Example 1, where the mass of the object is concentrated in two balls at the ends of the
rod. The moment of inertia of the rod is very small compared to that of the balls, and we
do not include it in our calculations. We also consider each ball to be concentrated at its
own center of mass when measuring its distance from the axis of rotation (marked by
the ×). This is a reasonable approximation when the size of an object is small relative to
its distance from the axis.
Not all situations lend themselves to such simplifications. For instance, let’s assume we
want to calculate the moment of inertia of a CD spinning about its center. In this case
the mass is uniformly distributed across the entire CD. In such a case, we need to use
calculus to sum up the contribution that each particle of mass makes to the moment, or
we must take advantage of a table that tells us the moment of inertia for a disk rotating
around its center.
I = Ȉmr^2
I = moment of inertia
m = mass of a particle
r = distance of particle from axis
Units: kg·m^2
What is the system's moment of
inertia? Ignore the rod's mass.
I = Ȉmr^2 = m 1 r 12 + m 2 r 22
I = (2.3 kg)(1.3 m)^2 + (1.2 kg)(1.1 m)^2
I = 5.3 kg·m^2
10.4 - A table of moments of inertia
Sets of objects are shown in the illustrations above and to the right. Above each object is a description of it and its axis of rotation. Below each
object is a formula for calculating its moment of inertia, I.
The variable M represents the object’s mass. It is assumed that the mass is distributed uniformly throughout each object.
If you look at the formulas in each table, they will confirm an important principle underlying moments of inertia: The distribution of the mass
relative to the axis of rotation matters. For instance, consider the equations for the hollow and solid spheres, each of which is rotating about an
axis through its center. A hollow sphere with the same mass and radius as a solid sphere has a greater moment of inertia. Why? Because the
mass of the hollow sphere is on average farther from its axis of rotation than that of the solid sphere.
Note also that the moment of inertia for an object depends on the location of the axis of rotation. The same object will have different moments
of inertia when rotated around differing axes. As shown on the right, a thin rod rotated around its center has one-fourth the moment of inertia as
the same rod rotated around one end. Again, the difference is due to the distribution of mass relative to the axis of rotation. On average, the
mass of the rod is further away from the axis when it is rotated around one end.
Moments of inertia
Cylinders