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Derive the differential eq, governing the motion of the system, use x as the generalizing co-ordinate, assume small x and determine the natural frequency of the system.
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$KE = (KE)_{rod} + (KE)_M$

$= \frac{1}{2} \ IG \dot{\theta} ^2 \ + \ \frac{1}{2} \ M \ (\dot{x})^2 $

$= \frac{1}{2} [ \frac{mL^2}{12}] \ [ \frac{2 \dot{x}}{L}]^2 \ + \ \frac{1}{2} \ M{\dot{x}}^2$

$= \frac{1}{2} \ [ \frac{m}{3} + M] {\dot{x}}^2$

$PE \ = \ \frac{1}{2} \ k \ (x)^2$

$K_{eq}$ = K

$W.D. \ = \ - \int \ cy \ dy$

$y = \frac{L}{2} \ \theta = x$

$\dot{y} = \frac{L}{2} \ \dot{\theta} = \dot{x}$

= $- \int \ c \dot{x} \ dx$

$C_{eq}$ = C

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