What is Barycentric frame?

Definition 19.4 University Physics — Year 1 · Chapter 19 — Systems of Points; Introduction to Rigid Bodies

The barycentric frame $\mathcal R^*$ has its origin at $G$ and is in translation with respect to the inertial frame $\mathcal R$ (its axes keep fixed directions). Quantities measured in it are starred: $\vect v_i^* = \vect v_i - \vect v_G$ , and $\sum m_i\vect v_i^* = \vect 0$ .

Examples

Example 19.6 (A rolling wheel)

A wheel of mass $M$ , radius $R$ , moment of inertia $J$ about its axle, rolling without slipping at speed $v$ : the contact point is at rest, so $\omega = v/R$ , and $E_k = \tfrac12 Mv^2 + \tfrac12 J\omega^2 = \tfrac12 Mv^2(1 + J/MR^2)$ — three quarters of $Mv^2$ for a uniform disk ( $J = \tfrac12 MR^2$ ), $Mv^2$ for a hoop. The second term is why a bicycle’s wheels are made light: their rotation costs energy that the frame’s mass does not.

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