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Question Number 24447 by Tinkutara last updated on 18/Nov/17

F i g u r e s h o w s t w o d i s c s o f s a m e m a s s

m . T h e y a r e r i g i d l y a t t a c h e d t o a s p r i n g

o f s t i f f n e s s k . T h e s y s t e m i s i n

e q u i l i b r i u m . F r o m t h i s e q u i l i b r i u m

p o s i t i o n, t h e u p p e r d i s c i s p r e s s e d

d o w n s l o w l y b y a d i s t a n c e x a n d

r e l e a s e d . F i n d t h e m i n i m u m v a l u e o f

x, i f t h e l o w e r d i s c i s j u s t l i f t e d o f f

t h e g r o u n d .

Commented by Tinkutara last updated on 18/Nov/17

Commented by ajfour last updated on 18/Nov/17

m g = {k x}_{0} \dots . (i)

E_{i} = \frac{1}{2} k {(x_{0} + x)}^{2} \dots . (i i)

E_{f} = \frac{1}{2} {k y}^{2} + m g (y + x + x_{0}) . . (i i i)

k y = m g \dots \dots (i v)

E_{i} = E_{f} g i v e s

{(x + x_{0})}^{2} = y^{2} + \frac{2 m g}{k} (y + x + x_{0})

u s i n g (i v) :

{(x + x_{0})}^{2} = {(\frac{m g}{k})}^{2} + \frac{2 m g}{k} (\frac{m g}{k} + x + x_{0})

\Rightarrow {(x + x_{0})}^{2} = 3 {(\frac{m g}{k})}^{2} + (\frac{2 m g}{k}) (x + x_{0})

\Rightarrow {(x + x_{0} - \frac{m g}{k})}^{2} = 4 {(\frac{m g}{k})}^{2}

\Rightarrow x + x_{0} = \frac{3 m g}{k}

r e c a l l i n g f r o m e q n (i) : {k x}_{0} = m g

w e h a v e x + \frac{m g}{k} = \frac{3 m g}{k}

H e n c e x_{m i n} = \frac{2 m g}{k} .

Commented by Tinkutara last updated on 18/Nov/17

W h y n o t E_{i} = \frac{1}{2} {k x}_{0}^{2} a n d h o w i t b e c o m e

\frac{1}{2} {(x + x_{0})}^{2} ? W e p r e s s e d i t d o w n s o i t

s h o u l d b e s u b t r a c t e d .

Commented by ajfour last updated on 18/Nov/17

t h e s p r i n g w a s c o m p r e s s e d b y

x_{0} d u e t o w e i g h t o f u p p e r d i s c

w h e n i t w a s i n e q u i l i b r i u m .

P r e s s i n g i t d o w n f u r t h e r b y x

i n c r e a s e s s p r i n g p o t e n t i a l e n e r g y

t o \frac{1}{2} k {(x + x_{0})}^{2} . T h i s i n s t a n t

i s t a k e n a s t h e i n i t i a l p o s i t i o n

i n m y s o l u t i o n . G r a v i t a t i o n a l

p o t e n t i a l e n e r g y i s t a k e n z e r o h e r e .

F r o m h e r e u p p e r d i s c r i s e s

t h r o u g h a h e i g h t = y - (- x_{0} - x)

= y + x_{0} + x

y i s t h e m x i m u m e x t e n s i o n i n

s p r i n g t h a t c o m e s a b o u t i n t h e

s p r i n g . F o r m i n i m u m x,

u p p e r d i s c i s a t i t s m a x i m u m

h e i g h t t h e n . S p r i n g p u l l s b o t h

d i s c s t o w a r d s e a c h o t h e r, a n d

t h i s p u l l i s j u s t s u f f i c i e n t t o

m a k e t h e n o r m a l r e a c t i o n (o f

g r o u n d) t o t h e l o w e r d i s c b e c o m e

z e r o, t h a t i s t o s a y, t h e l o w e r

d i s c i s l i f t e d o f f t h e g r o u n d f o r

a n y v a l u e o f x s u p e r i o r t o \frac{2 m g}{k} .

Commented by Tinkutara last updated on 18/Nov/17

B u t w h y t h e r e y - (- x - x_{0}) ? I t s h o u l d

b e y - (x + x_{0}) s i n c e s p r i n g h a s a l r e a d y

s t r e a c h e d x + x_{0} s o w e h a v e t o s u b t r a c t

t h i s f r o m m a x i m u m c o m p r e s s i o n,

i . e . y - x - x_{0} . W h y w r o n g ?

Commented by ajfour last updated on 18/Nov/17

Commented by ajfour last updated on 18/Nov/17

y i s e x t e n s i o n o f s p r i n g a b o v e

e q u i l i b r i u m p o s i t i o n; x_{0} a n d x

a r e c o m r e s s i o n s a n d a r e b e l o w i t .

Commented by Tinkutara last updated on 18/Nov/17

Thank you very much Sir!