A uniform chain of mass M and length L is hanging from the table. The chain is in limiting equilibrium when l length of chain over hangs. It is slightly disturbed from this position. Find the speed of the chain just after it completely comes off the table.


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Question Number 23288 by Tinkutara last updated on 28/Oct/17

$A uniform chain of mass M and length$ $L is hanging from the table . The chain$ $is in limiting equilibrium when l length$ $of chain over hangs . It is slightly$ $disturbed from this position . Find the$ $speed of the chain just after it completely$ $comes off the table .$
Commented by Tinkutara last updated on 28/Oct/17

Commented by Physics lover last updated on 28/Oct/17

$I s i t$ $\sqrt{g (L - l)} ?$
Commented by Physics lover last updated on 28/Oct/17

$T h e a n s w e r g i v e n i n t h e b o o k i s$ $\sqrt{\frac{g L}{(1 + μ)}}$ $u t t h e a n s w e r w h i c h i g o t i s s a m e .$ $μ = \frac{l}{L - l} \Rightarrow l = \frac{μ L}{(1 + μ)}$ $i f y o u s u b s t i t u t e l i n m y a n s w e r$ $\sqrt{g (L - l)}$ $= \sqrt{g {L - \frac{μ L}{(1 + μ)}}}$ $= \sqrt{g {\frac{L + μ L - μ L}{(1 + μ)}}}$ $= \sqrt{\frac{g L}{(1 + μ)}}$
	Answered by Physics lover last updated on 28/Oct/17

	$λ = l i n e a r m a s s d e n s i t y = \frac{M}{L}$ $\Rightarrow μ g (L - l) λ = λ g l$ $\Rightarrow μ = \frac{l}{(L - l)}$ $l e t a n e l e m e n t d x a t a d i s t a n c e$ $x f r o m e d g e o f t a b l e .$ ${d W}_{f r i c t i o n} = (- μ g λ d x) \cdot x$ $a s f r i c t i o n^{'} s d i r e c t i o n i s o p p o s i t e$ $t o t h a t o f d i s t c o v e r e d .$ $\Rightarrow \int_{0}^{W} d W = - \underset{0}{\int^{(L - l)}} μ g (\frac{M}{L}) x \cdot d x$ $\Rightarrow W = - \frac{M g l (L - l)}{2 L}$ $U s i n g E n e r g y c o n s e r v a t i o n$ $\Rightarrow ∣ Δ K . E . ∣ = ∣ Δ P . E . ∣ + W_{f r i c t i o n}$ $\Rightarrow \frac{1}{2} {m v}^{2} = ∣ g (λ l) (\frac{l}{2}) - g (λ L) (\frac{L}{2}) ∣ - \frac{M g l (L - l)}{2 L}$ $s u b t i t u t e λ = \frac{M}{L}$ $o n s o l v i n g$ $v = \sqrt{g (L - l)}$
	Commented by Physics lover last updated on 28/Oct/17

	Commented by Tinkutara last updated on 28/Oct/17

	$Thank you very much Sir!$
	Commented by Physics lover last updated on 28/Oct/17

	$y o u a r e w e l c o m e .$ $B T W, d i d y o u t r y q u e s t i o n n o . 14$ $H s e c t i o n, l a w s o f m o t i o n . I t s$ $a g o o d o n e t o o .$
	Commented by Physics lover last updated on 28/Oct/17

	$h m m m .$
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