Smugglers. Rumor has it that a company has been smuggling gold out of the country using sealed, cylindrical barrels with hollow walls. They pour molten gold into the hollows, and then fill the remainder of the barrel's internal volume with packing peanuts. The total mass of the gold-walled barrel was designed so that it exactly matches those used to transport a volatile chemical that cannot be exposed to air (and therefore the barrel cannot be opened and checked). The X-ray machine usually used to screen containers is suspiciously damaged and not available. (a) There are 20 barrels total, and they are all identical: mass $m=50.0 \mathrm{kg}$, height $h=1.20 \mathrm{m},$ and diameter $D=0.25 \mathrm{m} .$ How do you determine which ones have walls filled with gold (and are essentially hollow on the interior except for packing peanuts) and those completely filled with the volatile chemical (a tightly-packed powder) where the mass is uniformly distributed? Hint: apply the concepts of moment of inertia. Assume that, in the case of the gold-filled barrels, the entire mass is concentrated at the outer wall of the barrel and, in the case of the barrels filled with the chemical, the mass is distributed evenly throughout the volume of the cylinder. You can neglect the circular bottoms and the lids of the barrels, and assume there is no slipping. (b) What is the acceleration of the center of mass of each of the barrels as they roll down a $30^{\circ}$ inclined plane? (c) How much time does it take each barrel to roll $10.0 \mathrm{m}$ down the $30^{\circ}$ plane?