54 APPLIED MECHANICS 
(4) Circle (Fig. 57) of radius R about an axis passing through its 
centre and perpendicular to its plane. 
Consider an element of the form of a ring con- 
eentric with the circle and having a width dr and : 
a radius r, The area of this element is 2rrdr, 
its moment of inertia is 27r%dr, and the total 
moment of inertia 
R R 4 
h={ ee an an arnt : 
0 
0 Fiq, 57. 
(5) Circle of radius R about a diameter. If I is the moment of 
inertia of the circle about a diameter XOX, then I will also be the 
moment of inertia of the circle about a diameter YOY at right angles 
to XOX. But the moment of inertia about an axis through the centre 
O and perpendicular to the plane of the circle is by Theorem I, Art. 
68, p. 51, equal to I1+I=2I, and by the preceding example this is 
7R* aRt 
equal to 5 therefore [= —_. 
(6) A right prism or right cylinder of any cross-section about an 
axis X,X, (Fig. 58) in the plane of the 
base and passing through G, the centre 
of gravity of the base. 
Let a=area.of base, 7=length of 
solid, I,=moment of inertia of base 
about axis X,X,. Consider a thin 
parallel slice of thickness dz parallel 
to the base and at a distance x from it. 
‘Phe centre of gravity G of this slice 
will lie on the line G,G,, joining the 
centres of gravity of the ends. Take 
an axis XX through G and parallel to X,X,. Then, moment of 
inertia of slice about XX =I,dz, and ralomaant) oF inertia of slice about 
X,X,=1,de+au*dx. Hence I,, the moment of inertia of the whole 
soli Shout X,X,, is 
= [tates J ax*de =1 Jeet al sade, 1+ Zal’, 
(7) A solid of revolution about its axis. Fig. 59 shows the section 
of a solid wheel or pulley. Take a parallel strip of this section parallel 
to the axis XX of the solid. The. 
distances of the outside and inside of 
this strip from XX are R and ¢ re- 
spectively, and its mean width is 
AB=«a. Consider this strip as the sec- 
tion of a ring whose axis is XX. The 
moment of inertia of this ring about 
XX is approximately 5 (Bt —7*)x. 
If the ends of the ring are parallel 
its moment of inertia is. exactly X 
5(Rt - 14), and when the ends are © 
