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Derive the differential equation governing the motion of the system, using $\theta$ as the generalized co-ordinate.
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1] $K E \ = \ \frac{1}{2} \ I_o \dot{θ}^2$

$= \frac{1}{2} \ [ \frac{mL^2}{12} \ + \ \frac{ml^2}{16}]^ \ \dot{θ}^2$

$Ieq \ = \ \frac{ml^2}{12} \ + \ \frac{ml^2}{16}$

2] $PE \ = \ \frac{1}{2} \ \times \ k \ \times \ (\frac{3L}{4} \ \theta)^2$

$= \frac{1}{2} \ ( k \ \times \ \frac{9L^2}{16}) \ \theta^2$

$Keq \ = \ \frac{9KL^2}{16}$

3] Work done = $- \ \int \ c \dot{y} \ dy$

$= \ - \int \ C. \ \frac{L}{4} \ \dot{θ}. \ d \ (\frac{L}{4} \theta)$

$= \int \ \frac{CL^2}{16} \ \dot{θ} \ d \ \theta$

$Ceq \ = \ \frac{CL^2}{16}$

4] $Meq \ = \ Fo. \ \frac{L}{4} \ - \ Mo$

$Ieq \ \ddot{θ} \ + \ C \ eq \ \dot{θ} \ + \ K \ eq . \ \theta \ = \ Meq . \ sin \ wt$

$0.145 \ mL^2 \ \ddot{θ} \ + \ \frac{CL^2}{16} \ \theta \ + \ \frac{9kL^2}{16} \ \theta = [ \frac{foL}{4} \ = \ Mo] \ sin \ wt$

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