Resource Lesson
Thin Rods: Moment of Inertia
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For linear, or translational, motion an object's resistance to a change in its state of motion is called its
inertia
and it is measured in terms of its mass, (kg). When a rigid, extended body is rotated, its resistance to a change in its state of rotation is called its
rotational inertia
, or
moment of inertia
. This resistance has a two-fold property.
the amount of mass present in the object and
the distribution of that mass about the chosen axis of rotation
Recall that the equation used to calculate the moment of inertia of a collection of discrete masses about an arbitrary axis of rotation is
where
r
is the perpendicular distance from the axis of rotation to each mass. We can use this same process for a continuous, uniform thin rod having a
mass per unit length (kg/m), λ
. To begin, let's divide our rod into sections having a constant mass, Δm
_{i}
, each a distance r
_{i}
from the pivot point.
Now let's make the Δm sections smaller and smaller, that is, let's take the limit as Δm → 0.
In general, for a continuous, rigid body, the moment of inertia is calculated with the equation
Unfortunately we cannot calculate the given integral because we can't integrate x
^{2}
with respect to "dm." We must either express
x in term of m
or
dm in terms of dx
. We will use the rod's uniform
mass per unit length (kg/m), λ
, to facilitate this substitution.
Now let's use this process to calculate the moment of inertia of a uniform, thin rod, rotated about its center of mass.
Below is a series of diagrams for a thin rod illustrating how the moment of inertia for the same object can change with the placement of the axis of rotation. Notice, that the farther the pivot point is from the object's center of mass, the greater its moment of inertia.
axis: far end of a thin rod
axis: one-fourth of the way from
the end of a thin rod
axis: center of a thin rod
These results would indicate that a thin rod would be most easily rotated about an axis through its center of mass (
I
= 4/48 mL
^{2}
= 1/12 mL
^{2}
) than about one of its far ends (
I
= 16/48 mL
^{2}
= 1/3 mL
^{2}
). Consider a majorette. If she twirls her baton about its center of gravity, for the same amount of torque she will achieve a greater rate of angular acceleration than if she twirls the same baton about a pivot closer to one of its ends. Consequently, a drum major maneuvering his mace or a member of the band's flag corps spinning their flag pole have to deal with larger moments of inertia and therefore have more difficulty accelerating their respective apparatus.
Parallel Axis Theorem
Using the
parallel axis theorem
,
we can calculate the rod’s moment of inertia about any point as long as we know that position's distance (
h
) from the object's center of mass. Let's practice by calculating the moment of inertia of a thin rod about its left end.
Notice that this is the same result that we would have obtained had we integrated our basic definition for the moment of inertia.
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