Assume that the function x$($t$)$x$($t$)$ represents the length of tape that has unwound as a function of time. Find $($t$)$$($t$),$ the angle $($in radians$)$ through which the drum will have rotated, as a function of time.

You found that the total kinetic energy is the sum of the kinetic energy in the center of mass plus the kinetic energy of the center of mass. A similar decomposition exists for angular and linear momentum. There are also related decompositions that work for systems of masses, not just rigid bodies like a dumbbell.

It is important to understand the applicability of the formula Ktot$=$Kr$+$KtKtot$=$Kr$+$Kt$.$ Which of the following conditions are necessary for the formula to be valid?