Density - Online Tutor, Practice Problems & Exam Prep

Fluids, including liquids and gases, can change shape to fit their containers, characterized by density (ρ), defined as mass (m) divided by volume (v). Common densities include freshwater (1000 kg/m³) and air (1.2 kg/m³). Specific gravity (SG) relates a material's density to freshwater's density, expressed as SG = ρ_material / 1000. Understanding these concepts aids in solving problems involving buoyancy and weight, where weight (W) is calculated using W = mg, with g as the acceleration due to gravity.

Intro to Density

Everyone, in this video, we're going to start talking about fluids, which is sometimes called fluid mechanics, fluid dynamics, or sometimes just called liquids. Now, the first concept you'll need to know really well is the concept of density. So, we're just going to jump right in and get started.

Liquids and gases are types of what we call fluids. And the reason we use the term fluids is because they often share similar properties. Instead of repeating "liquids and gases" over again, we're just going to call these things fluids. Remember that the main characteristic of fluids is that unlike solids, which always have the same shape, fluids can actually change shape to fill their container. For example, water can fit inside of a cup, and you can pour it out into a bowl, and it'll change shape to fit the container.

You may remember density from an earlier science class. It's basically just a measure of how tightly the molecules are packed inside of a small volume. For example, I've got these two boxes right here. They're identical, which means they have the same exact volume. But in one of them, there are fewer molecules, and in the other, there are more of them, and they're more tightly packed together. So because this one has fewer molecules in the same space, this one has lower density and because this one has a lot of stuff in the same space, this one has high density.

Density itself is represented by the letter that we use is this one over here, which is the Greek letter rho (ρ). The definition of density is that it's the mass (m) divided by the volume (V),m/V. The units for it are kilograms for mass, and for volume, it's cubic meters (m³). The volume of any shape is generally the form of length times width times height.

In many problems, you'll be given the density of a material and the dimensions of a material. Then you can always rearrange to find what the mass is by using the formula m=ρV. Objects of the same material are going to have the same density. Thus, if you have a small block of wood and a big one, they both have the same exact density.

Additionally, you also have a lot of problems in which liquids will mix inside a container and settle into layers. The higher density liquids are going to settle at the bottom. This is because higher density means these things are more compact, and gravity pulls them lower.

Let's run through a quick example. We're going to calculate the total weight of air molecules inside of a really large warehouse. We model this warehouse as a rectangle like this. Its dimensions are 100 meters by 100 meters by 10 meters. We have the density of air and also with gravity as g = 10 m/s². To calculate the total weight of air, remember that weight is not mass, weight (W) is W=mg. We use the equation for density ρ=mV to find the mass as m=ρV, substitute the values, and we get the weight as 1,225,000 Newtons. That's it for this example, folks. Let me know if you have any questions.

Density Values & Conversions

Everyone, so now that we've seen the basics of density, a lot of problems in this chapter are going to involve one of these materials: water, saltwater, blood, air, or even oil and wood. So what I want to do in this video is go over some very common density values that you'll see throughout the chapter, and also some conversions because it will help you balance between the different types of units that you'll see expressed for density. So I'm going to go ahead and show you how to do that, and we'll do an example together. Alright, let's check it out. Some very common density values and units that you'll need to know. Let's check it out. The first one, probably the most common that you'll see, is going to be freshwater. Freshwater has a density of 1000 kilograms per meter cubed. Now, what that means here is that for every 1 meter cubed of volume, it has a mass of 1000 kilograms. Alright. Now, oftentimes, I kind of struggle to visualize that. So the best thing I can come up with is if you've ever moved, so you packed up your house or apartment or whatever, and you've had to put your clothes in one of those big wardrobe boxes, ones that one of those ones that kind of goes up to your waist. That's a box that is about 1 meter or about 3 feet length width and height. So basically what this means is if you were to empty it out and you were to fill it with purely water all the way up to the top, this volume here, it would weigh or it would have a mass of about 1000 kilograms of water. Alright, Here's a person for reference. A person is, I think, with the average height of a human is like 1.8 meters or something like that. Alright? Now there's other ways to express this density. 1000 kilograms per meter cubes can also be rewritten as 1 kilogram per liter, and sometimes we use this. We use liters for volume because it's a little bit more readily identifiable as a volume, and it's also more useful for everyday objects. Like if you go to a store and you pick up a 2-liter bottle of water or soda or something like that, that 2-liter bottle is going to be about 2 kilograms. That's how much mass it has. Alright, Now, the last sort of that's the second sort of pair of units or the other way you might see density. The third one is grams per centimeters cube. This is one that's usually used by chemists because they're using very small amounts of liquids and solids and things like that. Alright, so the centimeters cubed would be like if you had a box, but instead of it being 1 meter in length, width, and height, it would actually be 1 centimeter in length, width, and height. So it would be about the size of like a pencil eraser at the end of a wooden pencil. All right. So this is 1 grams per centimeter cubed. All right. So these are the units for freshwater and they're really important because they basically can help you jump between some different ways that you might see density expressed. What I mean by this is if you see a 1000 kilograms per meter cube, you know that's going to be 1 kilogram per liter. Now, if you had a 1030 kilograms per meter cube like you have for saltwater, which is a little bit denser than water, then basically according to the same pattern, all you have to do to convert to the other one is just move the decimal place back by 3 units. Right? So see how we move the decimal place back, and then it just turned into a one. We do the same exact thing for saltwater. So this would be expressed as 1.03 kilograms per liter and this would also be expressed as 1.03 grams per centimeter cube. Notice how we have 1 and then we have 1 as well. Alright. Now for whole blood, it would basically be the same thing instead of 1.03, we'd have 1.06, and then so on and so forth. So we're taking advantage of these conversion factors, which I want to go over really quickly here. There are 2 main ones that you need to know. 1 centimeter cubed is equal to 1 milliliter, and then 1 meter cubed is equal to 1000 liters. This conversion factor is actually how we got from this unit, these pairs of units, meters cubed over to liters. All we did was we just divided by 1000, and that's why we went from 1000 to 1. All right, so these are the 2 important conversion factors. Now there's one more thing I want to mention here. Sometimes in your problems, you are going to be given that you're dealing with water from a lake or a river or house water or tap water or something like that. That's an indication that you're dealing with freshwater. And if you're dealing with seawater or the ocean, or you're out in the middle of the ocean or sea or something like that, you can usually use this density. If you're ever not sure, the default water is always going to be freshwater. You can always default and use those numbers. Alright, and now the last thing I want to talk about is air, which is at sea level. If you're never not at sea level, the problem will tell you. If not, you can always assume that you are, and it has a density of 1.2 kilograms per meters cubed. Alright. Now, this isn't a typo. You'll notice that this number is way smaller than all of these other numbers here, and that's not a typo. It's because air is actually 800 times less dense than water, and that makes sense, right? We're standing in the air. We can move through it and all that stuff, and that's not very dense. Now the last one is oil and wood. Sometimes you might see these in a problem as well. Basically, these are going to be slightly less dense than water. Usually, they'll probably tell you what the density is, or you'll be able to figure it out, but a good rule of thumb is it's about 800 kilograms per meter cubed. You might need to know that, not sure, but it's a little bit less than the density of freshwater, which is 1000. Alright? So that's really all there is to it, guys. So let's go ahead and take a look at an example. Alright? So we have 500 milliliters of some kind of liquid, and we're told that it's 2.2 grams per centimeters cube, right? And we want to figure out how much does it weigh. So remember, we're not looking for mass, that's the mass. How much it weighs is going to be w and remember the w is equal to mg. So if we want to calculate w and we can just assume that g equals 10 for just to make the calculations a little bit simpler, really what we're looking for in this problem is what is m? Alright, so that's the whole game here. So how do we figure out m? Well, according to our equation for density, we know that m is going to equal rho times the density. Right? That just comes from the definition of density, mass over volume. Okay? So, we're going to have mass is equal to now what's the density of this liquid here? So we've got 500 milliliters, but remember that's a volume. Right? Milliliters, liters, that's all volumes and then we've got this number over here. This is grams per centimeters cube. This is a mass divided by a volume. So even though the problem doesn't explicitly tell you that's the density, you can figure it out from the units and sometimes you'll have to do that, right? So this is going to be the density. This is 2.2, and remember, you want to put this, you want to rearrange this so that you have your units as fractions. So we have grams per centimeter cubed. Now we're going to multiply by the volume, which is 500 milliliters. Alright. So can we cancel out the units and multiply? Well, actually, we can't because we have grams per centimeter cubed and then we have milliliters. Right. So we can't convert, we can't multiply those things and hope that they'll cancel. So how do we get rid of this? We are going to have to go up and look for our common conversion factors. Remember, there are 2 of them. One of them converts centimeters cube to milliliters. The other one is meters cube to liters. Alright? And really this is just, you're just off by factors of 10. Right? A centimeter cubed is actually 1,000,000 times smaller than a meter cube, and that's why you have this as 1 milliliter and this is 1000 liters. Right? So it's all really powers of 10. So with this conversion factor in particular, this one tells us that 1 centimeter cubed and 1 milliliter are actually the same exact thing. There's no difference between a centimeter cubed and a milliliter, and that's why sometimes you'll see this in glass vials or something like that. So even though this says centimeters cubed, this actually is also the same thing as a milliliter. Centimeters cubed or milliliters, it's equivalent. They're the same exact thing. So actually, you can cancel those things out because you don't have to insert a conversion factor. So once you multiply this, what happens is your milliliters cancel, and you're only left with grams on the left side. So be very careful with what units you are left with. So this is going to be 1100, but again, be careful because some of you might be tempted to put kilograms there. 9 times out of 10 when you make a mistake in these problems, it's going to have to do with the units and, you know, you stick at kilograms or you multiplied by grams or something like that. So just be very careful and follow your units. Alright? So this is 1100 grams, but remember when we plug into our equations for mg, we have to have this mass in SI units, which is kilograms. So this 1100 grams is 1.1 kilograms, because we're just shifting the decimal place. So now this is the number we plug back into our weight equation. And so last but not least here, we're going to have that the weight is equal to 1.1 times we're going to use 10 just make a little bit simpler, and this is going to be 11 Newtons. So that's the weight of 500 milliliters of this liquid here. Alright, so there we have it folks. That's for this one. Let me know if you have any questions.

Specific Gravity

Everyone, so now that we've seen density, we're going to talk about a slightly less familiar term which is called specific gravity. Now, if you've never heard that or seen that in your classes, you can probably go ahead and skip this video. But some professors might want you to know it, so it's a good thing to know. It's actually very straightforward, so let's just jump in. Right? So we have the specific gravity which doesn't really have a variable, so we just abbreviate it as SG. It is a term that's sort of related to the density of a material, but it's not exactly density. The specific gravity of any material is defined as the density of that material divided by the density of fresh water. Now, fresh water is always going to be at the bottom and we know that there is a fixed value. So really quickly, what here what happens here is that if you're dividing 2 densities, you're going to divide the units. Right? Kilogram over meters cubed, kilogram over meters cubed. And whenever you divide 2 variables that are the same, they cancel out. So this is just a number that has no units. It's just like 2 or 5 or 0.5 or something like that. Alright? So just to put it into an equation, you know, just to make it a little bit simpler, the specific gravity of any material, which I'm just going to call x, is equal to the density of that x, that material divided by the density of fresh water. Now, again, like we said, the density of fresh water is always a fixed value, it's 1,000. So one really simple way we can sort of phrase this or write this equation is the density over a1000. Alright. So pretty straightforward. So let me just go ahead and run you through some quick examples. If you know that the density of some material is equal to 2,000 kilograms per meter cubed, then that means that the specific gravity of that material is 2,000 over 1,000. So, in other words, it's just 2. It's twice as dense as water. It also works the other way around. If you know what the specific gravity for a material is, I'm going to say that the specific gravity of some material y is equal to 0.7. Right? It'd be less than 1 or more than 1, then that means that the density of that material is equal to 0.7 times a 1000, which is equal to 700. Right? So it works both ways. If you know the density, you could figure out the specific gravity and vice versa. Alright. So really, what we can see here is that the specific gravity doesn't really have anything to do with gravity. It's kind of one of those important terms that later on we figured out that it doesn't really have to do with gravity. But it's sort of what we're stuck with. Right? But it's just a relative number. Right? So 2 or 0.7 or something like that for how many times denser a material is relative to freshwater. Alright. So that's all it is. It's just a number that says how much times denser it is than water. So that's all there is to it. Let me let us go ahead and just work out this example. Alright? So we're going to calculate the volume of some wooden cube. So, in other words, we're going to calculate v, that's the volume, for a specific gravity of 0.8. So we have SG that equals 0.8, and it weighs 16,000 newtons. Now be careful here because we have newtons. So you might think it's mass, but it's actually weight. So, in other words, the w, which is equal to mg, is equal to 16,000 Newtons. Alright? So then how do we calculate the volume? We're going to have to relate this back to the density equation. We have rho that equals m over v. If you rearrange this, you see that v is equal to m over rho. Right? These two things swap places. So in order to calculate the volume, I'm going to need the mass, which I don't know. That's the mass, and I'm going to need the density. So the way I'm going to do this is that this SG here is going to give me density. Right? Because remember that's the relationship between specific gravity and density. And then this equation here for weight is going to give me the mass. Alright? So let me go ahead and just do this one first. So I'm just going to bring this down here. So we have that m equals w over g, which is 16,000. And we're just going to use 10 for g just to make it really simple here. So you can knock off one of the zeros and this just becomes, this is actually 1,600 and this is kilograms. Alright. So that's your mass. And then on the other side, we have the specific gravity. So we have SG is equal to 0.8. So what that means here is that the density is going to be 0.8 times 1,000. Right? It's 0.8 times the density of fresh water. So, in other words, you're going to get 800 kilograms per meter cubed. This makes sense. We got a number that was less than 1, which means that it's less dense than water and we got 800, so that makes sense. Alright? Oh, Actually, I did this wrong, so this is yellow and this should be blue. Alright, so now I'm just going to plug both of these into our equation here and we're going to get that v is equal to 1,600 divided by the density, which is 800, and you just get 2, and the units for that are going to be in meters cubed. Right? So we have kilograms and then kilograms per meter cubed. So you have SI units. So what you should end up with are also SI units. So that is your final answer. So pretty straightforward. Let me know if you guys have any questions.

A wooden door is 1 m wide, 2.5 m tall, 6 cm thick, and weighs 400 N. What is the density of the wood in g/cm³? (use g = 10 m/s²)

• A. 0.0267 g/cm³
• B. 0.267 g/cm³
• C. 267 g/cm³
• D. 2670 g/cm³

Suppose an 80 kg (176 lb) person has 5.5 L of blood (1,060 kg/m³ ) in their body. How much of this person's total mass consists of blood? What percentage of the person's total mass is blood?

• A. 0.0519 kg
• B. 0.0583 kg
• C. 5.19 kg
• D. 5.83 kg

You want to verify if a 70-g crown is in fact made of pure gold (19.32 g/cm³ ), so you lower it by a string into a deep bucket of water that is filled to the top. When the crown is completely submerged, you measure that 3.62 mL of water has overflown. Is the crown made of pure gold?

• A. Yes
• B. No

Here’s what students ask on this topic:

What is the formula for density and how is it used in calculations?

The formula for density (ρ) is defined as the mass (m) of an object divided by its volume (V). Mathematically, it is expressed as:

$\rho =\frac{m}{V}$

In calculations, if you know the mass and volume of an object, you can determine its density by dividing the mass by the volume. For example, if an object has a mass of 10 kg and a volume of 2 m³, its density would be:

$\rho =\frac{10}{2}=5 \mathrm{kg/m³}$

How do you convert between different units of density?

To convert between different units of density, you need to use appropriate conversion factors. Common conversions include:

• 1 kg/m³ = 0.001 g/cm³
• 1 g/cm³ = 1000 kg/m³
• 1 kg/m³ = 0.001 kg/L

For example, to convert 1000 kg/m³ to g/cm³, you multiply by 0.001:

$1000 \mathrm{kg/m³} \times 0.001 \mathrm{g/cm³} = 1 \mathrm{g/cm³}$

What is specific gravity and how is it calculated?

Specific gravity (SG) is a dimensionless quantity that compares the density of a material to the density of freshwater. It is calculated using the formula:

$\mathrm{SG}=\frac{\mathrm{\rho \_material}}{1000}$

For example, if a material has a density of 2000 kg/m³, its specific gravity would be:

$\mathrm{SG}=\frac{2000}{1000}=2$

This means the material is twice as dense as freshwater.

How does density affect the buoyancy of an object?

Density plays a crucial role in determining the buoyancy of an object. An object will float in a fluid if its density is less than the density of the fluid. Conversely, it will sink if its density is greater. This is because buoyant force, which acts upward, is equal to the weight of the fluid displaced by the object. Mathematically, buoyant force (F_b) is given by:

$\mathrm{F\_b}=\mathrm{\rho \_fluid} \mathrm{V\_displaced} g$

Where ρ_fluid is the density of the fluid, V_displaced is the volume of fluid displaced, and g is the acceleration due to gravity.

What are some common densities of materials and their units?

Common densities of materials include:

• Freshwater: 1000 kg/m³
• Saltwater: 1030 kg/m³
• Air: 1.2 kg/m³
• Oil: ~800 kg/m³
• Wood: ~600-800 kg/m³

These densities are typically expressed in kilograms per cubic meter (kg/m³), but can also be converted to grams per cubic centimeter (g/cm³) or kilograms per liter (kg/L) using appropriate conversion factors.