A hollow metal sphere of outer radius 6.0cm is constructed from a material of density ρm=7.80gcm−3. The sphere contains a concentric spherical cavity of unknown radius and encloses only air. When placed in fresh water, the sphere floats with exactly one-third of its volume above the waterline.
(a) Using the observed floating condition, determine the average density of the sphere (metal shell plus enclosed air).
(b) Calculate the radius of the internal cavity required to achieve the observed floating condition.
(c) The same sphere is now placed in a tank of oil of density 0.85gcm−3. Predict quantitatively whether the sphere will float or sink in the oil, and determine the fraction of its volume that will be submerged if it floats.
(d) Suppose a small hole were drilled to allow water (but not oil) to enter the cavity when the sphere is in fresh water. Explain qualitatively how this modification would affect the sphere’s ability to float in both fluids.