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Density

Density is an important characteristic of substances. It is crucial in determining whether an object sinks or floats in a fluid.

Density is the mass per unit volume of an object.

In equation form, density is usually symbolized by ρ (Greek letter rho,) or p (English and Latin letter pee.) .

     𝜌 =𝑚/𝑉           or      d = m/v

 

Do you understand the basic concept?

Which weighs more, a ton of feathers or a ton of bricks?

Don’t confuse mass and density.

Bricks have much greater density than feathers. Let’s see what happens:

feathers bricks mass and density

Figure 11.4 A ton of feathers and a ton of bricks have the same mass, but the feathers make a much bigger pile because they have a much lower density.

What is the source of density?

The source, ultimately, is how much mass or matter is in a given volume. For our purposes, we’re defining matter as anything made of atoms – and atoms are mostly empty space made of much smaller particles.

So let’s look at a sample.

density depends on

* the spacing between atoms

* the number of nucleons per atom

Here we see the spacing between the same type of atoms changing the density:

Let’s look at the same thing in 3d.

Density changes if the number of nucleons per atom changes.

Consider 2 blocks of solid matter, with same number of atoms, taking up the same volume.

two blocks metal

On the left is iron, on the right is phosphorus.

Which block would be denser – and why?

Deep thoughts: Why may we ignore the number of electrons when thinking about an object’s density?

http://quatr.us/chemistry/atoms/iron.htm

Density of liquids

Density of gases

Not all gases have the same density.

Here we see a demonstration of sulphur hexaflouride gas.

Density sulphur hexaflouride gas GIF

Density of solids and liquids 

Not all metals are the same density. Here an iron cannonball is demonstrated to be less dense than liquid mercury metal.

Cannonball in Mercury Density GIF

Density lab

  1. Use electronic balance to measure the mass of an object.

  2. Use a ruler, and use geometry, to measure the volume of several objects (length, or radius)

  3. Immerse an object in a liquid to measure the change in volume. This technique is called displacement.

  4. Explore the origin of experimental error through these different measurement techniques. (The two measurements of volume should be exactly the same, in principle.)

  5. Calculate the density in both cases.

Movie physics!

This next section from Physics in the realm of Hollywood, Adam Weiner, Science on NBC News

In “Willy Wonka and the Chocolate Factory,” when Charlie and Grandpa Joe sip a bit of Wonka’s “fizzy lifting drink” on the sly, they are immediately lifted off their feet and into the air, floating among the bubbles. This, presumably, is the result of all that carbonation inside their stomachs, increasing Grandpa and Charlie’s buoyancy to the point where it can overcome the force of their own weight, lifting them into the air.

Thanks to good old Archimedes’ principle, we can calculate the amount of air that would need to be displaced to perform the lifting, and thus the necessary increase in volume due to the drink’s carbonation of the bodies of Charlie and Grandpa Joe.

As you can see, to counteract the force of his mass (here we estimate his mass to be 70 kilograms), Grandpa would have to swell up to a massive 54 cubic meters-if he was a sphere, he would be five meters across (that’s over 15 feet).

For a more familiar reference, that’s at least twice as big as poor Violet’s sudden rotundity after sampling Wonka’s experimental three-course gum. Which, if you’re interested, would need to have a density of 6 x 109 kg/m3 to contain enough juice to fill Violet to the size depicted — that’s four or five thousand times the density of an average metal. Watch your fillings!

This next section from Sorry Girls, “Titanic” Doors Were Made of Oak, American Physical Society, 9/25/2012

Buoyancy is the force that makes something float. It depends on the volume of the floating object and the density of the liquid in which it is floating. For something to remain above the water level, the bouncy force must be greater than the force of gravity pulling down on the object.

In the case of Rose, Jack and the door, the buoyancy force of the ice cold salt water pushing up on them must be greater than the force of their combined weight. The force pushing up depends on the volume of the object submerged and the density of the water in which it is floating.

So lets see how that stacks up. Looking at the raft in stills from the movie and looking up Kate Winslet’s height, we can estimate that the raft is about 6’x3’x5″ and the density of ice cold salt water is 1000 kg/m^3.

Our heroes would survive only if the top of the raft were at least at the water level. So let’s assume the volume submerged is that of the full door, 0.254 m^3 (keeping it all in metric). Multiply this by the density of salt water and the pull of gravity and you find that the buoyant force is 2490N.

If the weight of Jack+Rose+door is greater than 2490N, they are all in hot water (or, I guess freezing cold water). Seeing as there was much controversy of Kate Winslet’s weight, it was easy to find out that at the time of the movie she weighed 125 lbs, or 549N. It was a little harder to track down Leo’s weight, but he topped out at a whopping 161 lbs or 715N.

Finding the weight of the door is a bit trickier. Weight is volume times density times the pull of gravity, but its not clear what the door is made of.

There were three types of wood commonly used on the Titanic, teak, oak and pine with densities of 980kg/m^3, 770 kg/m^3 and 420 kg/m^3 respectively. If the door were teak, the weight would be 2,440N, oak would be 1,147N and pine tops out at 617N.

Teak would barely float on its own so Rose and Jack would be headed into an eternity of sappy music together. If the door were pine, the total force of Jack+Rose+Door would be 2,313 N and all would have been well in the world of middle school girls across the globe.

But darn you Mr. Cameron, pine was simply not good enough for your movie! The door was most likely oak which has a weight of 1,920 N so adding the adorable couple would give a weight of 3,185 N, just a little too heavy.

Subtract Jack and you get a force of 2,470 N, just light enough to float and allow Rose to go on and live a long and happy life as Jack’s frozen body spent continued to bob in the ocean. Before he died in his melodramatic, tear-jerking manner, he made her make him a promise.

“You must do me this honor. Promise me you’ll survive. That you won’t give up, no matter what happens, no matter how hopeless. Promise me now, Rose, and never let go of that promise.”

She wouldn’t have had to make that promise if James Cameron had just used pine!

http://physicsbuzz.physicscentral.com/2012/09/sorry-girls-titanic-doors-were-made-of.html

We next may consider the physics of buoyancy in a Pirates of the Caribbean movie,

http://physics.stackexchange.com/questions/106723/can-a-submerged-capsized-vessel-be-turned-into-a-sort-of-diving-bell-and-propell

And of course, let’s look at the Buoyancy of balloons in the movie “Up”

Practical problem solving (standard level)

tba

Practical problem solving (Honors)

This problem from College Physics, OpenStax, Rice University, by Paul Peter Urone

Calculating the mass of a reservoir from its volume

 A reservoir has a surface area of 50 square km and an average depth of 40.0 m. What mass of water is held behind the dam?

Strategy: We can calculate the volume V of the reservoir from its dimensions, and find the density of water from a data table.

Then the mass can  be found from the definition of density  𝜌 =𝑚/𝑉

density reservoir problem calculation

Related discussion: A large reservoir contains a very large mass of water. In this example, the weight of the water in the reservoir is 𝑚𝑔=1.96×1013 Newtons , where 𝑔 is the acceleration due to the Earth’s gravity.

It is reasonable to ask whether the dam must supply a force equal to this tremendous weight. The answer is no. As we shall see in the following sections, the force the dam must supply can be much smaller than the weight of the water it holds back.

 

Online textbook from American Chemical Society

Chapter 3: Density

Students experiment with objects that have the same volume but different mass and other objects that have the same mass but different volume to develop a meaning of density. Students also experiment with density in the context of sinking and floating and look at substances on the molecular level to discover why one substance is more or less dense than another.

  1. What is Density?
  2. Finding Volume—The Water Displacement Method
  3. Density of Water
  4. Density—Sink and Float for Solids
  5. Density—Sink and Float for Liquids
  6. Temperature and Density

Learning Standards

Massachusetts Science and Technology/Engineering Standards

8.MS-PS1-2. Analyze and interpret data on the properties of substances before and after the substances interact to determine if a chemical reaction has occurred… Properties of substances include density, melting point, boiling point, solubility, flammability, and odor.

HS-PS1-3. Cite evidence to relate physical properties of substances at the bulk scale to spatial arrangements, movement, and strength of electrostatic forces among ions, small molecules, or regions of large molecules in the substances….Examples of bulk properties of substances to compare include melting point and boiling point, density

HS-PS1-11(MA). Design strategies to identify and separate the components of a mixture based on relevant chemical and physical properties…. Relevant chemical and physical properties can include melting point, boiling point, conductivity, and density.

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