The Energy of One Moving Link

The viewer can (1) explain why T = ½m‖ξ̇‖² + ½J_Cω² needs the frame at the CoM, (2) derive v_C = v + ωρ̃, (3) expand ½m‖v + ωρ̃‖² + ½J_Cω² and collect it into ½𝒱ᵀ𝔗𝒱, (4) read 𝔗: parallel-axis corner, spin–slide coupling, diagonal when ρ = 0.