For
a system of more than one particle, the net torque on the system is the
vector sum of the net torques for each particle. The system is assumed to
move as a unit, so that each particle has the same angular acceleration.
In that case it is clear from the second law that the moment of inertia of
the system is the sum of the individual moments of inertia.
If the system is a solid object, the moment of inertia may be calculated as an integral over the object's volume, as shown.
The simple expression in which the moment of inertia is a sum of scalar quantities applies to a limited set of situations: The position vectors and the force vectors on each of the particles must all lie in the same plane. The torques must all lie along the same axis. In more complex situations, the moment of inertia becomes a tensor.