## Modelling the motion of a floating object

Consider a homogeneous sphere floating in a large body of water. There are two forces acting on the sphere. First, there is the force of gravity acting downwards with a force equal to mg where m is the mass of the sphere and g is the accelaration due to gravity. According to Archimedes, the second force arises due to the water displaced by that portion of the sphere below the water line. This force is equal to m'g where m' is the mass of water which would be contained in the volume of the sphere below the water line. Let y(t) represent the height of the centre of the sphere above the water line as a function of time t. (Because it is being assumed that the body of water in which the sphere is floating is large, it may be assumed that the water line is constant). Using Newton's Second Law of Motion F = ma it follows that the following differential equation describes the motion of the sphere

where is the mass of the sphere, R is the radius of the sphere and is the density of water. Here the formula is used for the volume of a segment of a sphere of height H.

Analyse the motion of the sphere for different inital configurations and masses of the sphere. Use DEplot command which can be loaded by the command with(DEtools,DEplot);. Is the motion described always realistic? For which values of y(t) does the differential equation make sense? Consider what happens when the sphere is deeply submerged. How can the formula be modified so that it applies to all values of y(t)? The modification will involve a function defined by cases.

Next, add a resistive force acting against the direction of motion which is proportional to the velocity times the surface area in the direction of motion. This last term will be taken to be the radius of the sphere minus y(t) in order to keep things relatively simple. Of course, the surface area can be no greater than twice the radius of the sphere and no smaller than 0. This requires introducing a new function. Use this to set up a new differential equation modelling this situation and analyze the motion of the sphere under various initial conditions. A hollow sphere of radius 0.3 meters weighs 0.2 kilograms. How deeply must it be submerged so that it rises to a height of 0.2 meters above the water line?

Juris Steprans