(12 points) In this question, we will compare the masses, mg and mE, and moments of...

 

Transcribed Image Text:

(12 points) In this question, we will compare the masses, mg and mE, and moments of inertia, Ig and Ig, of both a solid sphere B = {r? + y² + 22 < 1} and a solid truncated sphere (that is, sliced to remove a spherical cap) E = {r² + y? + 2? < 1}n {x < } of uniform densities, pg and pE, that are precessing about the r-axis. (a) (5 points) Show that the masses of the unit and truncated spheres are mg = s0T and mg = (b) (5 points) The moment of inertia Iy of an object V rotating about the r-axis is Iy = (² + 2*)p(x, y, z) dV. Using cylindrical coordinates, compute the moments of inertia Ig and Ig of the unit sphere and the truncated sphere. (e) (2 points) If unit sphere B and truncated sphere E have the same mass M, but different uniform densities, then which object will have the larger moment of inertia? Which will roll down an inclined plane fastest? (12 points) In this question, we will compare the masses, mg and mE, and moments of inertia, Ig and Ig, of both a solid sphere B = {r? + y² + 22 < 1} and a solid truncated sphere (that is, sliced to remove a spherical cap) E = {r² + y? + 2? < 1}n {x < } of uniform densities, pg and pE, that are precessing about the r-axis. (a) (5 points) Show that the masses of the unit and truncated spheres are mg = s0T and mg = (b) (5 points) The moment of inertia Iy of an object V rotating about the r-axis is Iy = (² + 2*)p(x, y, z) dV. Using cylindrical coordinates, compute the moments of inertia Ig and Ig of the unit sphere and the truncated sphere. (e) (2 points) If unit sphere B and truncated sphere E have the same mass M, but different uniform densities, then which object will have the larger moment of inertia? Which will roll down an inclined plane fastest?