<Mentor's note: Moved from a technical forum and thus no template.> A ring's kinetic energy is integral of 0.5v2 dm. Distance X is rΘ, and Θ is defined as distance traveled/radius, so X is r*distance traveled / r. Velocity V is X divided by time, so V is r*distance traveled / rt, and I define omega w as distance traveled / rt. Plugging into integral of 0.5v2dm, I get 0.5(r*distance traveled / rt)2dm, and I get 0.5mr2(distance traveled2/r2t2), which is equal to 0.5mr2w2. I find that my KE of the ring is 0.5mr2w2. For the approach to a disk, I can add the kinetic energy of concentric rings. So KE of ring is 0.5mr2w2, and w is distance traveled / rt, so I get 0.5mr2(distance traveled2/r2t2). Canceling the r2 gives me 0.5m(distance traveled2/t2). Integrating this from radius of 0 to r gives me 0.5mr(distance traveled2/t2). If I multiply by r2/r2, I get 0.5mr*r2(distance traveled2/t2r2), and (distance traveled2/t2r2) is w2, so my KE of the disk is 0.5mr3w2. Is this correct?