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Question Number 53824 by ajfour last updated on 26/Jan/19

Commented by ajfour last updated on 26/Jan/19

$C o n e i s s o l i d, w a l l i s s m o o t h . F i n d$ $m a x i m u m l e n g t h o f s t r i n g p o s s i b l e$ $t o h o l d t h e c o n e a g a i n s t t h e w a l l .$
	Answered by mr W last updated on 26/Jan/19

	Commented by mr W last updated on 26/Jan/19

	$i f t h e w a l l i s s m o o t h, t h e m a x i m u m$ $s t r i n g l e n g t h i s w h e n t h e s t r i n g$ $p a s s e s t h e p o i n t G .$ $r = h \tan α$ $e = \frac{h}{4} = c e n t r o i d o f c o n e$ $\tan θ = \frac{h - e}{r} = \frac{3}{4 \tan α}$ $\Rightarrow l_{m a x} = \frac{h}{\sin θ} = h \sqrt{1 + {(\frac{4 \tan α}{3})}^{2}}$
	Commented by ajfour last updated on 26/Jan/19

	$c o r r e c t a n s w e r S i r; e l e g a n t m e t h o d,$ $i a m n o t g e t t i n g t h i s r i g h t a n s w e r .$
	Commented by ajfour last updated on 26/Jan/19

	$T \cos θ = M g$ $M g (\frac{h}{4}) = T h \cos θ - T r \sin θ$ $\Rightarrow \frac{h}{4} = h - r \tan θ = h - h \tan α \tan θ$ $\Rightarrow \tan θ = \frac{3}{4 \tan α}$ $l = h / \sin θ = h \sqrt{1 + \frac{16}{9} \tan^{2} α} .$
	Commented by mr W last updated on 26/Jan/19

	$e x a c t l y!$
	Answered by mr W last updated on 26/Jan/19

	Commented by mr W last updated on 26/Jan/19

	$t h e s a m e q u e s t i o n b u t w i t h f r i c t i o n$ $o n t h e w a l l : μ$ $f = μ N$ $\tan φ = \frac{f}{N} = μ$ $i n t h i s c a s e t h e m a x i m u m s t r i n g$ $l e n g t h i s w h e n s t r i n g p a s s e s p o i n t F .$ $r = h \tan α$ $e = \frac{h}{4}$ $\tan θ = \frac{h - e}{r + e \tan φ} = \frac{3 h}{4 (h \tan α + \frac{h μ}{4})} = \frac{3}{4 \tan α + μ}$ $\sin θ = \frac{3}{\sqrt{3^{2} + {(4 \tan α + μ)}^{2}}} = \frac{1}{\sqrt{1 + {(\frac{μ + 4 \tan α}{3})}^{2}}}$ $\Rightarrow l_{m a x} = \frac{h}{\sin θ} = h \sqrt{1 + {(\frac{μ + 4 \tan α}{3})}^{2}}$
	Commented by ajfour last updated on 26/Jan/19

	$n e e d s o m e t i m e c o m p r e h e n d i n g$ $t h e m, S i r .$
	Commented by mr W last updated on 26/Jan/19

	$t h e p r i n c i p l e i s : w h e n a b o d y i s i n$ $e q u i l i b r i u m u n d e r t h r e e f o r c e s, t h e n$ $t h e s e t h r e e f o r c e s m u s t i n t e r s e c t a t$ $o n e p o i n t .$ $s u c h t h a t t h e s t r i n g i s a s l o n g a s$ $p o s s i b l e, t h e n o r m a l f o r c e f r o m t h e$ $w a l l s h o u l d b e a s h i g h a s p o s s i b l e,$ $t h e u p m o s t p o s i t i o n o f t h e n o r m a l$ $f o r c e i s t h e u p m o s t p o i n t o f t h e$ $b a s e o f t h e c o n e .$
	Commented by ajfour last updated on 26/Jan/19

	$T h a n k s S i r, v e r y u s e f u l c o n c e p t;$ $u n d e r s t a n d y o u r s o l u t i o n b e t t e r n o w .$
	Commented by Otchere Abdullai last updated on 26/Jan/19

	$t h u s w h y i s a i d t h i s m a n W i s a$ $p r o f e s s o r!$
	Commented by ajfour last updated on 27/Jan/19

	$m o r e t h a n t h a t, a n I d e a l o n e!$
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