= Spring-mounted rolling cylinder
{title2=$\left(m+I/R^2\right)\ddot x+b\dot x+kx=0$}
Attach a <spring> of <spring constant> $k$ to the axle of a round body of <mass> $m$, radius $R$ and axial <moment of inertia> $I$. Assume <rolling without slipping>, a horizontal <drag force> $-b\dot x$ applied through the axle, and no additional resisting <torque>. The translational <force> equation is $m\ddot x=-kx-b\dot x+f$, where $f$ is signed <static friction>. The rotational equation and rolling constraint give $f=-I\ddot x/R^2$, yielding the displayed equation. Equivalently, the <kinetic energy> is $(m+I/R^2)\dot x^2/2$. This added effective inertia lowers the undamped <angular frequency> to $\sqrt{k/(m+I/R^2)}$. A distributed viscous <torque> would add a separate damping term and cannot be inferred from the drag force alone.
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