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Looks like you accidentally dropped a factor when you simplified the final kinetic energy term: after plugging in \(I_f = mr^2 + \tfrac{2}{5}MR^2\) and \(\omega_f = \tfrac{mr^2\,\omega_0}{mr^2 + \tfrac{2}{5}MR^2}\), you should get \(K_f = \tfrac12\bigl(mr^2 + \tfrac{2}{5}MR^2\bigr)\bigl[\tfrac{mr^2\,\omega_0}{mr^2 + \tfrac{2}{5}MR^2}\bigr]^2 = \tfrac12\,\tfrac{m^2r^4\omega_0^2}{mr^2 + \tfrac{2}{5}MR^2}\). When you then do \(K_i - K_f\), make sure you carefully factor out common terms; the difference should end up with \(\tfrac12\,\omega_0^2\bigl[\tfrac{2}{5}\,mMr^2R^2/(mr^2 + \tfrac{2}{5}MR^2)\bigr]\), so any discrepancy means you likely lost or gained a factor in the denominator or numerator while combining terms.
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Fri, 14 Feb 2025 21:28 GMT