Reflected inertia of a no-slip wheel and roller ($K=I+J(R/S{)}^2$)

If a fixed-axis wheel of radius $R$ drives a fixed-axis roller of radius $S$ without slip, their angular-speed magnitudes satisfy $\sigma =(R/S)\omega$. Their kinetic energy is $\frac{1}{2}[I+J(R/S{)}^2]{\omega }^2$, so the displayed reflected moment of inertia governs the wheel coordinate. A roller resisting torque of magnitude $\gamma {\sigma }^2$ consumes power $\gamma {\sigma }^3$ and therefore reflects to wheel torque $\gamma (R/S{)}^3{\omega }^2$. The squared ratio belongs to inertia; the cubed ratio belongs to this quadratic resisting torque.