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Find the frequency of small oscillatinos of a thin uniform vertical rod of mass ``m and length `l` hinged at the point `O` (figure). The combined stiffness fo the springs is equal to `x`. The mass of the springs is negligible. |
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Answer» Let us locate the rod at the position when it makes an angle `theta` from the vertical . In this problem both, the gravity and spring forces are restoring conservative forces, thus from the conservation of mechanical energy of oscillation of the oscillation system `:` `(1)/(2)(ml^(2))/(3)(theta)^(2)+mg(l)/(2)(1-cos theta)+(1)/(2)k(l theta)^(2)=` constant Differentiating w.r.t. time, we get `: ` `(1)/(2) (ml^(2))/(3)2 dot (theta)ddot(theta)+(mgl)/(2)sin thetadot(theta)+(1)/(2)k l^(2)2 theta dot (theta)=0` Thus for very small `theta` `ddot(theta)=-(3g)/(2l)(1+(kl)/(mg))theta` Hence, `omega_(0)=sqrt((3g)/(2l)(1+(kl)/(mg)))` |
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