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A Moment of Inertia, Please!

Just as in the translational [straight-line motion] part of Mechanics there is an inertial factor  m  which determines how much  p  you get for a given   $v \equiv \dot{x}$  and how much   $a \equiv \dot{v} \equiv \ddot{x}$  you get for a given  F, so in rotational Mechanics there is an angular analogue of the inertial factor that determines how much  L_O  you get for a given   $\omega \equiv \dot{\theta}$  and how much   $\alpha \equiv \dot{\omega}$  you get for a given  $\tau_O$. This angular inertial factor is called the moment of inertia about  O  [we must always specify the origin about which we are defining torques and angular momentum] and is written  I_O  with the prescription

$$I_O = \int r_{\perp}^2 \, dm$$ (11.21)

where the integral represents a summation over all little ``bits'' of mass  dm  [we call these ``mass elements''] which are distances  $r_{\perp}$  away from an axis through the point  O. Here we discover a slight complication:  $r_{\perp}$  is measured from the axis, not from  O  itself. Thus a mass element  dm  that is a long way from  O  but right on the axis will contribute nothing to  I_O. This continues to get more complicated until we have a complete description of Rotational Mechanics with  I_O  as a tensor of inertia and lots of other stuff I will never use again in this course. I believe I will stop here and leave the finer points of Rotational Mechanics for later Physics courses!