This is the prelab theory and design for L21.

You're given a stack of ancient coins, all apparently identical but of unknown composition. Your problem is to determine if the coins could contain gold. You know that gold is very dense. So if you can measure the density of the coins, you can determine if the density is great enough for them to be at least partially composed of gold. Since the coins have value as artifacts whether or not they contain gold, you can't scratch them or treat them in solutions that would harm them. At most, you can put them in water for short periods of time. Your problem is to devise a method to determine the density of the coins and to use the physics that you've learned this semester to show that the method will work. You are provided with the following items and may use no other items.

• a graduated cylinder
• a plastic canister that that can be sealed to prevent water from coming in when the canister floats in water
• an overflow can (shown to the right). In use, water is filled to the bottom of the spout. When something is placed in the can, the amount of water displaced is measured by the volume of water that flows into the graduated cylinder. (Think of this as a small version of Archimedes' bathtub.)
• a supply of water

You're also told that the coins will all fit into the plastic canister and that the canister will fit in the overflow can. You also know the density of water and of gold.

Do the following:

1. Subject to the constraints given above, describe in paragraph form the method that you would use to determine the density of the coins.

2. Write a list of the symbols that you will use in your proof (see item 3) and describe each symbol in a phrase. Use clearly descriptive, subscripted symbols for mass, volume, and density. The symbols m, V, ρ without subscripts are unacceptable, as there are potentially more than one mass, volume, and density in this problem.

3. Present the theory of your method. That is, apply laws of physics that you know to derive a formula for the density of the coins in terms of things that you can measure using the equipment above. Present your theory as a proof in two columns, giving equations in the left-hand column and justification in words in the right-hand column.