If we have a system which consists of a number (i) of
masses, each with mass m_{i} with a distance from the axis of
rotation, r_{i},
the moment of inertia of the system is defined as

.
(1)

In words, multiply each mass by the square of its
distance from the axis of rotation and sum all of these factors.
Thus we may calculate the moment of inertia of a collection of objects.

Alternatively, we can measure the moment of inertia
of an object (or a system of masses) by measuring the angular acceleration a that results from a given applied torque
t. The
rotational analog of Newton’s second law of motion then gives

I = t/a.
(2)

In this lab you will experimentally measure the
moment of inertia of an object using Equation 2, and then we will check that
measurement against the theoretical value, Equation 1.
You will use the smart pulley to measure the linear acceleration of an
object subjected to a constant applied torque.
You will use the same apparatus you used to measure centripetal force,
except that the mass and spring have been removed and the crossarm has been
replaced by a threaded rod. Four wing nuts on the threaded rod are used to clamp two
masses near its ends. The
threaded rod and its attached masses are set into rotation by wrapping a
string around the vertical shaft, running the string over the pulley,
attaching the string to a weight hanger and allowing the weight hanger to
fall.

The constant tension in the string applied **tangent**
to the vertical shaft gives rise to a constant torque and consequently to a
constant angular acceleration. This
acceleration will be obtained by measuring the angular velocity of rotation and
using the relation:

v
= v_{0}
+ at
(3)

We can plot **v** as a function of **t**,
yielding a straight line of slope **a** as indicated in the equation.
(Or you can plot the angular velocity as a function of time and then
relate the linear acceleration to the angular acceleration by a=aR where R is the radius of the vertical shaft.) You will then be asked
to compare the moment of inertia obtained from the acceleration measurement, as
in Equation 2, with the moment of inertia obtained by the use of Equation 1.