342 r r a2 cos2 t 2 where 2 349 r note that

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Unformatted text preview: E, 2 where E ≡ K0 − mr0 ω 2 (which isn’t the energy). Therefore, 1 2 1 2 2 2 2 1 2 1 2 2 (330) 
 78 motion with frequency k/mCHAPTER the equilibrium value of r is METHOD, . Note that 6. THE LAGRANGIAN r0 = mg/k so this frequency may be written as g /r0 . ˙˙ Equating order frequencies we have found set θ ˙ To secondthe twoin small quantities, we can gives˙α cos(45◦ + θ − α) ≈ θα cos 45◦ . Also, cos(45◦ − θ) + cos(45◦ + θ) = 2 cos 45◦ cos θ. Keeping only the second-order √ terms, and ignoring additive gr0 = constants, g have r0 = R . we =⇒ (339) R − r0 r0 2 √ 2 ˙2 2 ˙α) − mgR(θ2 + α2 )/ 2. L = mR (θ + α + θ ˙ ˙ (346) 6.43. Oscillating hoop The equations of motion are If the hoop has rotated through an angle α counterclockwise, then the positions of √ ¨ 2R θ + R α = − 2g θ , ¨ the two masses are √ ¨ R θ + 2 R α = − 2g α . ¨ (347) (x, y )1 = R − sin(θ − α), − cos(θ − α) , Adding and subtracting these gives (x, y )2 = R sin(θ + α), − cos(θ + α) . (340) √ 2 d 2g (θ is Rα The ma...
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