calculation of the moi

Calculation of the MOI

Calculation of the MOI

[1

Let's now apply [1] to a thin uniform rod shown in Figure 1. The MOI of the rod about the Y axis is

Figure 1
[2]

since

[3]

where r = the density of the rod, m = the mass of the rod, and L = the length of the rod.

The moment of inertia of the uniform circular ring shown in Figure 2 about the Z axis (the symmetry axis) is

Figure 2
[4]

since

[5]

where dl = the length of the arc formed by dq. [5] is also applicable to a circular cylinder.

A uniform circular disc of radius R can be considered as a cascade of uniform circular rings as shown in Figure 3. Thus, from [5], the moment of inertia about the Z axis (the symmetry axis) becomes

Figure 3
[6]

since

[7]

[6] can be used in computing the MOI of a circular bar about its longitudinal axis as well.

A uniform sphere can be considered as a cascade of uniform circular discs as shown in Figure 4. From [6], the moment of inertia about the vertical axis (Z axis) is

Figure 4
[8]

since

[9]