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mechanics

Easy

Question

$A space s m a l l space b l o c k space o f space m a s s space m space equals space 1 space k g space i s space p l a c e d space o v e r space a space w e d g e space o f space m a s s space M space equals space 4 space k g space a s space space s h o w n space i n space f i g u r e. space M a s s space m space i s space r e l e a s e d space f r o m space r e s t space A l l space s u r f a c e s space a r e space s m o o t h. space O r i g i n space O space i s space a s space s h o w n comma space left parenthesis g space equals space 10 space m divided by s squared right parenthesis F i n a l space v e l o c i t y space o f space t h e space w e d g e space i s.......... m divided by s$

$\frac{1}{\sqrt{3}}$
$\sqrt{3}$
$\frac{1}{\sqrt{2}}$
$\sqrt{2}$

The correct answer is: $\sqrt{2}$

$text At an instant when end text m text is just leaving end text M comma text let end text v subscript 1 text and end text v subscript 2 text be the end text text velocities of end text m text and end text M text (both horizontal). end text text From conservation of linear momentum end text 1 cross times v subscript 1 equals 4 v subscript 2 text From conservation of mechanical energy. end text 1 cross times 10 cross times left parenthesis 4 minus 2 right parenthesis equals 1 half cross times 1 cross times v subscript 1 superscript 2 plus 1 half cross times 4 cross times v subscript 2 superscript 2 o r space v subscript 1 superscript 2 plus 4 v subscript 2 superscript 2 equals 40 text Solving these two equations, we get end text table attributes columnalign left columnspacing 1em rowspacing 4 pt end attributes row cell v subscript 2 equals square root of 2 straight m divided by straight s end cell row cell a n d space v subscript 1 equals 4 square root of 2 straight m divided by straight s end cell end table$

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$bold italic P bold italic a bold italic s bold italic s bold italic a bold italic g bold italic e bold colon A space b l o c k space o f space m a s s space 1 space k g space i s space m o v i n g space t o w a r d s space a space m o v a b l e space w e d g e space o f space m a s s space 2 space k g space a s space s h o w n space i n space f i g u r e. space A l l space s u r f a c e s space a r e space s m o o t h. space W h e n space t h e space b l o c k space l e a v e s space t h e space w e d g e f r o m space t o p comma space i t s space v e l o c i t y space i s space m a k i n g space a n space a n g l e space theta space equals space 30 º space w i t h space h o r i z o n t a l. T h e space v a l u e space o f space v subscript 0 space i n space m divided by s space i s space$

$bold italic P bold italic a bold italic s bold italic s bold italic a bold italic g bold italic e bold colon A space b l o c k space o f space m a s s space 1 space k g space i s space m o v i n g space t o w a r d s space a space m o v a b l e space w e d g e space o f space m a s s space 2 space k g space a s space s h o w n space i n space f i g u r e. space A l l space s u r f a c e s space a r e space s m o o t h. space W h e n space t h e space b l o c k space l e a v e s space t h e space w e d g e f r o m space t o p comma space i t s space v e l o c i t y space i s space m a k i n g space a n space a n g l e space theta space equals space 30 º space w i t h space h o r i z o n t a l. T h e space v a l u e space o f space v subscript 0 space i n space m divided by s space i s space$

PhysicsMechanics

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

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

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$A space b o d y space o f space m a s s space 5 space k g space i s space m o v i n g space a l o n g space a space s t r a i g h t space l i n e space w i t h space a space v e l o c i t y space o f space 2 space m s to the power of – 1 end exponent. space A space v a r i a b l e space f o r c e space F space s t a r t e d space a c t i n g space o n space i t space f o r space a space t i m e space p e r i o d space o f space 0.5 s space a s space s h o w n space i n space f o r c e minus t i m e c u r v e. space T h e space f i n a l space v e l o c i t y space o f space t h e space b l o c k space i s$

$A space b o d y space o f space m a s s space 5 space k g space i s space m o v i n g space a l o n g space a space s t r a i g h t space l i n e space w i t h space a space v e l o c i t y space o f space 2 space m s to the power of – 1 end exponent. space A space v a r i a b l e space f o r c e space F space s t a r t e d space a c t i n g space o n space i t space f o r space a space t i m e space p e r i o d space o f space 0.5 s space a s space s h o w n space i n space f o r c e minus t i m e c u r v e. space T h e space f i n a l space v e l o c i t y space o f space t h e space b l o c k space i s$

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$T w o space p o i n t space m a s s e s space c o n n e c t e d space b y space a n space i d e a l space s t r i n g space a r e space p l a c e d space o n space a space s m o o t h space h o r i z o n t a l space s u r f a c e space a s space s h o w n space i n space t h e space d i a g r a m. space A space s h a r p space i m p u l s e space o f space 10 space k g minus m divided by s space i s space g i v e n space t o space t h e 5 space k g space m a s s. space T h e space v e l o c i t y space o f space t h e space 10 space k g space m a s s space w i l l space b e$

$T w o space p o i n t space m a s s e s space c o n n e c t e d space b y space a n space i d e a l space s t r i n g space a r e space p l a c e d space o n space a space s m o o t h space h o r i z o n t a l space s u r f a c e space a s space s h o w n space i n space t h e space d i a g r a m. space A space s h a r p space i m p u l s e space o f space 10 space k g minus m divided by s space i s space g i v e n space t o space t h e 5 space k g space m a s s. space T h e space v e l o c i t y space o f space t h e space 10 space k g space m a s s space w i l l space b e$

PhysicsMechanics

Mechanics

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$T h e space m a g n i t u d e space o f space t h e space f o r c e space left parenthesis i n space N right parenthesis space a c t i n g space o n space a space b o d y space v a r i e s space w i t h space t i m e space t space left parenthesis i n space m s right parenthesis space a s space s h o w n. space A B comma space B C space a n d space C D space a r e space s t r a i g h t space l i n e space s e g m e n t s. space T h e space m a g n i t u d e space o f space t h e space t o t a l space i m p u l s e space o f space t h e f o r c e space o n space t h e space b o d y space f r o m space t space equals space 4 space mu s space t o space t space equals space 16 space mu s space i s space$

$T h e space m a g n i t u d e space o f space t h e space f o r c e space left parenthesis i n space N right parenthesis space a c t i n g space o n space a space b o d y space v a r i e s space w i t h space t i m e space t space left parenthesis i n space m s right parenthesis space a s space s h o w n. space A B comma space B C space a n d space C D space a r e space s t r a i g h t space l i n e space s e g m e n t s. space T h e space m a g n i t u d e space o f space t h e space t o t a l space i m p u l s e space o f space t h e f o r c e space o n space t h e space b o d y space f r o m space t space equals space 4 space mu s space t o space t space equals space 16 space mu s space i s space$

PhysicsMechanics

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

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

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$I n space t h e space f i g u r e space s h o w n space a space p e n d u l u m space h a v i n g space b o b space o f space m a s s space m space equals space 1 space k g space i s space h a n g i n g space f r o m space t h e space r o o f space o f space a space b o x space o f space m a s s space M comma space k e p t space o v e r space a space s m o o t h space h o r i z o n t a l space p l a n e. space B o x space a n d space a space b l o c k space o f space m a s s space m ¢ space i s space c o n n e c t e d space b y space a space l i g h t space s t r i n g space g o i n g space o v e r space a space m a s s l e s s space l i g h t space p u l l e y. space W h e n space s y s t e m space i s space i n space m o t i o n comma space s t r i n g space m a k e s space a n g l e space 370 space f r o m space t h e space v e r t i c a l space a t space e q u i l i b r i u m space w i t h space r e s p e c t space t o space b o x. space F i n d space o u t space t h e space v a l u e space o f space m ¢ space i n space k g$

$I n space t h e space f i g u r e space s h o w n space a space p e n d u l u m space h a v i n g space b o b space o f space m a s s space m space equals space 1 space k g space i s space h a n g i n g space f r o m space t h e space r o o f space o f space a space b o x space o f space m a s s space M comma space k e p t space o v e r space a space s m o o t h space h o r i z o n t a l space p l a n e. space B o x space a n d space a space b l o c k space o f space m a s s space m ¢ space i s space c o n n e c t e d space b y space a space l i g h t space s t r i n g space g o i n g space o v e r space a space m a s s l e s s space l i g h t space p u l l e y. space W h e n space s y s t e m space i s space i n space m o t i o n comma space s t r i n g space m a k e s space a n g l e space 370 space f r o m space t h e space v e r t i c a l space a t space e q u i l i b r i u m space w i t h space r e s p e c t space t o space b o x. space F i n d space o u t space t h e space v a l u e space o f space m ¢ space i n space k g$

PhysicsMechanics

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$A space m o n k e y space o f space m a s s space m space i s space s i t t i n g space o n space a space p l a t f o r m space o f space m a s s space M. space M o n k e y space c a n space j u m p space w i t h space a space v e l o c i t y space o f space 5 space m divided by s space m a k i n g space a n space a n g l e space 370 space w i t h space t h e space h o r i z o n t a l space w i t h space r e s p e c t space t o space p l a t f o r m. space I f space v a l u e space o f space m divided by M space i s space x space ´ space 10 to the power of – 1 end exponent comma space t o space j u m p space t h e space m o n k e y space 1 space m e t e r space w i t h space r e s p e c t space t o space t h e space g r o u n d. space F i n d space o u t space t h e space v a l u e space o f space x$

$A space m o n k e y space o f space m a s s space m space i s space s i t t i n g space o n space a space p l a t f o r m space o f space m a s s space M. space M o n k e y space c a n space j u m p space w i t h space a space v e l o c i t y space o f space 5 space m divided by s space m a k i n g space a n space a n g l e space 370 space w i t h space t h e space h o r i z o n t a l space w i t h space r e s p e c t space t o space p l a t f o r m. space I f space v a l u e space o f space m divided by M space i s space x space ´ space 10 to the power of – 1 end exponent comma space t o space j u m p space t h e space m o n k e y space 1 space m e t e r space w i t h space r e s p e c t space t o space t h e space g r o u n d. space F i n d space o u t space t h e space v a l u e space o f space x$

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$A space p a r t i c l e space o f space m a s s space 500 space g space i s space p r o j e c t e d space a l o n g space x minus a x i s space w i t h space a space v e l o c i t y space 6 space m divided by s. space I t space i s space a c t e d space u p o n space b y space a space v a r i a b l e space f o r c e space a c t i n g space a l o n g space y minus a x i s space a s space s h o w n space i n space f i g u r e. space W h a t space i s space t h e space v e l o c i t y space o f space t h e space p a r t i c l e space a t space 8 space s ?$

$A space p a r t i c l e space o f space m a s s space 500 space g space i s space p r o j e c t e d space a l o n g space x minus a x i s space w i t h space a space v e l o c i t y space 6 space m divided by s. space I t space i s space a c t e d space u p o n space b y space a space v a r i a b l e space f o r c e space a c t i n g space a l o n g space y minus a x i s space a s space s h o w n space i n space f i g u r e. space W h a t space i s space t h e space v e l o c i t y space o f space t h e space p a r t i c l e space a t space 8 space s ?$

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$I n space t h e space a r r a n g e m e n t space s h o w n space i n space f i g u r e comma space space k g space a n d space space k g. space S t r i n g space i s space l i g h t space a n d space i n e x t e n s i b l e. space F i n d space t h e space a c c e l e r a t i o n space o f space c e n t r e space o f space m a s s space o f space b o t h space t h e space b l o c k s$

$I n space t h e space a r r a n g e m e n t space s h o w n space i n space f i g u r e comma space space k g space a n d space space k g. space S t r i n g space i s space l i g h t space a n d space i n e x t e n s i b l e. space F i n d space t h e space a c c e l e r a t i o n space o f space c e n t r e space o f space m a s s space o f space b o t h space t h e space b l o c k s$

PhysicsMechanics

Mechanics

Physics

$A space p r o j e c t i l e space o f space m a s s space m space i s space t h r o w n space w i t h space v e l o c i t y space v space m a k i n g space a n space a n g l e space o f space 30 degree space w i t h space v e r t i c a l. space N e g l e c t i n g space a i r space r e s i s tan c e space t h e space m a g n i t u d e space o f space c h a n g e space i n space m o m e n t u m space b e t w e e n space t h e space s t a r t i n g space p o i n t space a n d space a t space t h e space m a x i m u m space h e i g h t space i s space$

$A space p r o j e c t i l e space o f space m a s s space m space i s space t h r o w n space w i t h space v e l o c i t y space v space m a k i n g space a n space a n g l e space o f space 30 degree space w i t h space v e r t i c a l. space N e g l e c t i n g space a i r space r e s i s tan c e space t h e space m a g n i t u d e space o f space c h a n g e space i n space m o m e n t u m space b e t w e e n space t h e space s t a r t i n g space p o i n t space a n d space a t space t h e space m a x i m u m space h e i g h t space i s space$

PhysicsMechanics

Mechanics

Physics

$1. T w o space b l o c k s space o f space m a s s e s space m subscript 1 space end subscript a n d space m subscript 2 space a r e space space c o n n e c t e d space b y space a space s p r i n g space o f space s p r i n g space c o n s tan t space k. space T h e space b l o c k space o f space m a s s space m subscript 2 space i s space g i v e n space a space s h a r p space i m p u l s e space s o space t h a t space i t space a c q u i r e s space a space v e l o c i t y space v subscript 0 space t o w a r d space r i g h t. space I f space s p r i n g space b r e a k space w h e n space v e l o c i t y space o f space m subscript 2 space i s space fraction numerator 3 v subscript 0 over denominator 4 end fraction space t h e space v e l o c i t y space o f space o t h e r space b l o c k space w i l l space b e space$

$1. T w o space b l o c k s space o f space m a s s e s space m subscript 1 space end subscript a n d space m subscript 2 space a r e space space c o n n e c t e d space b y space a space s p r i n g space o f space s p r i n g space c o n s tan t space k. space T h e space b l o c k space o f space m a s s space m subscript 2 space i s space g i v e n space a space s h a r p space i m p u l s e space s o space t h a t space i t space a c q u i r e s space a space v e l o c i t y space v subscript 0 space t o w a r d space r i g h t. space I f space s p r i n g space b r e a k space w h e n space v e l o c i t y space o f space m subscript 2 space i s space fraction numerator 3 v subscript 0 over denominator 4 end fraction space t h e space v e l o c i t y space o f space o t h e r space b l o c k space w i l l space b e space$

PhysicsMechanics

Mechanics

Physics

$text Two blocks of masses m end text subscript text 1 end text end subscript text and m end text subscript text 2 end text end subscript text are connected by a spring of spring constant k. end text text The block of mass m end text subscript text 2 end text end subscript text is given a sharp impulse so that it acquires a velocity v end text subscript text 0 end text end subscript text toward right end text text Find maximum elongation of spring end text$

$text Two blocks of masses m end text subscript text 1 end text end subscript text and m end text subscript text 2 end text end subscript text are connected by a spring of spring constant k. end text text The block of mass m end text subscript text 2 end text end subscript text is given a sharp impulse so that it acquires a velocity v end text subscript text 0 end text end subscript text toward right end text text Find maximum elongation of spring end text$

PhysicsMechanics

Mechanics

Physics

$1. T w o space b l o c k s space o f space m a s s e s space m subscript 1 space a n d space m subscript 2 space a r e space space c o n n e c t e d space b y space a space s p r i n g space o f space s p r i n g space c o n s tan t space k. space T h e space b l o c k space o f space m a s s space m subscript 2 space i s space g i v e n space a space s h a r p space i m p u l s e space s o space t h a t space i t space a c q u i r e s space a space v e l o c i t y space v subscript 0 space t o w a r d space r i g h t. space space space V e l o c i t y space o f space c e n t r e space o f space m a s s space w i l l space b e space space$

$1. T w o space b l o c k s space o f space m a s s e s space m subscript 1 space a n d space m subscript 2 space a r e space space c o n n e c t e d space b y space a space s p r i n g space o f space s p r i n g space c o n s tan t space k. space T h e space b l o c k space o f space m a s s space m subscript 2 space i s space g i v e n space a space s h a r p space i m p u l s e space s o space t h a t space i t space a c q u i r e s space a space v e l o c i t y space v subscript 0 space t o w a r d space r i g h t. space space space V e l o c i t y space o f space c e n t r e space o f space m a s s space w i l l space b e space space$

PhysicsMechanics

Mechanics

Physics

A projectile is fixed with velocity v₀ = 2m/s at angle 60º with horizontal. At top of its trajectory it explodes into three fragment of equal mass. First fragment retraces the path, second moves vertically upward with speed $fraction numerator 3 straight V subscript 0 over denominator 2 end fraction$ . The speed of third fragment (in m/s) is –

A projectile is fixed with velocity v₀ = 2m/s at angle 60º with horizontal. At top of its trajectory it explodes into three fragment of equal mass. First fragment retraces the path, second moves vertically upward with speed $fraction numerator 3 straight V subscript 0 over denominator 2 end fraction$ . The speed of third fragment (in m/s) is –

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