# Archimedes

## Cartesian Diver

Ever wondered how a submarine sinks and floats? The demonstration here can be used to explain the changes in forces involved and is going to impress most people who see it for the first time. It consists of a floating object inside a sealed plastic bottle that sinks when the bottle is given a tight squeeze and floats again when the squeeze is released.

I have seen the Cartesian diver being made with something else, such as a packet of ketchup or a dropper. The method given below works better than those and uses things that are easily available around the house.

Materials

1. A plastic water-bottle
2. A pen cap
3. Some modelling clay
4. Water

Procedure

1. The first step is to attach some modelling clay on the tail of the pen cap to serve as weight so that when placed into water, the pen cap floats upright. There has to be just enough weight added so that the pen cap will “just float”. That is, if any more is added, the cap will sink. It takes some time to find the balance and the best way to do so is to test it in a basin of water.
2. Once the correct weight is attached to the pen cap, place it upright into the filled water bottle and close the cap.
3. Test it out by giving the bottle a tight squeeze. (If it remains afloat even when you have given it the tightest squeeze, take the pen cap out and add more weight.
4. If it sinks straightaway, remove some weight. This should not be necessary if we have already carried ou the t test in the basin.)

Physics Principles Explained

There are two ways to explain this demonstration, one for those who cannot be bothered with equations, and the other for those who are keen on delving deeper.

Using the simple idea of density, we can explain that when the bottle is squeezed, some of the water enters the pen cap and compresses the air trapped within. Hence, the collective density of the submerged pen cap, together with its air and water content, increases. (Note that we are not referring to the density of the pen cap alone, which is a constant.) When this density exceeds that of the water around it, the pen cap sinks. The action is reversed when the squeeze is released.

Some would prefer an alternative explanation. This invokes the idea of forces acting on the pen cap, namely, upthrust and weight. Archimedes’ principle, otherwise known as the law of buoyancy, states that the any object that is partially or fully submerged in a fluid (liquid or gas) experiences an upward force known as the upthrust that is equal in magnitude to the weight of the fluid which is displaced. In mathematical terms,

$$U=\rho Vg$$

where $\rho$ is the density of the fluid, V is the volume of the fluid that is displaced and g is the acceleration of free-fall.

This force opposes the weight of the object and the result determines the direction that the object will move.

For the case of the Carteesian diver, upthrust is varied by changing the volume of fluid, V, that is displaced by the air within the pen cap. When the bottle is squeezed, part of the original volume of air is now occupied by the water which enters due to a higher pressure. This means that the volume of fluid displaced decreases, and as a result, upthrust decreases.