1) The volume of the raft is the product between the lenghts of its three dimensions:
2) The mass of the raft is the product between its density, d, and its volume, V:
3) The weight of the raft is the product between its mass m and the gravitational acceleration,
:
4) The apparent weight is equal to the difference between the weight of the raft and the buoyancy (the weight of the displaced fluid):
where
is the water density and
is the volume of displaced fluid.
The density of the raft (
) is smaller than the water density (
), this means that initially the buoyancy (which has upward direction) is larger than the weight (downward direction) and so the raft is pushed upward, until it reaches a condition of equilibrium and it floats. At equilibrium, the weight and the buoyancy are equal and opposite in sign:
and therefore, the apparent weight will be zero:
5) The buoyant force B is the weight of the displaced fluid, as said in step 4):
When the raft is completely immersed in the water, the volume of fluid displaced
is equal to the volume of the raft,
. Therefore the buoyancy in this situation is
However, as we said in point 4), the raft is pushed upward until it reaches equilibrium and it floats. At equilibrium, the buoyancy will be equal to the weight of the raft (because the raft is in equilibrium), so:
6) At equilibrium, the mass of the displaced water is equal to the mass of the object. In fact, at equilibrium we have W=B, and this can be rewritten as
where
is the mass of the displaced water. From the previous equation, we obtain that
.
7) Since we know that the mass of displaced water is equal to the mass of the raft, using the relationship
we can rewrite
as:
and so
8) The volume of water displaced is (point 7)
. This volume is now "filled" with part of the volume of the raft, therefore
is also the volume of the raft below the water level. We can calculate the fraction of raft's volume below water level, with respect to the total volume of the raft,
:
Viceversa, the volume of raft above the water level is
. Therefore, the fraction of volume of the raft above water level is
9) Let's repeat steps 5-8 replacing
, the water density, with
, the ethanol density.
9-5) The buoyant force is given by:
when the raft is completely submerged. Then it goes upward until it reaches equilibrium and it floats: in this condition, B=W, so the buoyancy is equal to the weight of the raft.
9-6) Similarly as in point 6), the mass of the displaced ethanol is equal to the mass of the raft:
9-7) Using the relationship
, we can find the volume of displaced ethanol:
9-8) The volume of raft below the ethanol level is equal to the volume of ethanol displaced:
. Therefore, the fraction of raft's volume below the ethanol level is
Consequently, the raft's volume above the ethanol level is
and the fraction of volume above the ethanol level is
