25.3: Rolling Without Slipping - Two Views
- Page ID
- 30534
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Think of a hoop, mass \(M\) radius \(R\), rolling along a flat plane at speed \(V\). It has translational kinetic energy \(\frac{1}{2} M V^{2}\), angular velocity \(\Omega=V / R\), and moment of inertia \(I=M R^{2}\) so its angular kinetic energy \(\frac{1}{2} I \Omega^{2}=\frac{1}{2} M V^{2}\) and its total kinetic energy is \(M V^{2}\).
But we could also have thought of it as rotating about the point of contact—remember, that point of the hoop is momentarily at rest. The angular velocity would again be \(\Omega\), but now with moment of inertia, from the parallel axes theorem, \(I=M R^{2}+M R^{2}=2 M R^{2}\), giving same total kinetic energy, but now all rotational.