You may have seen those "dumping buckets" at a pool or water park. There is a water source filling a bucket. At some point when the bucket becomes full it will suddenly tilt, splashing peo$-$ ple beneath. The idea of this worksheet is to analyze the physics of the dumping bucket us$-$ ing calculus. For purposes of this problem the bucket will be assumed to have a height of $0\mathrm{.}5$ m$.$ The vertical coordinate will be denoted by X$,$ and the pivot axis for the bucket is halfway up$,$ at height x $=0\mathrm{.}25$ m$.$ The cross$-$sectional ra$-$ dius at bottom, c $=0,$ is $0\mathrm{.}125$m and the cross$-$ sectional radius at top, x $=0\mathrm{.}5$m$,$ is $0\mathrm{.}5$m$.$ Our idealized bucket. Problem $1($The Bucket$).$ a$)$ If the bucket is a section of a circular cone then the radius of the cross$-$section grows linearly with height. Find an expression for the radius as a function of height. b$)$ Using the formula V $=|$ A$($y$)$dy$,$ where A$($y$)$ is the cross$-$sectional area at height y$,$ find the total volume of the bucket in liters. c$)$ Find the volume of water if the bucket is only partially filler, to a height $2.$ This will, of course, be a function of c$.$ The basic physics of these is that the buckets are on a pivot, and initially the center of mass of the bucket lies below the axis of the pivot and the bucket is stable. As the bucket fills the center mass of mass moves upward. At some point the center of mass lies above the pivot point, the bucket becomes unstable, and it tips over and splashes the people underneath. In the analysis below we are going to assume that the weight of the bucket is much less than the weight of the water, and can be ignored. Problem $2.($The Center of Mass$)$ a$)$ The center of mass is just the "average$$ vertical coordinate of the water, defined to be S yA$($Y$)$dy XCOM $=|$ A$($y$)$dy Explain in words why this represents an $$average$$ vertical coordinate. $($There are two more coordinates of the center of mass but they are less important $$ by symmetry they lie on the centerline of the cone.$)$ b$)$ Compute XCOM assuming that the bucket is filled to height $2.$ What fraction of the way up the bucket is the vertical center of mass when the bucket is completely filled x $=0\mathrm{.}5$ m$.$ Is it more or less than halfway up$?$ Explain why this makes sense. c$)$ Assume that the pivot point is located half$-$way up the cone, at x $=0\mathrm{.}25$ m$,$ and that the water is filling the bucket at $0\mathrm{.}5$ liter$/$s$.$ How long will it take for the bucket to dump? Is this number optimal for splashing fun? Discuss!