Chemistry Reference
In-Depth Information
In this case it is commonly known as the torque. The concept has been extended
to more abstract applications such as the moment of an area with respect to a
plane and moments of statistical distributions. It is then referred to as the appro-
priate first moment (the term torque is not used).
The second moment of force about the same axis is the product of the force
and the square of the distance between its line of action and the axis. This is the
moment of inertia. The most direct example of its use is possibly connected with
the motion of a rotating body, for which the rotational acceleration caused by an
applied torque is calculated by dividing the torque by the moment of inertia of
the body. The concept of a second moment has been extended to other less read-
ily pictured applications such as computation of stresses in beams from second
moments of cross-sectional areas about particular axes.
By extending the above examples we can say that a moment in mechanics is
generally defined as
U
j
5
Fd
j
(2-15)
where U
j
is the jth moment, about a specified line or plane a of a vector or scalar
quantity F (for example, force, weight, mass, area), d is the distance from F to
the reference line or plane, and j is a number. The moment is named according to
the power j to which d is raised. If F is composed of elements F
i
each located a
distance d
i
from the same reference, the moment is the sum of the individual
moments of each element
X
F
i
d
i
U
j
5
(2-16)
i
Mathematically, there is no restriction on the choice of F or j, but use of
moments to solve practical mechanics problems usually confines F to the exam-
ples listed above and j to values of 1 or 2. The reference line or plane must be
specified when the value of the moment is quoted.
In polymer science the mathematical formulation for moments corresponds to
that in
Eq. (2-16)
. While the reference line may be located anywhere, the useful-
ness of choosing the ordinate (M
5
0) in the graph of the molecular weight distri-
bution (
Figs. 2.2 and 2.4
) is so great that this reference is usually not mentioned
explicitly. The distance d from the reference line is measured along the abscissa
in terms of the molecular weight M, and the quantity F is replaced by f
i
, the pro-
portion of the polymer with molecular weight M
i
. As a matter of utility, j assumes
a wider range of values in polymer science than in mechanics. With these differ-
ences, which are mainly matters of emphasis, the concepts of moments corre-
spond closely in both disciplines. A general definition of a statistical moment of a
molecular weight distribution taken about zero is then
X
q
i
M
i
U
j
(2-17)
