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Physics

Mechanics

Easy

Question

A uniform solid sphere of mass m and radius R is kept on the rough horizontal surface with co-efficient of friction $mu$ = 1/3 as shown in figure. An impulse of $J equals m v subscript 0$ is applied tangentially on the sphere at an angle 37° with the horizontal as shown in the figure. (assuming J >> mgdt, g = 10 m/s², $open tan invisible function application 37 to the power of ring operator equals 3 divided by 4 close parentheses$ )

Find the velocity of centre of mass of the sphere just after impulse applied.

$\frac{4}{5} v_{0}$
$\frac{6}{5} v_{0}$
$\frac{7}{5} v_{0}$
$v_{0}$

The correct answer is: $v subscript 0$

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Find maximum elongation of spring

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Two blocks of masses m₁ and m₂ are connected by a spring of spring constant k. The block of mass m₂ is given a sharp impulse so that it acquires a velocity v₀ toward right. Velocity of centre of mass will be

Two blocks of masses m₁ and m₂ are connected by a spring of spring constant k. The block of mass m₂ is given a sharp impulse so that it acquires a velocity v₀ toward right. Velocity of centre of mass will be

PhysicsMechanics

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$F i g u r e space s h o w space a space b l o c k space A space o f space m a s s space 9 space m space h a v i n g space a space s m o o t h space s e m i c i r c u l a r space g r o o v e space o f space r a d i u s R space p l a c e d space o n space a space s m o o t h space h o r i z o n t a l space s u r f a c e. space A space b l o c k space B space o f space m a s s space m space i s space r e l e a s e d space f r o m space a space p o s i t i o n space i n space g r o o v e space w h e r e space i t s space r a d i u s space i s space h o r i z o n t a l. space F i n d space t h e space s p e e d space o f space b i g g e r space b l o c k space w h e n space s m a l l e r space b l o c k space r e a c h e s space i t s space b o t t o m m o s t space p o s i t i o n.$

$F i g u r e space s h o w space a space b l o c k space A space o f space m a s s space 9 space m space h a v i n g space a space s m o o t h space s e m i c i r c u l a r space g r o o v e space o f space r a d i u s R space p l a c e d space o n space a space s m o o t h space h o r i z o n t a l space s u r f a c e. space A space b l o c k space B space o f space m a s s space m space i s space r e l e a s e d space f r o m space a space p o s i t i o n space i n space g r o o v e space w h e r e space i t s space r a d i u s space i s space h o r i z o n t a l. space F i n d space t h e space s p e e d space o f space b i g g e r space b l o c k space w h e n space s m a l l e r space b l o c k space r e a c h e s space i t s space b o t t o m m o s t space p o s i t i o n.$

PhysicsMechanics

$A space s m a l l space b l o c k space o f space m a s s space m space i s space p r o j e c t e d space o n space a space l o n g space p l a n k space o f space m a s s space 3 space m space w i t h space v e l o c i t y v subscript 0 space a s space s h o w n space i n space f i g u r e. space F i n d space t h e space w o r k space d o n e space b y space f r i c t i o n space w h e n space s l i p p i n g space s t o p s.$

$A space s m a l l space b l o c k space o f space m a s s space m space i s space p r o j e c t e d space o n space a space l o n g space p l a n k space o f space m a s s space 3 space m space w i t h space v e l o c i t y v subscript 0 space a s space s h o w n space i n space f i g u r e. space F i n d space t h e space w o r k space d o n e space b y space f r i c t i o n space w h e n space s l i p p i n g space s t o p s.$

PhysicsMechanics

$T w o space t r o l l e y s space A space a n d space B space a r e space f r e e space t o space m o v e space o n space a space l e v e l space f r i c t i o n l e s s space t r a c k comma space a n d space a r e space i n i t i a l l y space s t a t i o n a r y. space A space m a n space o n space t r o l l e y space A space t h r o w s space a space b a g space o f space m a s s space 10 space k g space w i t h space a space h o r i z o n t a l space v e l o c i t y space o f space 4 space m s to the power of negative 1 end exponent space w i t h space r e s p e c t space t o space h i m s e l f space o n space t o space t r o l l e y space B space o f space m a s s space 100 space k g. space T h e space c o m b i n e d space m a s s space o f space t r o l l e y space A space left parenthesis e x c l u d i n g space b a g right parenthesis space a n d space t h e space m a n space i s space 140 space k g. space F i n d space t h e space r a t i o space o f space v e l o c i t i e s space o f space t r o l l e y s space A space a n d B comma space j u s t space a f t e r space t h e space b a g space l a n d s space o n space t r o l l e y space B.$

$T w o space t r o l l e y s space A space a n d space B space a r e space f r e e space t o space m o v e space o n space a space l e v e l space f r i c t i o n l e s s space t r a c k comma space a n d space a r e space i n i t i a l l y space s t a t i o n a r y. space A space m a n space o n space t r o l l e y space A space t h r o w s space a space b a g space o f space m a s s space 10 space k g space w i t h space a space h o r i z o n t a l space v e l o c i t y space o f space 4 space m s to the power of negative 1 end exponent space w i t h space r e s p e c t space t o space h i m s e l f space o n space t o space t r o l l e y space B space o f space m a s s space 100 space k g. space T h e space c o m b i n e d space m a s s space o f space t r o l l e y space A space left parenthesis e x c l u d i n g space b a g right parenthesis space a n d space t h e space m a n space i s space 140 space k g. space F i n d space t h e space r a t i o space o f space v e l o c i t i e s space o f space t r o l l e y s space A space a n d B comma space j u s t space a f t e r space t h e space b a g space l a n d s space o n space t r o l l e y space B.$

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$A space m u d space b a l l space a t space r e s t space e x p l o d e s space i n t o space t h r e e space f r a g e m e n t s space o f space m a s s e s space i n space t h e space r a t i o space l colon 2 colon 1. space T h e space t w o space e q u a l space m a s s e s space m o v e space w i t h space v e l o c i t i e s space 2 space i with hat on top plus 5 space j with hat on top minus 6 space k with hat on top space a n d space space i with hat on top plus 3 space j with hat on top plus 2 space k with hat on top. F i n d space t h e space v e l o c i t y space o f space t h e space t h i r d space m a s s.$

$A space m u d space b a l l space a t space r e s t space e x p l o d e s space i n t o space t h r e e space f r a g e m e n t s space o f space m a s s e s space i n space t h e space r a t i o space l colon 2 colon 1. space T h e space t w o space e q u a l space m a s s e s space m o v e space w i t h space v e l o c i t i e s space 2 space i with hat on top plus 5 space j with hat on top minus 6 space k with hat on top space a n d space space i with hat on top plus 3 space j with hat on top plus 2 space k with hat on top. F i n d space t h e space v e l o c i t y space o f space t h e space t h i r d space m a s s.$

PhysicsMechanics

$A space s h e l l space i s space f i r e d space f r o m space a space c a n n o n space w i t h space a space s p e e d space o f space 100 m divided by s space a t space a n space a n g l e space 30 to the power of 0 space w i t h space t h e space v e r t i c a l space left parenthesis y minus d i r e c t i o n right parenthesis. space A t space t h e space h i g h e s t space p o i n t space o f space i t s space t r a j e c t o r y comma space t h e space s h e l l space e x p l o d e s space i n t o space t w o space f r a g m e n t s space o f space m a s s e s space i n space t h e space r a t i o space 1 colon 2. space T h e space l i g h t e r f r a g e m e n t space m o v e s space v e r t i c a l l y space u p w a r d s space w i t h space a n space i n i t i a l space s p e e d space o f space 200 space m divided by s. space W h a t i s space t h e space s p e e d space o f space t h e space h e a v i e r space f r a g e m e n t space a t space t h e space t i m e space o f space e x p l o s i o n ?$

$A space s h e l l space i s space f i r e d space f r o m space a space c a n n o n space w i t h space a space s p e e d space o f space 100 m divided by s space a t space a n space a n g l e space 30 to the power of 0 space w i t h space t h e space v e r t i c a l space left parenthesis y minus d i r e c t i o n right parenthesis. space A t space t h e space h i g h e s t space p o i n t space o f space i t s space t r a j e c t o r y comma space t h e space s h e l l space e x p l o d e s space i n t o space t w o space f r a g m e n t s space o f space m a s s e s space i n space t h e space r a t i o space 1 colon 2. space T h e space l i g h t e r f r a g e m e n t space m o v e s space v e r t i c a l l y space u p w a r d s space w i t h space a n space i n i t i a l space s p e e d space o f space 200 space m divided by s. space W h a t i s space t h e space s p e e d space o f space t h e space h e a v i e r space f r a g e m e n t space a t space t h e space t i m e space o f space e x p l o s i o n ?$

PhysicsMechanics

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

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

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

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

PhysicsMechanics

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

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

PhysicsMechanics

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

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

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$A n space e x p l o s i o n space b l o w s space a space r o c k space i n t o space t h r e e space p a r t s. space T w o space p i e c e s space g o space o f f space a t space r i g h t space a n g l e s space t o space e a c h space o t h e r semicolon space 1.0 space k g space p i e c e space w i t h space a space v e l o c i t y space o f space 12 space m space divided by space s space a n d space o t h e r space 2.0 space k g space p i e c e space w i t h space a space v e l o c i t y space o f space 8 space m space divided by space s. space I f space t h e space t h i r d space p i e c e space f l i e s space o f f space w i t h space a space v e l o c i t y space o f space 40 space m space divided by space s comma space c o m p u t e space t h e space m a s s space o f space t h e space t h i r d space p i e c e.$

$A n space e x p l o s i o n space b l o w s space a space r o c k space i n t o space t h r e e space p a r t s. space T w o space p i e c e s space g o space o f f space a t space r i g h t space a n g l e s space t o space e a c h space o t h e r semicolon space 1.0 space k g space p i e c e space w i t h space a space v e l o c i t y space o f space 12 space m space divided by space s space a n d space o t h e r space 2.0 space k g space p i e c e space w i t h space a space v e l o c i t y space o f space 8 space m space divided by space s. space I f space t h e space t h i r d space p i e c e space f l i e s space o f f space w i t h space a space v e l o c i t y space o f space 40 space m space divided by space s comma space c o m p u t e space t h e space m a s s space o f space t h e space t h i r d space p i e c e.$

PhysicsMechanics

$A space b o d y space o f space m a s s space 5 space m space e x p l o d e s space a t space r e s t space b r e a k space u p space i n t o space t h r e e space p a r t s. space I f space t w o space p a r t s space h a v i n g space e q u a l space m a s s e s space f l y space o f t space p e r p e n d i c u l a r l y space t o space e a c h space o t h e r space w i t h space a space v e l o c i t y space o f space 18 space m space divided by space s comma space t h e n space c a l c u l a t e space t h e space v e l o c i t y space o f space t h e space t h i r d space p a r t space w h i c h space h a s space a space m a s s space 3 space t i m e s space t h e space m a s s space o f space e q u a l space p a r t s.$

$A space b o d y space o f space m a s s space 5 space m space e x p l o d e s space a t space r e s t space b r e a k space u p space i n t o space t h r e e space p a r t s. space I f space t w o space p a r t s space h a v i n g space e q u a l space m a s s e s space f l y space o f t space p e r p e n d i c u l a r l y space t o space e a c h space o t h e r space w i t h space a space v e l o c i t y space o f space 18 space m space divided by space s comma space t h e n space c a l c u l a t e space t h e space v e l o c i t y space o f space t h e space t h i r d space p a r t space w h i c h space h a s space a space m a s s space 3 space t i m e s space t h e space m a s s space o f space e q u a l space p a r t s.$

PhysicsMechanics

$A space 5.0 space g space b u l l e t space m o v i n g space a t space 100 space m space divided by space s space s t r i k e s space a space log. space A s s u m e space t h a t space t h e space b u l l e t space u n d e r g o e s space u n i f o r space d e c e l e r a t i o n space a n d space s t o p s space i n space 6.0 space c m. space F i n d space t h e space t i m e space t a k e n space f o r space t h e space b u l l e t space t o space s t o p$

$A space 5.0 space g space b u l l e t space m o v i n g space a t space 100 space m space divided by space s space s t r i k e s space a space log. space A s s u m e space t h a t space t h e space b u l l e t space u n d e r g o e s space u n i f o r space d e c e l e r a t i o n space a n d space s t o p s space i n space 6.0 space c m. space F i n d space t h e space t i m e space t a k e n space f o r space t h e space b u l l e t space t o space s t o p$

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