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Question Number 29093 by ajfour last updated on 04/Feb/18

Commented by ajfour last updated on 04/Feb/18

$T w o r o d s m u t u a l l y j o i n e d a t$ $h i n g e H, l i e o n a f r i c t i o n l e s s$ $h o r i z o n t a l s u r f a c e, a r e h i t b y$ $t w o p o i n t m a s s e s i n e l a s t i c a l l y .$ $(a) F i n d a n g u l a r v e l o c i t y o f e a c h$ $r o d j u s t a f t e r c o l l i s i o n .$ $(b) F i n d a l s o t h e v e l o c i t y o f h i n g e$ $j u s t a f t e r c o l l i s i o n .$ $A s s u m e h i n g e H i s f r i c t i o n l e s s$ $a n d f r e e t o m o v e .$
	Answered by mrW2 last updated on 04/Feb/18

	$e = c o e f f i c i e n t o f r e s t i t u t i o n$ $v = v e l o c i t y o f h i n g e a f t e r h i t (↑)$ $ω = a n g u l a r s p e e d o f r o d s (↷↶)$ $u_{1} = v e l o c i t y o f m a s s a f t e r h i t$ $u_{0} = v e l o c i t y o f c o m o f r o d$ $u_{0} = v + \frac{ω l}{2}$ $u_{2} = v e l o c i t y o f f r e e e n d o f r o d$ $v_{2} = v + ω l$ $v_{2} - u_{1} = e u$ $\Rightarrow v + ω l - u_{1} = e u$ $\Rightarrow v_{1} = v + ω l - e u$ $M u = {M u}_{1} + {m u}_{0}$ $\Rightarrow M u = M (v + ω l - e u) + m (v + \frac{ω l}{2})$ $\Rightarrow (M + m) v + (M + \frac{m}{2}) ω l = M (1 + e) u . . . (i)$ $M u l = {M u}_{1} l + \frac{{m l}^{2}}{12} ω + \frac{m l}{2} v_{0}$ $M u l = {M u}_{1} l + \frac{{m l}^{2}}{12} ω + \frac{m l}{2} (v + \frac{ω l}{2})$ $M u l = {M u}_{1} l + \frac{{m l}^{2}}{3} ω + \frac{m v l}{2}$ $\Rightarrow M u = M (v + w l - e u) + \frac{m l}{3} ω + \frac{m v}{2}$ $\Rightarrow (M + \frac{m}{2}) v + (M + \frac{m}{3}) ω l = M (1 + e) u . . . (i i)$ $\Rightarrow [(M + m) (M + \frac{m}{3}) - (M + \frac{m}{2}) (M + \frac{m}{2})] v = [(M + \frac{m}{3}) - (M + \frac{m}{2})] M (1 + e) u$ $\Rightarrow v = - \frac{2 M (1 + e) u}{4 M + m} = - \frac{2 (1 + e) u}{5}$ $\Rightarrow [(M + m) (M + \frac{m}{3}) - (M + \frac{m}{2}) (M + \frac{m}{2})] ω l = [(M + m) - (M + \frac{m}{2})] M (1 + e) u$ $\overset{}{\Rightarrow} ω = \frac{6 M (1 + e) u}{(4 M + m) l} = \frac{6 (1 + e) u}{5 l}$
	Commented by ajfour last updated on 04/Feb/18

	$y o u h a v e p r o v e d i t g e n e r a l l y e n o u g h$ $S i r, i^{'} d m e a n t c o m p l e t e l y i n e l a s t i c$ $c o l l i s i o n . T h a n k y o u s o m u c h S i r!$
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