First, you must use Newton's Laws, applied to
translational and rotational motion, in order to derive the relationship between
moment of inertia and the parameters measured experimentally.

If
the radius of the vertical shaft is R and the tension in the cord is T, then the
torque applied to the shaft is simply TRsin(q) = TR (since T and R are **perpendicular**)
and Newton's Second Law for the motion of the shaft is

t
= TR = I_{total }a ,
(4)

where
I_{total} is the total moment of inertia of the rotating system, which we
will split up into two parts,

I_{total}
= I_{0} + I_{added}.

I_{o
} is the moment of inertia of
the shaft and threaded rod (which will only be measured) and I_{added} is
the contribution to the moment of inertia due to the masses attached to the
threaded rod, theoretically given
by

I_{added}
=M_{1}r_{1}^{2}+ M_{2}r_{2}^{2},
(5)

Note:
r_{1} and r_{2} are the distances from the __center__ of each
__mass__ to the __center__ of the vertical __shaft__.

Newton's
Second Law for the motion of the weight hanger is

(6)

where T is the tension in the cord, m is the mass of the
weight hanger and a is the acceleration of the weight hanger.
Downward has been chosen to be the positive direction (as usual).

Since
they are moving together, the linear acceleration a of the weight hanger and the
angular acceleration a
of the shaft are related by

a=
a
R
(7)

Combining
equations (4), (6), and (7) gives

I |

We
can use this equation, along with measurements of a
=a/R,
to experimentally determine the moments of inertia.