1. The problem statement, all variables and given/known data
See picture :
2. Relevant equations
##\sum M_{O}=I_{O}\ddot{\theta }##
3. The attempt at a solution
Consider the free-body diagram associated with an arbitrary positive angular displacement ##\theta##; The moment about point ##O## is given by
##\sum M_{O}=-k\left ( b\sin\theta \right )\left ( b\cos\theta \right )-k\left ( 2b\sin\theta \right )\left ( 2b\cos\theta \right )-\underbrace{\left ( 3mgb\sin\theta \right )\left ( 3b\cos\theta \right )}_{\mathit{Why \ shouldn't \ this \ be \ included?}}##
Further, by the parallel axis theorem, ##I_{0}=\overline{I}+md^2=0+m(3b)^2=9mb^2## and for small oscillations ##\sin\theta\simeq \theta \ \ \wedge \cos\theta\simeq 1## and ##\tau =\frac{2\pi}{\omega _{n}}=2\pi\sqrt{\frac{m}{k}}##. BUT why does not the mass of the sphere contribute to the moment?
See picture :
2. Relevant equations
##\sum M_{O}=I_{O}\ddot{\theta }##
3. The attempt at a solution
Consider the free-body diagram associated with an arbitrary positive angular displacement ##\theta##; The moment about point ##O## is given by
##\sum M_{O}=-k\left ( b\sin\theta \right )\left ( b\cos\theta \right )-k\left ( 2b\sin\theta \right )\left ( 2b\cos\theta \right )-\underbrace{\left ( 3mgb\sin\theta \right )\left ( 3b\cos\theta \right )}_{\mathit{Why \ shouldn't \ this \ be \ included?}}##
Further, by the parallel axis theorem, ##I_{0}=\overline{I}+md^2=0+m(3b)^2=9mb^2## and for small oscillations ##\sin\theta\simeq \theta \ \ \wedge \cos\theta\simeq 1## and ##\tau =\frac{2\pi}{\omega _{n}}=2\pi\sqrt{\frac{m}{k}}##. BUT why does not the mass of the sphere contribute to the moment?
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