Imagine a force
F acting on some object at a distance r from its axis of
rotation. We can break up the force into tangential (Ftan),
radial (Frad) (see Figure 1). (This is assuming a
twodimensional scenario. For three dimensions  a more realistic, but
also more complicated situation  we have three components of force: the
tangential component Ftan, the radial component Frad
and the zcomponent Fz. All components of force are
mutually perpendicular, or normal.)
From Newton's Second Law,
However, we know that angular acceleration, ,
and the tangential acceleration atan are related by:
atan = r
Then,
Ftan = m r
If we multiply both sides by r (the moment arm), the equation becomes
Ftan r = m r
Note that the radial component of the force goes through the axis of rotation,
and so has no contribution to torque. The left hand side of the equation
is torque. For a whole object, there may be many torques. So the sum of
the torques is equal to the moment of inertia (of a particle mass, which
is the assumption in this derivation), I = m r
multiplied by the angular acceleration, .

Figure 1 Radial and Tangential Components of Force, two dimensions
Figure 2 Radial, Tangential and zComponents of Force, three
dimensions 