Category: Physics

Dimensional Analysis: Definition, Examples, And Practice

If you’ve heard the term “dimensional analysis,” you might find it a bit overwhelming. While there’s a lot to “unpack” when learning about dimensional analysis, it’s a lot easier than you might think. Learn more about the basics and a few examples of how to utilize the unique method of conversion.

Dimensional Analysis: Definition, Examples, and Practice

As a student of Biology or any of the sciences, you will have to use math of some kind, and there’s a good chance that you will find dimensional analysis (or unit analysis) to be helpful. Math equations and other conversions can be overwhelming for some, but dimensional analysis doesn’t have to be; once you learn it, it’s relatively easy to use and understand.

We’ll give you the basics and give you some easy-to-understand examples that you might find on a dimensional analysis worksheet so that you can have a general understanding about what it is and how to use the technique in all types of applications as you continue to take science courses.

What Is Dimensional Analysis?

As we mentioned, you may hear dimensional analysis referred to as unit analysis; it is often also known as factor-label method or the unit factor method. A formal definition of dimensional analysis refers to a method of analysis “in which physical quantities are expressed in terms of their fundamental dimensions that is often used.”

Most people might agree that this definition needs to be broken down a bit and simplified. It might be easier to understand this method of analysis if we look at it as a method of solving problems by looking converting one thing to another.

While dimensional analysis may seem like just another equation, one of the unique (and important) parts of the equation is that the unit of measurement always plays a role in the equation (not just the numbers).

We use conversions in everyday life (such as when following a recipe) and in math class or in a biology course. When we think about dimensional analysis, we’re looking at units of measurement, and this could be anything from miles per gallon or pieces of pie per person.

Many people may “freeze up” when they see a dimensional analysis worksheet or hear about it in class, but if you’re struggling with some of the concepts, just remember that it’s about units of measurements and conversion. Dimensional analysis is used in a variety of applications and is frequently used by chemists and other scientists.

The Conversion Factor in Dimensional Analysis

One important thing to consider when using dimensional analysis is the conversion factor. A conversion factor, which is always equal to 1, is a fraction or numerical ratio that can help you express the measurement from one unit to the next.

When using a conversion factor, the values must represent the same quantity. For example, one yard is the same as three feet or seven days is the same as one week. Let’s do a quick example of a conversion factor.

Imagine you have 20 ink pens and you multiply that by 1; you still have the same amount of pens. You might want to find out how many packages of pens that 20 pens equal and to figure this out, you need your conversion factor.

Now, imagine that you found the packaging for a set of ink pens and the label says that there are 10 pens to each package. Your conversion factor ends up being your conversion factor. The equation might look something like this:

20 ink pens x 1 package of pens/10 pens = 2 packages of ink pens. We’ve canceled out the pens (as a unit) and ended up with the package of pens.

​While this is a basic scenario, and you probably wouldn’t need to use a conversion factor to figure out how many pens you have, it gives you an idea of what it does and how it works. As you can see, conversion factors work a lot like fractions (working with numerators and denominators)

Even though you’re more likely to work with more complex units of measurement while in chemistry, physics, or other science and math courses, you should have a better understanding of using the conversion factor in relation to the units of measurement.

Steps For Working Through A Problem Using Dimensional Analysis

Like many things, practice makes perfect and dimensional analysis is no exception. Before you tackle a dimensional analysis that your instructor hands to you, here are some tips to consider before you get started.

  • Read the problem carefully and take your time
  • Find out what unit should be your answer
  • Write down your problem in a way that you can understand
  • Consider a simple math equation and don’t forget the conversion factors
  • Remember, some of the units should cancel out, resulting in the unit you want
  • Double-check and retry if you have to
  • The answer you come up with should make sense to you

To help you understand the basic steps we are using an easy problem that you could probably figure out fairly quickly. The question is: How many seconds are in a day?

First, you need to read the question and determine the unit you want to end up with; in this case, you want to figure out “seconds in a day.” To turn this word problem into a math equation, you might decide to put seconds/day or sec/day.

The next step is to figure out what you already know. You know that there are 60 seconds to one minute and you also know that there are 24 hours in one day; all of these units work together, and you should be able to come up with your final unit of measurement. Again, it’s best to write down everything you know into an equation.

After you’ve done a little math, your starting factor might end up being 60 seconds/1 minute. Next, you will need to work your way into figuring out how many seconds per hour. This equation will be 60 seconds/1 minute x 60 minutes/1 hour. The minutes cancel themselves out, and you have seconds per hour.

Remember, you want to find out seconds per day so you’ll need to add another factor that will cancel out the hours. The equation should be 60 seconds/1 minute x 60 minutes/1 hour x 24 hours/1 day. All units but seconds per day should cancel out and if you’ve done your math correctly 86,400 seconds/1 day.

When doing a dimensional analysis problem, it’s more important to pay attention to the units and make sure you are canceling out the right ones to get the final product. Doing your math correctly important, but it’s easier to double-check than trying to backtrack and figure out how you ended up with the wrong unit.

Our example is relatively simple, and you probably had no problem getting the right answer or using the right units. As you work through your science courses, you will be faced with more difficult units to understand. While dimensional analysis will undoubtedly be more challenging, just keep your eye on the units, and you should be able to get through a problem just fine.

Why Use Dimensional Analysis?

As we’ve demonstrated, dimensional analysis can help you figure out problems that you may encounter in your everyday. While you’re likely to explore dimensional analysis a bit more as you take science courses, it can be particularly helpful for Biology students to learn more.

Some believe that dimensional analysis can help students in Biology have a “better feel for numbers” and help them transition more easily into courses like Organic Chemistry or even Physics (if you haven’t taken those courses yet).

Can you figure out a math equation or a word problem without dimensional analysis? Of course, and many people have their own ways of working through a problem. If you do it correctly, dimensional analysis can actually help you answer a problem more efficiently and accurately.

Ready To Test Your Dimensional Analysis Skills?

If you want to practice dimensional analysis, there are dozens of online dimensional analysis worksheets. While many of them are pretty basic or geared towards specific fields of study like Chemistry, we found a worksheet that has an interesting variety. Test out what we’ve talked about and check your answers when you’re done.

  • How many minutes are in 1 year?
  • Traveling at 65 miles/hour, how many minutes will it take to drive 125 miles to San Diego?
  • Convert 4.65 km to meters
  • Convert 9,474 mm to centimeters
  • Traveling at 65 miles/hour, how many feet can you travel in 22 minutes? (1 mile = 5280 feet)

Ready to check out your answers and see more questions? Click here.

Volume of Irregular Object Sample Lab 1

 

 

Volume of an Irregular Object

 

Introduction

Everything is made of matter, and matter has both chemical and physical properties. Physical properties of matter are observable and include mass, volume, and density. Mass is the amount of matter in an object and can be found by using an electronic balance or scale. Volume is the amount of space an object occupies and can be determined by measuring the amount of water that a submerged object displaces. The formula for determining density is:

Density = Mass of the object / Volume of the object

Hypothesis

Determining the volume of an irregularly shaped object can help determine its density.

Materials

The materials used in this lab included a rubber stopper, small rock, shell, 100 ml graduated cylinder, electronic balance, water, paper, and pencil.

Methods

Obtain three, irregularly shaped objects — a rubber stopper, a shell, and a rock. Estimate the volume and mass of each object and record this in the data table. Use an electronic balance to mass each object and record the masses in the data table. Fill a graduated cylinder approximately half way with water and record this initial volume. Place one of the objects into the graduated cylinder and record the final volume. Subtract the initial volume of the water from the final volume of water to find the volume of the water the object displaced. Record this as the object’s volume in the data table. Remove the first object and repeat this process of determining volume for the other two objects. Determine each object’s density by dividing their mass by their volume. Record each density in the data table.

Results

 

Data Table
Object estimated mass (g) estimated volume (ml) actual mass (g) initial volume of water (ml) final volume of water (ml) volume of object (ml) density (g/ml)
Rubber stopper 19.3 10 9.3 50 56 6 1.55
Shell 6.0 2 3.3 50 51.5 1.5 2.2
Rock 10.0 5 9.3 50 55 5 1.86

 

Discussion and Conclusion

Objects that are irregular in shape can have their volume determined by measuring the amount of water that they displaced when submerged in a known amount of water. The increase in volume from the initial water level to the final water level is the volume of the object. Once the volume is determined and the actual mass measured then the object’s density can also be determined. The formula for finding density is the mass of the object divided by the volume of the object. For example, since the rock had a mass of 9.3 grams and a volume of 5 ml, the density of the rock was determined to be 1.86 grams/ millimeters.

BACK

 

Water Properties Notes

 

Water Properties
States of Water
Adhesion and Cohesion
Surface Tension
Capillary Action

The States of Water

Water has three states. Below freezing water is a solid (ice or snowflakes), between freezing and boiling water is a liquid, and above its boiling point water is a gas. There are words scientists use to describe water changing from one state to another. Water changing from solid to liquid is said to be melting. When it changes from liquid to gas it is evaporating. Water changing from gas to liquid is called condensation (An example is the ‘dew’ that forms on the outside of a glass of cold soda). Frost formation is when water changes from gas directly to solid form. When water changes directly from solid to gas the process is called sublimation.

Gas
Liquid
Solid

Most liquids contract (get smaller) when they get colder. Water is different. Water contracts until it reaches 4 C then it expands until it is solid. Solid water is less dense that liquid water because of this. If water worked like other liquids, then there would be no such thing as an ice berg, the ice in your soft drink would sink to the bottom of the glass, and ponds would freeze from the bottom up!

Water is found on Earth in all three forms. This is because Earth is a very special planet with just the right range of temperatures and air pressures.

Adhesion and Cohesion

Water is attracted to other water. This is called cohesion. Water can also be attracted to other materials. This is called adhesion.

The oxygen end of water has a negative charge and the hydrogen end has a positive charge. The hydrogens of one water molecule are attracted to the oxygen from other water molecules. This attractive force is what gives water its cohesive and adhesive properties.

Surface Tension

Surface tension is the name we give to the cohesion of water molecules at the surface of a body of water. Try this at home: place a drop of water onto a piece of wax paper. Look closely at the drop. What shape is it? Why do you think it is this shape?

What is happening? Water is not attracted to wax paper (there is no adhesion between the drop and the wax paper). Each molecule in the water drop is attracted to the other water molecules in the drop. This causes the water to pull itself into a shape with the smallest amount of surface area, a bead (sphere). All the water molecules on the surface of the bead are ‘holding’ each other together or creating surface tension.

Surface tension allows water striders to ‘skate’ across the top of a pond. You can experiment with surface tension. Try floating a pin or a paperclip on the top if a glass of water. A metal pin or paper clip is heavier than water, but because of the surface tension the water is able to hold up the metal.

Surface tension is not the force that keeps boats floating.

Capillary Action

Surface tension is related to the cohesive properties of water. Capillary action however, is related to the adhesive properties of water. You can see capillary action ‘in action’ by placing a straw into a glass of water. The water ‘climbs’ up the straw. What is happening is that the water molecules are attracted to the straw molecules. When one water molecule moves closer to a the straw molecules the other water molecules (which are cohesively attracted to that water molecule) also move up into the straw. Capillary action is limited by gravity and the size of the straw. The thinner the straw or tube the higher up capillary action will pull the water (Can you make up an experiment to test this?).

Plants take advantage of capillary action to pull water from the into themselves. From the roots water is drawn through the plant by another force, transpiration.