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Question Number 41255 by ajfour last updated on 04/Aug/18

Commented by ajfour last updated on 04/Aug/18

$T h e n a t u r a l l e n g t h o f s p r i n g i s$ $l a n d i f r e a l e a s e d a s s h o w n, f i n d$ $s p e e d o f r i n g a s i t r e a c h e s t h e$ $g r o u n d . μ i s t h e f r i c t i o n c o e f f i c i e n t$ $b e t w e e n v e r t i c a l r o d a n d t h e r i n g .$
	Answered by ajfour last updated on 05/Aug/18

	$A t a n i n t e r m e d i a t e p o s i t i o n, l e t$ $t h e a n g l e o f s p r i n g l i n e w i t h$ $h o r i z o n t a l b e θ .$ $N o r m a l r e a c t i o n o n r i n g$ $N = k x \cos θ$ $x = l (\sec θ - 1)$ $f r i c t i o n f = μ k l (1 - \cos θ)$ $h e i g h t o f r i n g i s y = l \tan θ$ $d y = l \sec^{2} θ d θ$ $W_{f} = \int_{α}^{0} μ {k l}^{2} (1 - \cos θ) \sec^{2} θ d θ$ $w h e r e α = \tan^{- 1} (\frac{3}{4})$ $\Rightarrow W_{f} = - μ {k l}^{2} \int_{0}^{α} (\sec^{2} θ - \sec θ) d θ$ $= - μ {k l}^{2} [\tan α - \ln (\sec α + \tan α)]$ $= - μ {k l}^{2} [\frac{3}{4} - \ln (\frac{5}{4} + \frac{3}{4})]$ $= - μ {k l}^{2} (\frac{3}{4} - \ln 2)$ $\frac{1}{2} {m v}^{2} = m g (\frac{3 l}{4}) + \frac{1}{2} k {[l \sec α - l]}^{2}$ $- μ {k l}^{2} (\frac{3}{4} - \ln 2)$ $v^{2} = \frac{3 g l}{2} + \frac{{k l}^{2}}{16 m} - \frac{μ {k l}^{2}}{m} (\frac{3}{2} - 2 \ln 2)$ $v = \sqrt{\frac{3 g l}{2} + \frac{{k l}^{2}}{16 m} - \frac{μ {k l}^{2}}{m} (\frac{3}{2} - 2 \ln 2)} .$
	Commented by MrW3 last updated on 05/Aug/18

	$g r e a t j o b!$
	Answered by tanmay.chaudhury50@gmail.com last updated on 05/Aug/18
	$at any instant the ring is at a height y from the ground and make angle ∝ with vertical rod. the length of spring=(√(l^2 +y_ ^2 )) so elongation of spring=((√(l^2 +y^2 )) −l) as the spring elongated so it tries to go back to its original length..so a force k((√(l^2 +y^2 )) −l) act along the spring downward.the force k((√(l^2 +y^2 )) −l) has two components 1)k((√(l^2 +y^2 )) −l)cos∝ vertically down in the same direction of mg of ring 2)k((√(l^2 +y^2 )) −l)sin∝ horigontal normal forve on ring... 3)frction force=μk((√(l^2 +y^2 )) −)sin∝ eqn mv(dv/dy)=mg+k((√(l^2 +y^2 )) −l)cos∝−μk((√(l^2 +y^2 )) −l)sin∝ v(dv/dy)= mg+k((√(l^2 +y^2 )) −l).(y/(((√(l^2 +y^2 )))))−μk((√(l^2 +y_ ^2 )) −l)(l/(√(l^2 +y^2 ))) mg+k(y−((ly)/(√(l_ ^2 +y^2 ))))−μk(l−(l^2 /(√(l^2 +y^2 _ )))) v(dv/dy)= g+(k/m)(y−((ly)/(√(l^2 +y^2 ))))−((μk)/m)(l−(l^2 /(√(l^2 +y^2 )))) v(dv/dy)=g+(k/m)y−((μkl)/m)+((μkl^2 )/m).(1/(√(l^2 +y^2 )))−((kl)/m).(y/(√(l^2 +y^2 ))) (v^2 /2)=∣gy+(k/m).(y^2 /2)−((μkl)/m)y+((μkl^2 )/m).ln(y+(√(l^2 +y^2 ))) − ((kl)/(m ))(√(l^2 +y^(2 ) )) ∣_0 ^((3l)/4) (v^2 /2)=3g(l/4)+(k/m).((9l^2 )/(32))−((μkl)/m).((3l)/4)+((μkl^2 )/m){ln(((3l)/4)+((5l)/4))− ln(0+(√(l^2 +0)) )}−((kl)/m)((√(l^2 +((9l^2 )/(16)))) −l) (v^2 /2)=((3gl)/4)+((9kl^2 )/(32m))−((3μkl^2 )/(4m))+((μkl^2 )/m){ln2l−lnl}−((kl)/m)((l/4)) (v^2 /2)=((3gl)/4)+((kl^2 )/m)((9/(32))−(1/4))−((3μkl^2 )/(4m))+((μkl^2 )/m)ln2 (v^2 /2)=((3gl)/4)+((kl^2 )/m).(1/(32))+((μkl^2 )/m)(ln2 −(3/4)) v={((3gl)/2)+((kl^2 )/m).(1/(16))+((2μkl^2 )/m) (ln2 −(3/4))}^(1/2) ={((3gl)/2)+((kl^2 )/m).(1/(16))−((μkl^2 )/m)((3/2)−2ln2)}^(1/2)$
	$a t a n y i n s t a n t t h e r i n g i s a t a h e i g h t y f r o m t h e g r o u n d$ $a n d m a k e a n g l e \propto w i t h v e r t i c a l r o d .$ $t h e l e n g t h o f s p r i n g = \sqrt{l^{2} + y_{}^{2}} s o e l o n g a t i o n o f$ $s p r i n g = (\sqrt{l^{2} + y^{2}} - l)$ $a s t h e s p r i n g e l o n g a t e d s o i t t r i e s t o g o b a c k$ $t o i t s o r i g i n a l l e n g t h . . s o a f o r c e k (\sqrt{l^{2} + y^{2}} - l)$ $a c t a l o n g t h e s p r i n g d o w n w a r d . t h e f o r c e$ $k (\sqrt{l^{2} + y^{2}} - l) h a s t w o c o m p o n e n t s$ $1) k (\sqrt{l^{2} + y^{2}} - l) c o s \propto v e r t i c a l l y d o w n i n t h e$ $s a m e d i r e c t i o n o f m g o f r i n g$ $2) k (\sqrt{l^{2} + y^{2}} - l) s i n \propto h o r i g o n t a l n o r m a l f o r v e$ $o n r i n g . . .$ $3) f r c t i o n f o r c e = μ k (\sqrt{l^{2} + y^{2}} -) s i n \propto$ $e q n$ $m v \frac{d v}{d y} = m g + k (\sqrt{l^{2} + y^{2}} - l) c o s \propto - μ k (\sqrt{l^{2} + y^{2}} - l) s i n \propto$ $v \frac{d v}{d y} =$ $m g + k (\sqrt{l^{2} + y^{2}} - l) . \frac{y}{(\sqrt{l^{2} + y^{2})}} - μ k (\sqrt{l^{2} + y_{}^{2}} - l) \frac{l}{\sqrt{l^{2} + y^{2}}}$ $m g + k (y - \frac{l y}{\sqrt{l_{}^{2} + y^{2}}}) - μ k (l - \frac{l^{2}}{\sqrt{l^{2} + {\underset{}{y}}^{2}}})$ $v \frac{d v}{d y} =$ $g + \frac{k}{m} (y - \frac{l y}{\sqrt{l^{2} + y^{2}}}) - \frac{μ k}{m} (l - \frac{l^{2}}{\sqrt{l^{2} + y^{2}}})$ $v \frac{d v}{d y} = g + \frac{k}{m} y - \frac{μ k l}{m} + \frac{μ {k l}^{2}}{m} . \frac{1}{\sqrt{l^{2} + y^{2}}} - \frac{k l}{m} . \frac{y}{\sqrt{l^{2} + y^{2}}}$ $\frac{v^{2}}{2} =∣ g y + \frac{k}{m} . \frac{y^{2}}{2} - \frac{μ k l}{m} y + \frac{μ {k l}^{2}}{m} . l n (y + \sqrt{l^{2} + y^{2})} -$ $\frac{k l}{m} \sqrt{l^{2} + y^{2}} ∣_{0}^{\frac{3 l}{4}}$ $\frac{v^{2}}{2} = 3 g \frac{l}{4} + \frac{k}{m} . \frac{9 l^{2}}{32} - \frac{μ k l}{m} . \frac{3 l}{4} + \frac{μ {k l}^{2}}{m} {l n (\frac{3 l}{4} + \frac{5 l}{4}) -$ $l n (0 + \sqrt{l^{2} + 0})} - \frac{k l}{m} (\sqrt{l^{2} + \frac{9 l^{2}}{16}} - l)$ $\frac{v^{2}}{2} = \frac{3 g l}{4} + \frac{9 {k l}^{2}}{32 m} - \frac{3 μ {k l}^{2}}{4 m} + \frac{μ {k l}^{2}}{m} {l n 2 l - l n l} - \frac{k l}{m} (\frac{l}{4})$ $\frac{v^{2}}{2} = \frac{3 g l}{4} + \frac{{k l}^{2}}{m} (\frac{9}{32} - \frac{1}{4}) - \frac{3 μ {k l}^{2}}{4 m} + \frac{μ {k l}^{2}}{m} l n 2$ $\frac{v^{2}}{2} = \frac{3 g l}{4} + \frac{{k l}^{2}}{m} . \frac{1}{32} + \frac{μ {k l}^{2}}{m} (l n 2 - \frac{3}{4})$ $v = {\frac{3 g l}{2} + \frac{{k l}^{2}}{m} . \frac{1}{16} + \frac{2 μ {k l}^{2}}{m} (l n 2 - \frac{3}{4})}^{\frac{1}{2}}$ $= {\frac{3 g l}{2} + \frac{{k l}^{2}}{m} . \frac{1}{16} - \frac{μ {k l}^{2}}{m} (\frac{3}{2} - 2 l n 2)}^{\frac{1}{2}}$
	Commented by MrW3 last updated on 04/Aug/18

	$v e r y n i c e!$
	Commented by MrW3 last updated on 05/Aug/18

	$y o u m e a n t h e a n s w e r f r o m a j f o u r s i r .$ $y o u r a n s w e r a n d h i s a n s w e r a r e i d e n t i c a l .$ $You can't use 'macro parameter character #' in math mode$ $w o r k i n g a b o v e .$
	Commented by tanmay.chaudhury50@gmail.com last updated on 04/Aug/18

	$y e s s i r i h a v e r e c t i f i e d i t . . . y o u r s a m d m y s e l f$ $a n s w e r b e c o m e i d e n t i c a l . . . t h a n k y o u . . y o u h a h e$ $g i v e n y o u r t i m e t o g o t h r o u v h m y p o s t . . .$
	Commented by tanmay.chaudhury50@gmail.com last updated on 05/Aug/18

	$o k s i r l e t m e c h e c k . . .$
	Commented by tanmay.chaudhury50@gmail.com last updated on 05/Aug/18

	$p l s c h e c k$
	Commented by MrW3 last updated on 05/Aug/18

	$You can't use 'macro parameter character #' in math mode$ $s p r i n g = (\sqrt{l^{2} + y^{2}} - l)$
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