4.3 Solutions
153
Fig. 4.18
Then the moment of inertia about an axis parallel to the slice and passing
through the centre of mass is given by the parallel axis theorem, Fig. 4.18.
dI C =
1
4
dm R
2
+ dm z
2
= π R
2
ρ
R 2
4
+ z
2
dz
where dm = π R 2 ρdz is the mass of the slice and ρ is the density.
∴ I C =
dI C = π R
2
ρ
+h/2
−h/2
R 2
4
+ z
2
dz
= π R
2
ρ
R 2
4
h +
h 3
12
But ρ =
M
π R 2 h
∴ I C =
M
12
3R
2
+ h
2
4.5 The moment of inertia of the larger solid sphere of mass M
I 1 =
2
5
Ma
2
(1)
The moment of inertia of the smaller solid sphere of mass m, which is removed
to hollow the sphere, is
I 2 =
2
5
mb
2
(2)