Moment of Inertia Experiment Purpose: To determine the shape and ...

Moment of Inertia ExperimentPurpose: To determine the shape and material of the body inside of theblack sphere.Theory: There are two ways to approach the theory behind thisexperiment. One is to think about conservation of energy, and the otheris to think about adding up the forces involved.Energy: Conservation of Energy tells us that in the absence of friction, theenergy before, E i , =the energy after, E f . The apparatus can be spun bydropping a mass, m, from rest at a height, h. Use conservation of energyto derive the following relationship where r is the radius of the spool ofstring that turns the sphere:mga =I/r 2 + m or I = ⎝ ⎜⎛ ga -1 ⎠ ⎟⎞ mr2 .I is the moment of inertia of the sphere and its contents.When Ir2 >>m, we can approximate the relationship between the massacceleration as a linear one where a = mgr2IProcedure:Measure the acceleration for a varietyof masses. Determine the accelerationas the mass drops and theacceleration as the mass rises. I canbe determined by making a plot of aas a function of m for each axis; I isrelated to the slope of the line. Youcan make three such graphs, oneusing a up , one using a down , and oneusing a avg .Error!After you have determined the threed=2rmoments of inertia (one about eachaxis) and measured the mass of themasssphere select a shape that isconsistent with your data. The possible shapes are cylinder,parallelepiped, annulus, cube, or disk. Find the dimensions of the shape.

y solving simultaneous equations for the moment of inertia of each axis(subtract the moment of inertia of the sphere, I = 2.76 x 10 -3 kgm 2 , fromthe moment of inertia about each axis). Then calculate the density of thebody and compare it to brass (ρ = 8.47 x 10 3 kg/m 3 ) and aluminum(ρ = 2.7 x 10 3 kg/m 3 ). If it is not consistent with either density, pick a newshape and start over.Questions:1. Do your plots of mass vs. acceleration go through the origin. Why orwhy not?2. Is it possible to determine all of the variables for all of the shapes?If not, which shapes remain ambiguous? Why?Bonus: Analyze the theory involved in thinking about this system usingNewton's laws, ΣF = ma. Include friction, it is significant.The design of this experiment is based on a paper by Joe Amato:Amato, Joseph C., Roger E. Williams and Hugh Helm, "A 'black box'moment of inertia apparatus", Am. J. Phys., 63, 1995, pp. 891-894.