fisica1-youn-e-freedman-exercicios-resolvidos

9.99: a) Following the hint, the moment of inertia of a uniform sphere in terms of the
2 2 8π
5
mass density is I = ΜR = ρR ,
5
15
and so the difference in the moments of inertia of
two spheres with the same density ρ but different radii
5
R2 and R1
is Ι = ρ(8π
15)( R2
− R
5
1
).
b) A rather tedious calculation, summing the product of the densities times the
difference in the cubes of the radii that bound the regions and multiplying by
24
4π 3, gives M = 5.97 × 10 kg. c) A similar calculation, summing the product of the
densities times the difference in the fifth powers of the radii that bound the regions and
22 2
2
multiplying by 8π
15, gives I = 8.02 × 10 kg ⋅ m = 0.334MR
.
9.100: Following the procedure used in Example 9.14 (and using z as the coordinate
2 2
4
R R
πρ R 4
along the vertical axis) r(z) = z ,dm = πρ z dz and dΙ = z dz.
Then,
h
2
h
2
4
h
Ι
=
4
4
πρ R h
4 πρ R 5 h 1 4
∫ dΙ = ∫ z dz = [ z ]
0=
πρR h .
4
2
h
0
10
h
10
The volume of a right circular cone is V
=
2
πR
h,
the mass is
π
1 1 2
R h
3
3
and so
2
⎛ πρR
h ⎞ 2 3
Ι =
R = ΜR
10
⎜
3
⎟
⎝ ⎠ 10
3 2
.