by the formula P = AF ​, where P is in pascals, F is in newtons, and A is in square meters. Express pressure in terms of atmospheres, where 1 atm=101325 Pa1 atm=101325 Pa.

Feel free to try solving these problems, and I can provide you with the solutions if you need them!

How can I improve my proficiency in dimensional analysis?

To improve your proficiency in dimensional analysis, start by thoroughly understanding conversion factors and their applications. Practice solving a variety of problems involving different units and dimensions regularly. Seek out online resources such as tutorials or videos that provide step-by-step explanations and examples.

Is it necessary to memorize all the conversion factors?

While memorizing some common conversion factors can be helpful initially, it is not essential to memorize them all. It is more important to understand how conversion factors work so that you can derive them when needed using basic principles like unit equivalences or dimensional analysis.

Can dimensional analysis be used in everyday life?

Absolutely! Dimensional analysis is a useful tool in everyday life. It can help you convert units when cooking, following recipes, or planning trips using different systems of measurement. It also comes in handy when comparing prices, calculating distances, or understanding scientific information presented with different units.

Are there any online resources for practicing dimensional analysis?

Yes, there are numerous online resources available for practicing dimensional analysis. You can find practice problems, quizzes, and interactive tutorials on educational websites, math and science forums, and even mobile apps dedicated to unit conversions and problem-solving.

How long does it take to become proficient in dimensional analysis?

The time required to become proficient in dimensional analysis varies from person to person. With regular practice and dedication, you can start feeling confident within a few weeks or months. However, mastery comes with continued engagement and application of the principles over an extended period of time.

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5.6 Using Dimensional Analysis

A common method used to perform calculations with different units of measurement is called dimensional analysis. Dimensional analysis is a problem-solving technique where measurements are converted to equivalent units of measure by multiplying a given unit of measurement by a fractional form of 1 to obtain the desired unit of administration. This method is also referred to as creating proportions that state equivalent ratios. Equivalencies described in Section 5.7 are used to set up ratios with the fractional form of 1 to achieve the desired unit the problem is asking for. The units of measure that must be eliminated to solve the problem are set up on the diagonal so that they can be cancelled out. Lines are drawn during the problem-solving process to show that cancellation has occurred. [1]

When setting up a dosage calculation using dimensional analysis, it is important to begin by identifying the goal unit to be solved. After the goal unit is set, the remainder of the equation is set up using fractional forms of 1 and equivalencies to cancel out units to achieve the goal unit. It is important to understand that when using this problem-solving method, the numerator and denominator are interchangeable because they are expressing a relationship. [2]  Let’s practice using dimensional analysis to solve simple conversion problems of ounces to milliliters in Section 5.7 “Conversions” to demonstrate the technique.

Review of Dimensional Analysis on YouTube [3]

• Esser, P. (2019). Dimensional analysis in nursing. Southwest Technical College. https://swtcmathscience.wixsite.com/swtcmath/dimensional-analysis-in-nursing ↵
• RegisteredNurseRN. (2015, February 4). Dimensional analysis for nursing & nursing students for dosage calculations nursing school  [Video]. YouTube. All rights reserved. Used with permission. https://youtu.be/6dyM2puXbgc ↵

Dimensional analysis is a problem-solving technique where measurements are converted to a different (but equivalent) unit of measure by multiplying with a fractional form of 1 to obtain a desired unit of administration.

Nursing Skills - 2e Copyright © 2023 by Chippewa Valley Technical College is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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1.6: Dimensional Analysis

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Learning Objectives

• To be introduced to the dimensional analysis and how it can be used to aid basic chemistry problem solving.
• To use dimensional analysis to identify whether an equation is set up correctly in a numerical calculation
• To use dimensional analysis to facilitate the conversion of units.

Dimensional analysis is amongst the most valuable tools physical scientists use. Simply put, it is the conversion between an amount in one unit to the corresponding amount in a desired unit using various conversion factors. This is valuable because certain measurements are more accurate or easier to find than others.

A Macroscopic Example: Party Planning

If you have every planned a party, you have used dimensional analysis. The amount of beer and munchies you will need depends on the number of people you expect. For example, if you are planning a Friday night party and expect 30 people you might estimate you need to go out and buy 120 bottles of sodas and 10 large pizza's. How did you arrive at these numbers? The following indicates the type of dimensional analysis solution to party problem:

\[(30 \; \cancel{humans}) \times \left( \dfrac{\text{4 sodas}}{1 \; \cancel{human}} \right) = 120 \; \text{sodas} \label{Eq1} \]

\[(30 \; \cancel{humans}) \times \left( \dfrac{\text{0.333 pizzas}}{1 \; \cancel{human}} \right) = 10 \; \text{pizzas} \label{Eq2} \]

Notice that the units that canceled out are lined out and only the desired units are left (discussed more below). Finally, in going to buy the soda, you perform another dimensional analysis: should you buy the sodas in six-packs or in cases?

\[(120\; { sodas}) \times \left( \dfrac{\text{1 six pack}}{6\; {sodas}} \right) = 20 \; \text{six packs} \label{Eq3} \]

\[(120\; {sodas}) \times \left( \dfrac{\text{1 case }}{24\; {sodas}} \right) = 5 \; \text{cases} \label{Eq4} \]

Realizing that carrying around 20 six packs is a real headache, you get 5 cases of soda instead.

In this party problem, we have used dimensional analysis in two different ways:

• In the first application (Equations \(\ref{Eq1}\) and Equation \(\ref{Eq2}\)), dimensional analysis was used to calculate how much soda is needed need. This is based on knowing: (1) how much soda we need for one person and (2) how many people we expect; likewise for the pizza.
• In the second application (Equations \(\ref{Eq3}\) and \(\ref{Eq4}\)), dimensional analysis was used to convert units (i.e. from individual sodas to the equivalent amount of six packs or cases)

Using Dimensional Analysis to Convert Units

Consider the conversion in Equation \(\ref{Eq3}\):

\[(120\; {sodas}) \times \left( \dfrac{\text{1 six pack}}{6\; {sodas}} \right) = 20 \; \text{six packs} \label{Eq3a} \]

If we ignore the numbers for a moment, and just look at the units (i.e. dimensions ), we have:

\[\text{soda} \times \left(\dfrac{\text{six pack}}{\text{sodas}}\right) \nonumber \]

We can treat the dimensions in a similar fashion as other numerical analyses (i.e. any number divided by itself is 1). Therefore:

\[\text{soda} \times \left(\dfrac{\text{six pack}}{\text{sodas}}\right) = \cancel{\text{soda}} \times \left(\dfrac{\text{six pack}}{\cancel{\text{sodas}}}\right) \nonumber \]

So, the dimensions of the numerical answer will be "six packs".

How can we use dimensional analysis to be sure we have set up our equation correctly? Consider the following alternative way to set up the above unit conversion analysis:

\[ 120 \cancel{\text{soda}} \times \left(\dfrac{\text{6 sodas}}{\cancel{\text{six pack}}}\right) = 720 \; \dfrac{\text{sodas}^2}{\text{1 six pack}} \nonumber \]

• While it is correct that there are 6 sodas in one six pack, the above equation yields a value of 720 with units of sodas 2 /six pack .
• These rather bizarre units indicate that the equation has been setup incorrectly (and as a consequence you will have a ton of extra soda at the party).

Using Dimensional Analysis in Calculations

In the above case it was relatively straightforward keeping track of units during the calculation. What if the calculation involves powers, etc? For example, the equation relating kinetic energy to mass and velocity is:

\[E_{kinetics} = \dfrac{1}{2} \text{mass} \times \text{velocity}^2 \label{KE} \]

An example of units of mass is kilograms (kg) and velocity might be in meters/second (m/s). What are the dimensions of \(E_{kinetic}\)?

\[(kg) \times \left( \dfrac{m}{s} \right)^2 = \dfrac{kg \; m^2}{s^2} \nonumber \]

The \(\frac{1}{2}\) factor in Equation \ref{KE} is neglected since pure numbers have no units. Since the velocity is squared in Equation \ref{KE}, the dimensions associated with the numerical value of the velocity are also squared. We can double check this by knowing the the Joule (\(J\)) is a measure of energy, and as a composite unit can be decomposed thusly:

\[1\; J = kg \dfrac{m^2}{s^2} \nonumber \]

Units of Pressure

Pressure ( P ) is a measure of the Force ( F ) per unit area ( A ):

\[ P =\dfrac{F}{A} \nonumber \]

Force, in turn, is a measure of the acceleration (\(a\)) on a mass (\(m\)):

\[ F= m \times a \nonumber \]

Thus, pressure (\(P\)) can be written as:

\[ P= \dfrac{m \times a}{A} \nonumber \]

What are the units of pressure from this relationship? ( Note: acceleration is the change in velocity per unit time )

\[ P =\dfrac{kg \times \frac{\cancel{m}}{s^2}}{m^{\cancel{2}}} \nonumber \]

We can simplify this description of the units of Pressure by dividing numerator and denominator by \(m\):

\[ P =\dfrac{\frac{kg}{s^2}}{m}=\dfrac{kg}{m\; s^2} \nonumber \]

In fact, these are the units of a the composite Pascal ( Pa ) unit and is the SI measure of pressure.

Performing Dimensional Analysis

The use of units in a calculation to ensure that we obtain the final proper units is called dimensional analysis . For example, if we observe experimentally that an object’s potential energy is related to its mass, its height from the ground, and to a gravitational force, then when multiplied, the units of mass, height, and the force of gravity must give us units corresponding to those of energy.

Energy is typically measured in joules, calories, or electron volts (eV), defined by the following expressions:

• 1 J = 1 (kg·m 2 )/s 2 = 1 coulomb·volt
• 1 cal = 4.184 J
• 1 eV = 1.602 × 10 −19 J

Performing dimensional analysis begins with finding the appropriate conversion factors . Then, you simply multiply the values together such that the units cancel by having equal units in the numerator and the denominator. To understand this process, let us walk through a few examples.

Example \(\PageIndex{1}\)

Imagine that a chemist wants to measure out 0.214 mL of benzene, but lacks the equipment to accurately measure such a small volume. The chemist, however, is equipped with an analytical balance capable of measuring to \(\pm 0.0001 \;g\). Looking in a reference table, the chemist learns the density of benzene (\(\rho=0.8765 \;g/mL\)). How many grams of benzene should the chemist use?

\[0.214 \; \cancel{mL} \left( \dfrac{0.8765\; g}{1\;\cancel{mL}}\right)= 0.187571\; g \nonumber \]

Notice that the mL are being divided by mL, an equivalent unit. We can cancel these our, which results with the 0.187571 g. However, this is not our final answer, since this result has too many significant figures and must be rounded down to three significant digits. This is because 0.214 mL has three significant digits and the conversion factor had four significant digits. Since 5 is greater than or equal to 5, we must round the preceding 7 up to 8.

Hence, the chemist should weigh out 0.188 g of benzene to have 0.214 mL of benzene.

Example \(\PageIndex{2}\)

To illustrate the use of dimensional analysis to solve energy problems, let us calculate the kinetic energy in joules of a 320 g object traveling at 123 cm/s.

To obtain an answer in joules, we must convert grams to kilograms and centimeters to meters. Using Equation \ref{KE}, the calculation may be set up as follows:

\[ \begin{align*} KE &=\dfrac{1}{2}mv^2=\dfrac{1}{2}(g) \left(\dfrac{kg}{g}\right) \left[\left(\dfrac{cm}{s}\right)\left(\dfrac{m}{cm}\right) \right]^2 \\[4pt] &= (\cancel{g})\left(\dfrac{kg}{\cancel{g}}\right) \left(\dfrac{\cancel{m^2}}{s^2}\right) \left(\dfrac{m^2}{\cancel{cm^2}}\right) = \dfrac{kg⋅m^2}{s^2} \\[4pt] &=\dfrac{1}{2}320\; \cancel{g} \left( \dfrac{1\; kg}{1000\;\cancel{g}}\right) \left[\left(\dfrac{123\;\cancel{cm}}{1 \;s}\right) \left(\dfrac{1 \;m}{100\; \cancel{cm}}\right) \right]^2=\dfrac{0.320\; kg}{2}\left[\dfrac{123 m}{s(100)}\right]^2 \\[4pt] &=\dfrac{1}{2} 0.320\; kg \left[ \dfrac{(123)^2 m^2}{s^2(100)^2} \right]= 0.242 \dfrac{kg⋅m^2}{s^2} = 0.242\; J \end{align*} \nonumber \]

Alternatively, the conversions may be carried out in a stepwise manner:

Step 1: convert \(g\) to \(kg\)

\[320\; \cancel{g} \left( \dfrac{1\; kg}{1000\;\cancel{g}}\right) = 0.320 \; kg \nonumber \]

Step 2: convert \(cm\) to \(m\)

\[123\;\cancel{cm} \left(\dfrac{1 \;m}{100\; \cancel{cm}}\right) = 1.23\ m \nonumber \]

Now the natural units for calculating joules is used to get final results

\[ \begin{align*} KE &=\dfrac{1}{2} 0.320\; kg \left(1.23 \;ms\right)^2 \\[4pt] &=\dfrac{1}{2} 0.320\; kg \left(1.513 \dfrac{m^2}{s^2}\right)= 0.242\; \dfrac{kg⋅m^2}{s^2}= 0.242\; J \end{align*} \nonumber \]

Of course, steps 1 and 2 can be done in the opposite order with no effect on the final results. However, this second method involves an additional step.

Example \(\PageIndex{3}\)

Now suppose you wish to report the number of kilocalories of energy contained in a 7.00 oz piece of chocolate in units of kilojoules per gram.

To obtain an answer in kilojoules, we must convert 7.00 oz to grams and kilocalories to kilojoules. Food reported to contain a value in Calories actually contains that same value in kilocalories. If the chocolate wrapper lists the caloric content as 120 Calories, the chocolate contains 120 kcal of energy. If we choose to use multiple steps to obtain our answer, we can begin with the conversion of kilocalories to kilojoules:

\[120 \cancel{kcal} \left(\dfrac{1000 \;\cancel{cal}}{\cancel{kcal}}\right)\left(\dfrac{4.184 \;\cancel{J}}{1 \cancel{cal}}\right)\left(\dfrac{1 \;kJ}{1000 \cancel{J}}\right)= 502\; kJ \nonumber \]

We next convert the 7.00 oz of chocolate to grams:

\[7.00\;\cancel{oz} \left(\dfrac{28.35\; g}{1\; \cancel{oz}}\right)= 199\; g \nonumber \]

The number of kilojoules per gram is therefore

\[\dfrac{ 502 \;kJ}{199\; g}= 2.52\; kJ/g \nonumber \]

Alternatively, we could solve the problem in one step with all the conversions included:

\[\left(\dfrac{120\; \cancel{kcal}}{7.00\; \cancel{oz}}\right)\left(\dfrac{1000 \;\cancel{cal}}{1 \;\cancel{kcal}}\right)\left(\dfrac{4.184 \;\cancel{J}}{1 \; \cancel{cal}}\right)\left(\dfrac{1 \;kJ}{1000 \;\cancel{J}}\right)\left(\dfrac{1 \;\cancel{oz}}{28.35\; g}\right)= 2.53 \; kJ/g \nonumber \]

The discrepancy between the two answers is attributable to rounding to the correct number of significant figures for each step when carrying out the calculation in a stepwise manner. Recall that all digits in the calculator should be carried forward when carrying out a calculation using multiple steps. In this problem, we first converted kilocalories to kilojoules and then converted ounces to grams.

Converting Between Units: Converting Between Units, YouTube(opens in new window) [youtu.be]

Dimensional analysis is used in numerical calculations, and in converting units. It can help us identify whether an equation is set up correctly (i.e. the resulting units should be as expected). Units are treated similarly to the associated numerical values, i.e., if a variable in an equation is supposed to be squared, then the associated dimensions are squared, etc.