386 CHAPTER 20. THE INTEGRAL IN OTHER COORDINATES

9. Let R denote the finite region bounded by z = 9− x2− y2 and the xy plane. Find themoment of inertia of this shape about the x axis given the density equals 1.

10. Let B be a solid ball of constant density and radius R. Find the moment of inertiaabout a line through a diameter of the ball. You should get 2

5 R2M where M is themass..

11. Let B be a solid ball of density σ = ρ where ρ is the distance to the center of the ballwhich has radius R. Find the moment of inertia about a line through a diameter ofthe ball. Write your answer in terms of the total mass and the radius as was done inthe constant density case.

12. Let C be a solid cylinder of constant density and radius R. Find the moment of inertiaabout the axis of the cylinder

You should get 12 R2M where M is the mass.

13. Let C be a solid cylinder of constant density and radius R and mass M and let B be asolid ball of radius R and mass M. The cylinder and the ball are placed on the top ofan inclined plane and allowed to roll to the bottom. Which one will arrive first andwhy?

14. A ball of radius 4 has a cone taken out of the top which has an angle of π/2 and thena cone taken out of the bottom which has an angle of π/3. If the density is λ = ρ ,find the z component of the center of mass.

15. A ball of radius 4 has a cone taken out of the top which has an angle of π/2 and thena cone taken out of the bottom which has an angle of π/3. If the density is λ = ρ ,find the moment of inertia about the z axis.

16. Suppose a solid of mass M occupying the region B has moment of inertia, Il about aline, l which passes through the center of mass of M and let l1 be another line parallelto l and at a distance of a from l. Then the parallel axis theorem states Il1 = Il +a2M.Prove the parallel axis theorem. Hint: Choose axes such that the z axis is l and l1passes through the point (a,0) in the xy plane.

17. ∗ Using the parallel axis theorem find the moment of inertia of a solid ball of radiusR and mass M about an axis located at a distance of a from the center of the ball.Your answer should be Ma2 + 2

5 MR2.

18. Consider all axes in computing the moment of inertia of a solid. Will the smallestpossible moment of inertia always result from using an axis which goes through thecenter of mass?

19. Find the moment of inertia of a solid thin rod of length l, mass M, and constantdensity about an axis through the center of the rod perpendicular to the axis of therod. You should get 1

12 l2M.

20. Using the parallel axis theorem, find the moment of inertia of a solid thin rod oflength l, mass M, and constant density about an axis through an end of the rod per-pendicular to the axis of the rod. You should get 1

3 l2M.