A-rigid-body-is-made-of-three-identical-thin-rods-each-of-length-L-fastened-together-in-the-form-of-letter-H-The-body-is-free-to-rotate-about-a-horizontal-axis-that-runs-along-the-length-of-one-of-

Question Number 24873 by Tinkutara last updated on 28/Nov/17

A rigid body is made of three identical

thin rods, each of length L, fastened

together in the form of letter H . The

body is free to rotate about a horizontal

axis that runs along the length of one of

legs of H . The body is allowed to fall

from rest from a position in which plane

of H is horizontal . The angular speed

of body when plane of H is vertical is

Commented by Tinkutara last updated on 28/Nov/17

Commented by mrW1 last updated on 28/Nov/17

{l e t}^{'} s s a y e a c h l e g h a s a m a s s m .

I = \frac{{m L}^{2}}{3} + {m L}^{2} = \frac{4 {m L}^{2}}{3}

I α = m g (\frac{L}{2} + L) \cos θ = \frac{3 m g L}{2} \cos θ

\Rightarrow \frac{4 {m L}^{2}}{3} α = \frac{3 m g L}{2} \cos θ

\Rightarrow α = \frac{9 g \cos θ}{8 L} \dots (i)

\frac{1}{2} I ω^{2} = m g (\frac{L}{2} + L) \sin θ = \frac{3 m g L}{2} \sin θ

\Rightarrow \frac{1}{2} \times \frac{4 {m L}^{2}}{3} ω^{2} = \frac{3 m g L}{2} \sin θ

\Rightarrow ω^{2} = \frac{9 g}{4 L} \sin θ

\Rightarrow ω = (\frac{3}{2} \sqrt{\frac{g}{L}}) \sqrt{\sin θ} \dots (i i)

a t θ = \frac{π}{2} :

α = 0

ω = \frac{3}{2} \sqrt{\frac{g}{L}}

Commented by Tinkutara last updated on 28/Nov/17

W h a t i s θ ?

Commented by mrW1 last updated on 28/Nov/17

t h a t i s t h e p o s i t i o n o f t h e p l a n e :

θ = 0 m e a n s h o r i z o n t a l p o s i t i o n

θ = \frac{π}{2} m e a n s v e r t i c a l p o s i t i o n

Commented by mrW1 last updated on 28/Nov/17

Commented by Tinkutara last updated on 28/Nov/17

Thank you Sir!