156
4 Rotational Dynamics
I 1 =
m
6
(AB)
2
=
m
6
4
2
=
8m
3
I 2 =
m
6
(BC)
2
=
m
6
3
2
=
3m
2
I 3 =
m
6
(BD)
2
=
m
6
12
5
2
=
24m
25
∴ I 1 > I 2 > I 3
4.9 If the radius of the sphere is r then the volume of the sphere must be equal to
that of the disc:
4
3
π r
3
= π R
2 R
6
∴ r =
R
2
The moment of inertia of the disc I = I D = (1/2) m R 2
The moment of inertia of the sphere
I S =
2
5
mr
2
=
2
5
m
R 2
4
=
1
5
×
1
2
m R
2
=
1
5
I D
4.10 Consider a strip of radius r on the surface of the sphere symmetrical about the
z-axis and width Rdθ , where R is the radius of the hollow sphere, Fig. 4.22.
Fig. 4.22
Area of the strip is 2πr · Rd θ = 2π R 2 sin θ d θ . If σ is the surface mass density (mass per unit area) then the mass of the strip is dm = 2π R 2 σ sin θ d θ .
Moment of inertia of the elementary strip about the z-axis
dI = dm r
2
= 2π R
4
σ sin
3
θ d θ
Moment of inertia contributed by the entire surface will be