physics

mechanics

Easy

Question

$T w o space b l o c k s space o f space m a s s e s space m subscript 1 space a n d space m subscript 2 space end subscript space c o n n e c t e d space b y space a space m a s s l e s s space s p r i n g space o f space s p r i n g space c o n s tan t space K space o n space a space s m o o t h space h o r i z o n t a l space s u r f a c e. B l o c k space 2 space i s space s h i f t e d space b y space a space s m a l l space d i s tan c e space x space t o space t h e l e f t space a n d space t h e n space r e l e a s e d. space W h e n space b l o c k space 1 space b r e a k s space o f f space t h e space w a l l comma v e l o c i t y space o f space c e n t r e space o f space m a s s space o f space t h e space s y s t e m space i s space$

$\frac{\sqrt{k} x}{m_{1} + m_{2}}$
$\frac{\sqrt{k} m_{1}}{m_{1} + m_{2}}$
$\frac{m_{1} x}{\sqrt{k} (m_{1} + m_{2)}}$
$\frac{\sqrt{k}}{m_{1} x}$

The correct answer is: $\frac{\sqrt{k} x}{m_{1} + m_{2}}$

$A t space t h e space t i m e space o f space b r e a k i n g space o f f comma space m subscript 2 space w l l space b e space m o v i n g space t o w a r d s space r i g h t space w i t h space v e l o c i t y space v space s u c h t h a t space p o t e n t i a l space e n e r g y space o f space s p r i n g space d u e space t o space c o m p r e s s i o n space c o n v e r t s space i n t o space k i n e t i c space e n e r g y rightwards double arrow 1 half K x squared equals 1 half space m subscript 2 space v subscript 2 superscript 2 space a n d space v subscript 2 equals fraction numerator square root of K x over denominator m subscript 2 end fraction V subscript c m end subscript equals fraction numerator m subscript 1 left parenthesis 0 right parenthesis plus m subscript 1 plus v subscript 2 over denominator m subscript 1 plus m subscript 2 end fraction equals space fraction numerator square root of K x over denominator left parenthesis m subscript 1 plus m subscript 2 right parenthesis end fraction$

$A space s h e l l space i n space f l i g h t space e x p o l d e s space i n t o space n space e q u a l space f r a g m e n t s comma space k space o f space t h e space f r a g m e n t s space r e a c h space t h e space g r o u n d e a r l i e r space t h a n space t h e space o t h e r space f r a g m e n t s. space T h e space a c c e l e r a t i o n space o f space t h e i r space c n t r e space o f space m a s s space s u b s e q u e n t l y space w i l l space b e$

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Mechanics

Physics

$A space b o a t space i s space a space t space r e s t space i n space a space r i v e r space a n d space h a s space a space m a s s space o f space 98 space k g. space A space d o g space o f space m a s s space 2 k g space o n space t h e space b o a t space f r o m space o n e space e n d space t o space t h e space o t h e r space w i t h space a space v e l o c i t y space 20 m divided by s. space T h e n space v e l o c i t y space o f space b o a t space i n space m divided by s i s$

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Mechanics

Physics

$A space 10 k g space b o y space s tan d i n g space i n space a space 40 k g space b o a t space f l o a t i n g space o n space w a t e r space i s space 20 m space f r o m t h e space s h o r e space o f space t h e space r i v e r. space I f space h e space m o v e s space 8 space m e t r e s space o n space t h e space b o a t space t o w a r d s space t h e space s h o r e comma space t h e n space h o w f a r space i s space h e space f r o m space t h e space s h o r e space n o w ?$

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Mechanics

Physics

$A space l a d d e r space A P thin space o f space t e n g t h space 5 m space i n c l i n e d space t o space a space v e r t i c a l space w a l l space i s space s l i p p i n g o v e r space a space h o r i z o n t a l space s u r f a c e space w i t h space v e l o c i t y space o f space 2 m divided by s comma space W h e n space A space i s space a t space a space d i s tan c e space 3 m space f r o m space O comma space t h e space v e l o c i t y space o f space C M space a t space t h i s space m o m e n t space i s space$

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Mechanics

Physics

$T w o space b l o c k s space o f space m a s s e s space 10 k g space a r e space p l a c e d space o n space X minus a x i s. space T h e space f i r s t space m a s s space i s m o v e d space o n space t h e space a x i s space b y space a space d i s tan c e space o f space 2 space c m space r i g h t. space B y space w h a t space d i s tan c e space s h o u l d space t h e space s e c o n d m a s s space b e space m o v e d space t o space k e e p space t h e space p o s i t i o n space o f space c e n t r e space o f space m a s s space u n c h a n g e d. space$

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Mechanics

Physics

$T h e space d e n s i t y space o f space a space l i n e a r space r o d space o f space l e n g t h space L space v a r i e s space a s rho equals left parenthesis a minus b x right parenthesis space w h e r e space x space i s space t h e space d i s tan c e space f r o m space t h e space e n d space O. D i s tan c e space o f space c e n t r e space o f space m a s s space o f space t e h space r o d space f r o m space O space i s$

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Mechanics

Physics

$T h r e e space i d e n t i c a l space b r i c k s space e a c h space o f space l e n g t h space L space a r e space p l a c e d space a s T h e space c e n t r e space o f space m a s s space o f space t h e space s y s t e m space i s space f r o m space O space a t space a space d i s tan c e.$

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Mechanics

Physics

$text Two particles of equal mass have velocities end text v with not stretchy rightwards arrow on top subscript 1 equals 4 i with not stretchy hat on top text and end text v with not stretchy rightwards arrow on top subscript 2 equals 4 j with not stretchy hat on top m s to the power of negative 1 end exponent text First particle end text text has an acceleration end text a with not stretchy rightwards arrow on top subscript 1 equals left parenthesis 5 i with not stretchy hat on top plus 5 ȷ with not stretchy hat on top right parenthesis m s to the power of negative 2 end exponent text while the acceleration of the other particle is end text text zero. The centre of mass of the two particles moves in a path of end text$

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Mechanics

Physics

$space Two particles of masses space 100 gm text and end text 300 gm space space have at a given time velocities space left parenthesis 10 i with hat on top minus 7 j with hat on top minus 3 k with hat on top right parenthesis text end text text and end text left parenthesis 7 i with hat on top minus 9 j with hat on top plus 6 k with hat on top right parenthesis cms to the power of negative 1 end exponent text respectively. Then velocity of centre of mass of the system in end text cms to the power of negative 1 end exponent text is end text$

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Mechanics

Physics

$A space 30 k g space p r o j e c t i l e space m o v i n g space h o r i z o n t a l l y space w i t h space a space v e l o c i t y space v with rightwards arrow on top subscript 0 equals left parenthesis 120 m divided by s right parenthesis i with hat on top space e x p l o d e s space i n t o space t w o space f r a g m e n t s space A space a n d space B space o f space m a s s e s space 12 k g space a n d space 18 k g comma space r e s p e c t i v e l y. space T a k i n g space p o i n t space o f space e x p l o s i o n space a s space o r i g i n space a n d space k n o w l e d g e space t h a t space 3 s space l a t e r space t h e space p o s i t i o n space o f space f r a g m e n t space A space i s space left parenthesis 300 m comma space 24 m comma negative 48 m space right parenthesis comma space d e t e r m i n e space t h e space p o s i t i o n space o f space f r a g m e n t space B space a t space t h e space s a m e space i n s tan t. space$

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Mechanics

Physics

$A space b l o c k space i s space r e l e a s e d space o n space t h e space c o n v e x space s u r f a c e space o f space a space h e m i s p h e r i c a l space w e d g e space a s space s h o w n space i n space f i g u r e. space D e t e r m i n e space t h e space d i s p l a c e m e n t space o f space t h e space w e d g e space w h e n space t h e space b l o c k space r e a c h e s space t h e space a n g u l a r space p o s i t i o n space theta.$

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Mechanics

Physics

$A space r i n g space a n d space a space p a r t i c l e space o f space m a s s e s space 2 m space a n d space m space a r e space k e p t space a s space s h o w n space i n space t h e space f i g u r e comma space s u c h space t h a t space t h e space C M space o f space t h e space c o m b i n a t i o n space l i e s space a t space t h e space o r i g i n. space F i n d space x.$

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Mechanics

Physics

$F i g u r e space s h o w s space a space u n i f o r m space d i s c space o f space r a d i u s space R comma space f r o m space w h i c h space a space h o l e space o f space r a d i u s space R divided by 2 space h a s space b e e n space c u t space o u t space f r o m space l e f t o f space t h e space c e n t r e space a n d space i s space p l a c e d space o n space t h e space r i g h t space o f space t h e space c e n t r e space o f space t h e space d i s c. space F i n d space t h e space C M space o f space t h e space r e s u l t i n g space d i s c.$

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