Comparison of Linear (Translational) and Rotational Motion

Every physical quantity, and every governing equation, used across this entire chapter to describe the

rotation of a rigid body about a fixed axis has an EXACT counterpart already met, earlier in the course,

for straight-line (translational) motion of a single particle. The accompanying table gathers this

complete correspondence into one place, quantity for quantity and equation for equation.

The pattern behind every single row is the same, simple substitution: replace ordinary mass $m$ by moment

of inertia $I$, replace force $F$ by torque $\tau$, and replace every LINEAR kinematic quantity

(displacement, velocity, acceleration) by its corresponding ANGULAR quantity (angular displacement, angular

velocity, angular acceleration) -- and every formula from straight-line mechanics turns, without any further

change, into the matching formula for rotational mechanics about a fixed axis. Newton's second law $F=ma$

becomes $\tau = I\alpha$; linear momentum $p=mv$ becomes angular momentum $L=I\omega$; the three constant-

acceleration kinematic equations carry over unchanged in form; kinetic energy $\tfrac12mv^2$ becomes

rotational kinetic energy $\tfrac12I\omega^2$; and work done $W=Fs$ becomes $W=\tau\theta$.

This correspondence is far more than a convenient bookkeeping trick -- it reflects a single, deeper

mathematical fact: the equations governing rotation about a fixed axis are derived, throughout this

chapter, by literally the same reasoning used earlier for straight-line motion, merely applied to the

angular variables $\theta,\omega,\alpha$ in place of the linear variables $s,v,a$, and to torque and …