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*)?

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 relatuvely 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 rise
to a height of 0.2 meters above the water line?

Thu Mar 27 17:08:37 EST 1997