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.
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
= TR = Itotal a ,
Itotal is the total moment of inertia of the rotating system, which we
will split up into two parts,
= I0 + Iadded.
is the moment of inertia of
the shaft and threaded rod (which will only be measured) and Iadded is
the contribution to the moment of inertia due to the masses attached to the
threaded rod, theoretically given
r1 and r2 are the distances from the center of each
mass to the center of the vertical shaft.
Second Law for the motion of the weight hanger is
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).
they are moving together, the linear acceleration a of the weight hanger and the
angular acceleration a
of the shaft are related by
equations (4), (6), and (7) gives
can use this equation, along with measurements of a
to experimentally determine the moments of inertia.