Archimedes' principle can be put to practical use in medicine, where it is often desired to assess what percentage of a person's body is fat. Since fat is less dense than water and other constituents of the body, we intuitively expect bodies with higher fat percentages to float more (experience a greater buoyant force). The first thing to do is to measure the patient's average density. Suppose Kevin, the patient, is hoisted with a rope into a water tank deep enough to be completely submerged. The tension is adjusted so that Kevin remains static, then its magnitude is measured with a force gauge.
A. Draw a free-body diagram for Kevin. Use the following labels for your forces: FG → weight, FT → tension, FB → buoyant force.
B. The reading on the force gauge (that is, the tension) is 35 N. If Kevin's mass is 75 kg, what is the magnitude of the buoyant force (in Newtons)?
C. Using the magnitude of the buoyant force just calculated, find Kevin's average density in g/cm³. The value you'll get is very close to the density of water, so round to the nearest second decimal. (Hint: Since Kevin is completely submerged, the buoyant force is proportional to Kevin's volume and the density of water while the force of gravity on Kevin is proportional to Kevin's volume and his density).
D. Now that we have the density, we need to correct for other effects like the air trapped inside Kevin's lungs and gastrointestinal tract (there will also be some residual amount of air in Kevin's lungs even after exhaling as hard as he can). There is a handy formula incorporating these additional effects, allowing us to calculate a male's body fat percentage knowing only their density: Fat % = [4.950 / Density - 4.500] × 100, where the density is given in units of g/cm³. According to this formula, what is Kevin's fat percentage?
E. Initially, Kevin is suspended in the air above the water. As he is lowered and slowly submerged in the tank at a constant velocity, how does the buoyant force change over time?
F. As Kevin is being submerged in the tank at a constant velocity, the tension force is changing over time, too. Explain why.