Resource Lesson
A Chart of Common Moments 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 is measured in terms of its mass, in kg. When a rigid body is rotated, its resistance to a change in its state or rate of rotation is called its
rotational inertia
, which is measured in terms of its moment of inertia, in kg m
^{2}
. 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.
In general, the formula for a single object's moment of inertia is
I
_{cm}
= kmr
^{2}
where
k
is a constant whose value varies from 0 to 1. Different positions of the axis result in different moments of inertia for the same object; the further the mass is distributed from the axis of rotation, the greater the value of its moment of inertia.
That is, the smaller the coefficient of mr
^{2}
, the easier it is to accelerate the object. That is, spheres accelerate easier than cylinders, which accelerate easier than thin rings or hoops. Since an object's moment of inertia increases as its mass is moved further from its axis of rotation, hoops and rings would represent the greater inertia since all of their mass is concentrated at a constant distance, r, from the center of rotation.
Below is a series of diagrams illustrating how the moment of inertia for the same object can change with the placement of the axis of rotation. This is not an all inclusive list, but it is a "most used" list.
solid sphere
I
= 2/5 MR
^{2}
thin-walled sphere
I
= 2/3 MR
^{2}
thin rod
(perpendicular at end)
I
= 1/3 ML
^{2}
thin rod
(perpendicular at center)
I
= 1/12 ML
^{2}
solid cylinder
(about central axis)
I
= 1/2 MR
^{2}
thin-walled cylinder/hoop/ring
(about central axis)
I
= MR
^{2}
thick-walled cylinder
(about central axis)
I
= 1/2 M(R
_{1}
^{2}
+ R
_{2}
^{2}
)
solid cylinder
(perpendicular to central axis)
I
= 1/4 MR
^{2}
+ 1/12 ML
^{2}
thin-walled cylinder
(perpendicular to central axis)
I
= 1/2 MR
^{2}
+ 1/12 ML
^{2}
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