Which of the following correctly
shows the SI unit for the moment of inertia? (A) Kilograms per meter squared,
(B) kilograms squared times meters, (C) kilograms squared times meter squared, (D)
kilograms times meter squared, (E) the quantity kilograms times meters squared.
Considering this question, we can
see that we’re not talking about moment of inertia for a particular shape, rotating
about a particular axis. But instead we’re speaking of this
term in general; we want our answer to give the units that apply to all moments of
inertia. Along these lines, we can recall
that this term, moment of inertia, applies to a mass that’s rotating. In general, the equation for the
moment of inertia of a particular mass rotating about a particular axis depends on
those two values. But nonetheless, it is true that
all moments of inertia share the same SI base units. One way then to identify which of
our five options is correct is to recall the equation for the moment of inertia of
any shape rotating about any axis.
Perhaps the simplest moment of
inertia we can recall is that of a point mass, where that mass is in rotation about
an axis a distance 𝑟 away. The moment of inertia of this point
mass is its mass 𝑚 times that distance 𝑟 squared. Now, like we said, not all moments
of inertia have the same form. In fact, most are different. But they do all have the same units
that this one has. And considering the units of this
expression, we know that the SI base units of mass are kilograms and that the SI
base unit of distance is the meter. So we have some mass in kilograms
multiplied by some distance in meters squared, which means that our units for this
expression will be kilograms times meter squared.
And like we said, these units apply
not just to the moment of inertia of a point mass, but to all moments of
inertia. Looking over our answers, we see
that option (D) matches up with what we found. And so we choose this as our
answer. Kilograms times meter squared are
the correct SI units for moment of inertia.