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Question Number 47005 by ajfour last updated on 03/Nov/18

Commented by ajfour last updated on 03/Nov/18

$A s t i c k o f l e n g t h L i s h e l d b a l a n c e d$ $a g a i n s t t w o m u t u a l l y p e r p e n d i c u l a r$ $f r i c t i o n l e s s w a l l s, w i t h t h e h e l p$ $o f t w o s t r i n g s A C (l e n g t h b) a n d$ $B C (l e n g t h a) . F i n d T e n s i o n s$ $a n d N o r m a l r e a c t i o n s T_{1}, T_{2}, N_{1},$ $a n d N_{2} . (m a s s o f s t i c k i s m) .$
	Answered by MrW3 last updated on 04/Nov/18

	Commented by MrW3 last updated on 04/Nov/18

	${(a \cos α)}^{2} + {(b \cos β)}^{2} + {(b \sin β - a \sin α)}^{2} = L^{2}$ $a^{2} + b^{2} - 2 a b \sin α \sin β = L^{2}$ $\Rightarrow \sin α \sin β = \frac{a^{2} + b^{2} - L^{2}}{2 a b} . . . (i)$ $T_{a} \sin α = \frac{m g}{2}$ $\Rightarrow T_{a} = \frac{m g}{2 \sin α}$ $s i m i l a r l y$ $\Rightarrow T_{b} = \frac{m g}{2 \sin β}$ $N_{1} = T_{a} \cos α = \frac{m g \cos α}{2 \sin α} = \frac{m g}{2 \tan α}$ $N_{2} = T_{b} \cos β = \frac{m g \cos β}{2 \sin β} = \frac{m g}{2 \tan β}$ $N_{1} \times b \cos β = N_{2} \times a \cos α$ $\Rightarrow \frac{m g}{2 \tan α} \times b \cos β = \frac{m g}{2 \tan β} \times a \cos α$ $\Rightarrow \frac{b}{\sin α} = \frac{a}{\sin β} . . . (i i)$ $(i i) i n t o (i) :$ $\frac{a}{b} \sin^{2} α = \frac{a^{2} + b^{2} - L^{2}}{2 a b}$ $\sin α = \sqrt{\frac{a^{2} + b^{2} - L^{2}}{2 a^{2}}}$ $\Rightarrow α = \sin^{- 1} \sqrt{\frac{a^{2} + b^{2} - L^{2}}{2 a^{2}}}$ $s i m i l a r l y$ $\Rightarrow β = \sin^{- 1} \sqrt{\frac{a^{2} + b^{2} - L^{2}}{2 b^{2}}}$ $\Rightarrow T_{a} = \frac{m g}{2 \sin α} = m g \frac{a}{\sqrt{2 (a^{2} + b^{2} - L^{2})}}$ $\Rightarrow T_{b} = \frac{m g}{2 \sin β} = m g \frac{b}{\sqrt{2 (a^{2} + b^{2} - L^{2})}}$ $\Rightarrow N_{1} = \frac{m g}{2 \tan α} = \frac{m g}{2} \times \sqrt{\frac{1}{\sin^{2} α} - 1}$ $\Rightarrow N_{1} = \frac{m g}{2} \sqrt{\frac{a^{2} + L^{2} - b^{2}}{a^{2} + b^{2} - L^{2}}}$ $\Rightarrow N_{2} = \frac{m g}{2 \tan β} = \frac{m g}{2} \times \sqrt{\frac{1}{\sin^{2} β} - 1}$ $\Rightarrow N_{2} = \frac{m g}{2} \sqrt{\frac{b^{2} + L^{2} - a^{2}}{a^{2} + b^{2} - L^{2}}}$
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