Page 175 - Standard Handbook Of Petroleum & Natural Gas Engineering

160 General Engineering and Science

W

(a)

Figure 2-1 5. Diagram for Example 2-8.

ax = g(sin 30" - p cos 30")

ax = 13.31 ft/s'
Where unbalanced couples are involved, a rotational analog to Newton's second
law can be applied:

(2-25)

where M is the sum of all moments acting about the center of mass in the plane of
rotation, I is the mass moment of inertia about the center of mass, and a is the
angular acceleration of the body. The mass moment of inertia is defined by

I = r2dm = mk2 (2-26)

where r is the perpendicular distance from the axis of rotation to the differential
element of mass, dm. I is sometimes expressed in terms of k, the radius of gyration,
and m, the mass of the body. If the axis of rotation pas_ses through the center of mass,
then the mass moment of inertia is designated as I. Mass moments of inertia of
common shapes are compiled in Tables 2-6 and 2-7.
It is often convenient to sum the moments about some arbitrary point 0, other
than the mass center. In this case, Equation 2-25 becomes

C M, =h+m?id (2-27)
where m is the mass of the body, ?i is the linear acceleration of the mass center, and
d is the perpendicular distance between the vector ii and point 0.