# Buoyancy: Throwing rocks from boats (and weighing coins)

This is a simple demonstration of a classic buoyancy question about throwing a rock out of a boat (the full scale problem in discussed by Physics Girl here). The question is ‘if you are sitting in a boat holding a rock and throw the rock into the water will the water level go up or down’. The answer is that, when the rock is in the (floating) boat it displaces its weight but when it is thrown in the water (and sinks) it displaces its volume. Therefore, the water level drops when it is thrown out of the boat.

This is a simple scaled down version that clearly demonstrates the puzzle result but can also be used to weigh the heavy object (in this case coins).

Equipment

1. Small plastic cup
2. measuring cup
3. syringe (for small adjustments in the volume of water in the measuring cup
4. a hand full of identical coins (I used 9 US quarters).

Demonstration

1. fill the measuring cup so that the water level is one line below the top volume measurement when the empty cup is floating in the water
2. drop the coins in one by one until the water level rises up to the top line (record the number of coins needed)
3. pour the coins into the water and put the plastic cup back on the water.
4. note that the water level is below what it was when the coins were floating in the cup.

Analysis

The puzzle answer is fairly straightforward. However,the demonstration can also be used to estimate the weight of the individual coins (hence the need to use all the same coins). The change in volume recorded as the coins are added is equal to the volume displaced by the coins (see figure below). Therefore, the volume change multiplied by the density of water will equal the mass of the coins added. I measured a 50 ml change in volume when I added 9 quarters. The density of water is approximately 1 g/ml so the 9 coins have a mass of 50 g. The weight of the individual quarters is, therefore 50/9=5.56 g (which is very close to the actual standard mass of a quarter of 5.67 g). It is also possible to work out the volume of the coins by measuring the volume they displace when they sink (see figure). My measuring cup did not have enough resolution to get a good volume. The total volume of the 9 coins would be 7.5 ml based on US Mint dimensions.

Figure showing (from left to right) the cup displacing the coins weight, the empty cup, and the coins displacing their mass with the water level lower than when the coins are floating in the cup.

The other great thing about the demonstration is that it is easy to ask a lot of questions around the demonstration. Obviously you can pose the original question about the water level going up or down. There are also simple calculations that can be asked such as “what is the mass (or weight) of the coins?”, “what is the approximate density of the coins?”. The US mint website has information about size and metal content that can be used to calculate the actual density for comparison.

Ii is also really easy to have this as an activity in class. I used a version of it on the last day of class one semester using pennies.I used small and large plastic cups and gave the students the dimensions of the small cup and the equation for the cup volume (here). In this case you count how many pennies it takes to sink the cup so that the mass of pennies matches the density of water multiplied by the cup volume.  You will also need a roll of paper towel for the clean up. I did not explain the demonstration I just gave them the equipment and told the to calculate the weight of a penny. It was well received by the students though I had to give some groups a hint or two as to how it worked.

An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.