>>11299683Moment of inertia is the rotational analogue of mass. For linear motion, you have F=ma (F=force, m=mass, a=acceleration); for rotation you have ?=I? where ?=torque, I=moment of inertia, ?=angular acceleration.
Consider a point mass m rotating about an axis with rotational speed ?, where the distance between the mass and the axis is r. Its linear speed is v=r? and its linear acceleration a=dv/dt=d(r?)/dt=r(d?/dt)=r?. So the force is F=mr? and the torque is ?=Fr=mr^2?=I? where I=mr^2.
For a distributed (non-point) mass, you need to integrate over its volume, effectively summing infinitely many ?m.r^2 point masses. For a 1-dimensional object (conventionally termed a "rod"), you integrate ?r^2 w.r.t. r over [-l/2,l/2] where l is the length and ?=m/l is the mass per unit length. The antiderivative is ?r^3/3 so the definite integral is ?(l/2)^3/3-?(-l/2)^3/3 = ?l^3/12 = ml^2/12.
This can be used for objects with one dimension much larger than the others, so the (radial) distance from the centre of mass can be approximated as the linear distance measured along the "length" axis.
This also assumes that the axis of rotation is perpendicular to the length (i.e. you're basically dealing with rotation in the plane). In 3 dimensions, the moment of inertia is replaced with a matrix and the math becomes somewhat more complex (for an object not at rest, you have to allow for precession, which is what makes gyroscopes work).
See the wikipedia page for "moment of inertia" for more information.
