Deep point: In uniform circular motion, $a_t=0$, but $\veca \neq 0$ because direction changes – a common conceptual hurdle.
$y(x,t) = A\cos(kx - \omega t + \phi)$, with $k = 2\pi/\lambda$, $\omega = 2\pi f$, $v = \lambda f = \omega/k$. --- Fisica Para Ingenieria Y Ciencias Ohanian Vol.1 3ed Pdf
For fixed-axis rotation: $\theta$, $\omega = d\theta/dt$, $\alpha = d\omega/dt$. Relationship to linear: $s = r\theta$, $v_t = r\omega$, $a_t = r\alpha$, $a_c = r\omega^2$. Non-conservative (friction, drag): dissipate energy as heat