# Density

### Introduction

*Density* is something that affects many of our everyday decisions. Conciously or not, we make mental calculations of density every time we interract with the physical world around us. Can we slide that box? Can we lift that rock? This lab examines some of the ways density effects our everyday lives.

People are often confused about the difference between weight and density. There is an old riddle which highlights this confusion: “What weighs more – a pound of feathers or a pound of lead?” The answer, of course, is that both weight same – one pound. However, feathers are much less dense than lead, and therefore take up much more space. Density is the ratio of an object’s mass to its volume. This means that to find density, you must measure an object’s mass and divide it by the amount of space it takes up. The standard units of density are [kg/m^{3}], although other units are commonly used such as [g/ml], [g/cm^{3}], or [kg/l]. 1 ml has the same volume as 1 cm^{3}.

### Preliminary questions

- What two things do you need to know about a sample if you are to determine its density?
- What does density indicate?
- If you measured the density of a nail on the earth and then on the moon, would the densities vary? Why?
- If you measured the density of a gallon of water and then a teaspoon of water, would the densities vary? Why?
- Consider the concrete blocks below. Explain your answers.
- Which has the greatest volume?
- The greatest density?
- the greatest mass?

- Consider the balloons above. These balloons were the same size, but the second one has gotten smaller due to a change in temperature. Explain your answers.
- Which has the greatest volume?
- The greatest density?
- the greatest mass?

### Materials

- Copper sample
- Graduated cylinder
- Triple beam balance with platform
- Stack of post-1982 pennies

### Activity 1: Finding density using volume

- Design and describe an experiment that determines the density of water.
- Measure and record the mass of the copper sample.
- Determine the volume of the copper sample, first geometrically then by calculating the amount of water displaced in the graduated cylinder. Show your measured values and all your calculations.
- Which method to you think is more accurate? Why?
- Using the method you feel is more accurate, determine the density of copper.
- Using the table below, compare the value you just calculated to the accepted value using percent error.

Substance | Aluminum | Zinc | Tin | Iron | Copper | Silver | Lead | Gold |
---|---|---|---|---|---|---|---|---|

Density [g/cm3] |
2.70 | 7.08 | 7.31 | 7.87 | 8.92 | 10.50 | 11.34 | 19.32 |

### Activity 2: Finding density without measuring the volume

It is difficult to find the volume of an irregularly shaped object, e.g. an intricate golden crown. First of all, it is very difficult to determine the volume geometrically. Secondly, it is difficult to attain great precision by observing a change in water level. As observed in the density lab, volume is needed in order to determine the density of an object.

*ρ *=* m*/*V* (1)

If volume cannot be determined to any great accuracy, then how can one accurately determine the density of an object? Archimedes, according to legend, solved this problem while bathing. King Hieron had provided a quantity of pure gold to a smith to make into a crown. When the crown was complete, the king suspected the goldsmith of stealing some of the gold and substituting some other metal. The crown weighed the same as the original measure of gold, so Archimedes needed to know the density of the crown in order to determine whether there had been any foul play. He knew that the volume of the crown was equal to the amount of water it displaced, but needed a more precise method of measurement. While pondering this in the bath, Archimedes suddenly realized that he didn’t need to know the volume; only the weight of the water displaced. Since Archimedes was able to measure weight much more accurately than volume, this was very good news indeed!

*Archimedes’ Principle: **A body immersed in fluid is buoyed up by a force equal to the weight of the fluid displaced.*

*F*_{buoyant} = W_{water dispaced} (2)

Since the immersion of an object in water results in some water being lifted, then that action would cause an equal and opposite force back on the object (see Newton’s 3rd law). This is why an object in water seems to weigh less in water than out of the water. The observed difference in weight is equal to the weight of the water that the object displaces.

*W _{water displaced} = (W_{object out} – W_{object in}) *(3)

Now Archimedes could determine weight of the displaced water. He already knew the density of water (1.000 g/ml), and so he was now ready to calculate the volume of the displaced water, and then go on to calculate the density of the crown!

- How might Archimedes have determined the buoyant force (the difference in the weight of the crown in and out of the water)?
- How is the buoyant force related to the weight of the displaced water?
- If Archimedes knew the density of water and the weight of the displaced water, how could he then calculate the volume of the displace water?
- How is the volume of the displaced water related to the volume of the crown?
- If Archimedes knew the volume of the crown and the weight of the crown, how could he then calculate the density or the crown?
- As the tradition goes, Archimedes discovered that a quantity of silver had been mixed in with the gold, and so the goldsmith was exposed as a thief. What about his density results might have made Archimedes think that there could be some silver mixed in?
- Did Archimedes ever have to directly measure the volume of the crown or the volume of the displaced water?
- Given that the density of water is 1.000 g/ml, use Archimedes’ Principle to experimentally determine the density of the copper sample. When measuring the sample in the water, be sure that it is completely submerged and try to remove as many air bubbles as possible. Use the platform of the triple beam balance to support the cup while you measure the mass of the copper in and out of the water. Record these measurements and show all your calculations in determining the density.
- Compare your value of the density of copper to the accepted value using percent error.
- Which method of determining density of copper gave you the better result? The result from step #11 or from step #20?

### Activity 3: What are pennies really made of?

Before the middle of 1982, pennies were made of solid copper. Beginning about halfway into the year 1982, pennies started being made of a less dense metal core with very thin copper plating. You will determine what post-1982 pennies are made of.

- Is it better to use a single penny and determine its density, or it is better to combine multiple pennies together to determine their density as a unit? Why?
- What is your method for determining the density of a penny? Describe in detail.
- Calculate the density of a post-1982. Show all your work, and all your measured values.
- How will the small coating of copper on the outside of the mystery metal affect your measured density? How did you determine this?
- Compare your density calculation to the densities of the metals in the table above. What is your best estimate of the metal of the post-1982 pennies?

## Contact

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