A rigid body is made of three identical thin rods, each with length l fastened together in the form of a letter H. The body is free to rotate….

Q: A rigid body is made of three identical thin rods, each with length l fastened together in the form of a letter H. The body is free to rotate about a horizontal axis that runs along the length of one the lengths of the H. The body is allowed to fall from rest from a position in which the plane of the H is horizontal. What is the angular speed of the body when the plane of the H is vertical?

Numerical

(a) $ \displaystyle \frac{2}{3}\sqrt{\frac{g}{l}} $

(b) $\displaystyle \frac{1}{3}\sqrt{\frac{g}{l}} $

(c) $\displaystyle \frac{3}{2}\sqrt{\frac{g}{l}} $

(d) $ \displaystyle \frac{4}{3}\sqrt{\frac{g}{l}} $

Ans: (c)

Sol:
$ \displaystyle I_{zz’} = \frac{ml^2}{3}+ml^2 = \frac{4ml^2}{3}$

From conservation of energy

$ \displaystyle mgl/2 +mgl = \frac{1}{2}I\omega^2$

$ \displaystyle \frac{3}{2}mgl = \frac{1}{2}\frac{4}{3}ml^2\omega^2$

$\displaystyle \omega^2 = \frac{9g}{4l}$

$ \displaystyle \frac{3}{2}\sqrt{\frac{g}{l}} $

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