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Question Number 134249 by abdurehime last updated on 01/Mar/21

Answered by mr W last updated on 01/Mar/21

Commented by mr W last updated on 01/Mar/21

$$1:stretchedposition$$$$2:unstretchedpositionofspring$$$$3:restposition$$$$ky=mg\sin \theta$$$$\Rightarrow y=\frac{mg\sin \theta }{k}$$$$atposition1:$$$$E=mg(x+y)\sin \theta +\frac{1}{2}{kx}^2$$$$atposition3:$$$$E=\frac{1}{2}I{\omega }^2+\frac{1}{2}m{\left(R\omega \right)}^2+\frac{1}{2}{ky}^2$$$$$$$$\frac{1}{2}I{\omega }^2+\frac{1}{2}m{\left(R\omega \right)}^2+\frac{1}{2}{ky}^2=mg(x+y)\sin \theta +\frac{1}{2}{kx}^2$$$$(I+{mR}^2){\omega }^2=2mgx\sin \theta +{kx}^2+\frac{{\left(mg\sin \theta \right)}^2}{k}$$$$\omega =\sqrt{\frac{2mgx\sin \theta +{kx}^2+\frac{{\left(mg\sin \theta \right)}^2}{k}}{I+{mR}^2}}$$$$=\sqrt{\frac{2\times 0.5\times 9.71\times 0.2\times \sin 37°+50\times 0.2^2+\frac{{(0.5\times 9.81\times \sin 37°)}^2}{50}}{1+0.5\times 0.3^2}}$$$$=1.792rad/s$$

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