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Mechanics

Physics

Three identical spheres each of radius R are placed touching each other on a horizontal table as shown in figure. The co-ordinates of centre of mass are:

Three identical spheres each of radius R are placed touching each other on a horizontal table as shown in figure. The co-ordinates of centre of mass are:

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Mechanics

Physics

$A space m a n space left parenthesis m a s s space equals 50 space k g right parenthesis space a n d space h i s space s o n space left parenthesis m a s s space equals 20 space k g right parenthesis space a r e space s tan d i n g space o n space a space f r i c t i o n l e s s space s u r f a c e space f a c i n g space e a c h space o t h e r. space T h e space m a n space p u s h e s space h i s space s o n space s o space t h a t space h e space s t a r t s space m o v i n g space a t space a space s p e e d space o f space 0.70 space m s to the power of negative 1 end exponent space w i t h space r e s p e c t space t o space t h e space m a n. space T h e space s p e e d space o f space t h e space m a n space w i t h space r e s p e c t space t o space t h e space s u r f a c e space i s space colon$

$A space m a n space left parenthesis m a s s space equals 50 space k g right parenthesis space a n d space h i s space s o n space left parenthesis m a s s space equals 20 space k g right parenthesis space a r e space s tan d i n g space o n space a space f r i c t i o n l e s s space s u r f a c e space f a c i n g space e a c h space o t h e r. space T h e space m a n space p u s h e s space h i s space s o n space s o space t h a t space h e space s t a r t s space m o v i n g space a t space a space s p e e d space o f space 0.70 space m s to the power of negative 1 end exponent space w i t h space r e s p e c t space t o space t h e space m a n. space T h e space s p e e d space o f space t h e space m a n space w i t h space r e s p e c t space t o space t h e space s u r f a c e space i s space colon$

PhysicsMechanics

Mechanics

Physics

$T h e space t r a j e c t o r y space o f space a space p r o j e c t i l e space n e a r space t h e space s u r f a c e space o f space t h e space e a r t h space i s space g i v e n space a s space y equals 2 space x minus 9 space x squared. space I f space i t space w e r e space l a u n c h e d space a t space a n space a n g l e space theta subscript 0 space w i t h space s p e e d space v subscript 0 space t h e n space left parenthesis g equals 10 space m space s to the power of negative 2 end exponent right parenthesis colon$

$T h e space t r a j e c t o r y space o f space a space p r o j e c t i l e space n e a r space t h e space s u r f a c e space o f space t h e space e a r t h space i s space g i v e n space a s space y equals 2 space x minus 9 space x squared. space I f space i t space w e r e space l a u n c h e d space a t space a n space a n g l e space theta subscript 0 space w i t h space s p e e d space v subscript 0 space t h e n space left parenthesis g equals 10 space m space s to the power of negative 2 end exponent right parenthesis colon$

PhysicsMechanics

Mechanics

Physics

$A space u n i f o r m space r o d space o f space l e n g t h space ell space space i s space b e i n g space r o t a t e d space i n space a space h o r i z o n t a l space p l a n e space w i t h space a space c o n s tan t space a n g u l a r space s p e e d space a b o u t space a n space a x i s space p a s sin g space t h r o u g h space o n e space o f space i t s space e n d s. space I f space t h e space t e n s i o n space g e n e r a t e d space i n space t h e space r o d space d u e space t o space r o t a t i o n space i s space T left parenthesis x right parenthesis space a t space a space d i s tan c e space x space f r o m space t h e space a x i s comma space t h e n space w h i c h space o f space t h e space f o l l o w i n g space g r a p h s space d e p i c t s space i t space m o s t space c l o s e l y ?$

$A space u n i f o r m space r o d space o f space l e n g t h space ell space space i s space b e i n g space r o t a t e d space i n space a space h o r i z o n t a l space p l a n e space w i t h space a space c o n s tan t space a n g u l a r space s p e e d space a b o u t space a n space a x i s space p a s sin g space t h r o u g h space o n e space o f space i t s space e n d s. space I f space t h e space t e n s i o n space g e n e r a t e d space i n space t h e space r o d space d u e space t o space r o t a t i o n space i s space T left parenthesis x right parenthesis space a t space a space d i s tan c e space x space f r o m space t h e space a x i s comma space t h e n space w h i c h space o f space t h e space f o l l o w i n g space g r a p h s space d e p i c t s space i t space m o s t space c l o s e l y ?$

PhysicsMechanics

Mechanics

Physics

$A space c i r c u l a r space d i s c space o f space r a d i u s space b space h a s space a space h o l e space o f space r a d i u s space a space a t space i t s space c e n t r e space left parenthesis s e e space f i g u r e right parenthesis. space I f space t h e space m a s s space p e r space u n i t space a r e a space o f space t h e space d i s c space v a r i e s space a s space open parentheses sigma subscript 0 over r close parentheses comma space t h e n space t h e space r a d i u s space o f space g y r a t i o n space o f space t h e space d i s c space a b o u t space i t s space a x i s space p a s sin g space t h r o u g h space t h e space c e n t r e space i s colon$

$A space c i r c u l a r space d i s c space o f space r a d i u s space b space h a s space a space h o l e space o f space r a d i u s space a space a t space i t s space c e n t r e space left parenthesis s e e space f i g u r e right parenthesis. space I f space t h e space m a s s space p e r space u n i t space a r e a space o f space t h e space d i s c space v a r i e s space a s space open parentheses sigma subscript 0 over r close parentheses comma space t h e n space t h e space r a d i u s space o f space g y r a t i o n space o f space t h e space d i s c space a b o u t space i t s space a x i s space p a s sin g space t h r o u g h space t h e space c e n t r e space i s colon$

PhysicsMechanics

Mechanics

Physics

$A space s h e l l space i s space f i r e d space f r o m space a space f i x e d space a r t i l l e r y space g u n space w i t h space a n space i n i t i a l space s p e e d space u space s u c h space t h a t space i t space h i t s space t h e space t a r g e t space o n space t h e space g r o u n d space a t space a space d i s tan c e space R space f r o m space i t. space I f space t subscript 1 space a n d space t subscript 2 space a r e space t h e space v a l u e s space o f space t h e space t i m e space t a k e n space b y space i t space t o space h i t space t h e space t a r g e t space i n space t w o space p o s s i b l e space w a y s comma space t h e space p r o d u c t space t subscript 1 space t subscript 2 space i s space equals$

$A space s h e l l space i s space f i r e d space f r o m space a space f i x e d space a r t i l l e r y space g u n space w i t h space a n space i n i t i a l space s p e e d space u space s u c h space t h a t space i t space h i t s space t h e space t a r g e t space o n space t h e space g r o u n d space a t space a space d i s tan c e space R space f r o m space i t. space I f space t subscript 1 space a n d space t subscript 2 space a r e space t h e space v a l u e s space o f space t h e space t i m e space t a k e n space b y space i t space t o space h i t space t h e space t a r g e t space i n space t w o space p o s s i b l e space w a y s comma space t h e space p r o d u c t space t subscript 1 space t subscript 2 space i s space equals$

PhysicsMechanics

Mechanics

Physics

$A space p l a n e space i s space i n c l i n e d space a t space a n space a n g l e space alpha equals 30 degree space w i t h space r e s p e c t space t o space t h e space h o r i z o n t a l. space A space p a r t i c l e space i s space p r o j e c t e d space w i t h space a space s p e e d space u equals space 2 space m s space to the power of negative 1 end exponent comma space f r o m space t h e space b a s e space o f space t h e space p l a n e comma space m a k i n g space a n space a n g l e space theta equals 15 degree space w i t h space r e s p e c t space t o space t h e space p l a n e space a s space s h o w n space i n space t h e space f i g u r e. space t h e space d i s tan c e space f r o m space t h e space b a s e comma space a t space w h i c h space t h e space p a r t i c l e space h i t s space t h e space p l a n e space i s space c l o s e space t o space colon space left parenthesis T a k e space g equals 10 m s to the power of negative 2 end exponent right parenthesis$

$A space p l a n e space i s space i n c l i n e d space a t space a n space a n g l e space alpha equals 30 degree space w i t h space r e s p e c t space t o space t h e space h o r i z o n t a l. space A space p a r t i c l e space i s space p r o j e c t e d space w i t h space a space s p e e d space u equals space 2 space m s space to the power of negative 1 end exponent comma space f r o m space t h e space b a s e space o f space t h e space p l a n e comma space m a k i n g space a n space a n g l e space theta equals 15 degree space w i t h space r e s p e c t space t o space t h e space p l a n e space a s space s h o w n space i n space t h e space f i g u r e. space t h e space d i s tan c e space f r o m space t h e space b a s e comma space a t space w h i c h space t h e space p a r t i c l e space h i t s space t h e space p l a n e space i s space c l o s e space t o space colon space left parenthesis T a k e space g equals 10 m s to the power of negative 2 end exponent right parenthesis$

PhysicsMechanics

Mechanics

Physics

$T h e space t i m e space d e p e n d e n c e space o f space t h e space p o s i t i o n space o f space a space p a r t i c l e space o f space m a s s space m equals 2 space i s space g i v e n space b y space r with rightwards arrow on top left parenthesis t right parenthesis equals 2 t i with hat on top minus 3 t squared j with hat on top. space I t s space a n g u l a r space m o m e n t u m comma space w i t h space r e s p e c t space t o space t h e space o r i g i n comma space a t space t i m e space t equals 2 space i s$

$T h e space t i m e space d e p e n d e n c e space o f space t h e space p o s i t i o n space o f space a space p a r t i c l e space o f space m a s s space m equals 2 space i s space g i v e n space b y space r with rightwards arrow on top left parenthesis t right parenthesis equals 2 t i with hat on top minus 3 t squared j with hat on top. space I t s space a n g u l a r space m o m e n t u m comma space w i t h space r e s p e c t space t o space t h e space o r i g i n comma space a t space t i m e space t equals 2 space i s$

PhysicsMechanics

Mechanics

Physics

$A space s o l i d space s p h e r e space o f space m a s s space M space a n d space r a d i u s space R space i s space d i v i d e d space i n t o space t w o space u n e q u a l space p a r t s. space T h e space f i r s t space p a r t space h a s space a space m a s s space o f space fraction numerator 7 M over denominator 8 end fraction space a n d space i s space c o n v e r t e d space i n t o space a space u n i f o r m space d i s c space o f space r a d i u s space 2 space R. space T h e space s e c o n d space p a r t space i s space c o n v e r t e d space i n t o space a space u n i f o r m space s o l i d space s p h e r e. space L e t space I subscript 1 space b e space t h e space m o m e n t space o f space i n e r t i a space o f space t h e space d i s c space a b o u t space i t s space a x i s space a n d space I subscript 2 space b e space t h e space m o m e n t space o f space i n e r t i a space o f space t h e space n e w space s p h e r e space a b o u t space i t s space a x i s. space T h e space r a t i o space I subscript 1 space divided by space I subscript 2 space i s space g i v e n space b y colon$

$A space s o l i d space s p h e r e space o f space m a s s space M space a n d space r a d i u s space R space i s space d i v i d e d space i n t o space t w o space u n e q u a l space p a r t s. space T h e space f i r s t space p a r t space h a s space a space m a s s space o f space fraction numerator 7 M over denominator 8 end fraction space a n d space i s space c o n v e r t e d space i n t o space a space u n i f o r m space d i s c space o f space r a d i u s space 2 space R. space T h e space s e c o n d space p a r t space i s space c o n v e r t e d space i n t o space a space u n i f o r m space s o l i d space s p h e r e. space L e t space I subscript 1 space b e space t h e space m o m e n t space o f space i n e r t i a space o f space t h e space d i s c space a b o u t space i t s space a x i s space a n d space I subscript 2 space b e space t h e space m o m e n t space o f space i n e r t i a space o f space t h e space n e w space s p h e r e space a b o u t space i t s space a x i s. space T h e space r a t i o space I subscript 1 space divided by space I subscript 2 space i s space g i v e n space b y colon$

PhysicsMechanics

Mechanics

Physics

$A space m e t a l space c o i n space o f space m a s s space 5 space g space a n d space r a d i u s space 1 space c m space i s space f i x e d space t o space a space t h i n space s t i c k space A B space o f space n e g l i g i b l e space m a s s space a s space s h o w n space i n space t h e space f i g u r e. space T h e space s y s t e m space i s space i n i t i a l l y space a t space r e s t. space T h e space c o n s tan t space t o r q u e comma space t h a t space w i l l space m a k e space t h e space s y s t e m space r o t a t e space a b o u t space A B a t space 25 space r o t a t i o n s space p e r space s e c o n d space i n space 5 space s comma space i s space c l o s e space t o space colon$

$A space m e t a l space c o i n space o f space m a s s space 5 space g space a n d space r a d i u s space 1 space c m space i s space f i x e d space t o space a space t h i n space s t i c k space A B space o f space n e g l i g i b l e space m a s s space a s space s h o w n space i n space t h e space f i g u r e. space T h e space s y s t e m space i s space i n i t i a l l y space a t space r e s t. space T h e space c o n s tan t space t o r q u e comma space t h a t space w i l l space m a k e space t h e space s y s t e m space r o t a t e space a b o u t space A B a t space 25 space r o t a t i o n s space p e r space s e c o n d space i n space 5 space s comma space i s space c l o s e space t o space colon$

PhysicsMechanics

Mechanics

Physics

$T w o space c o a x i a l space d i s c s comma space h a v i n g space m o m e n t s space o f space i n e r t i a space space space straight I subscript 1 text and end text straight I subscript 1 over 2 space comma space a r e space r o t a t i n g space w i t h space r e s p e c t i v e space a n g u l a r space v e l o c i t i e s space omega subscript 1 a n d space omega subscript 1 over 2 space comma space a b o u t space t h e i r space c o m m o n space a x i s. space T h e y space a r e space b r o u g h t space i n space c o n t a c t space w i t h space e a c h space o t h e r space a n d space t h e r e a f t e r space t h e y space r o t a t e space w i t h space a space c o m m o n space a n g u l a r space v e l o c i t y. space I f space straight E subscript straight f space space a n d space E subscript i space a r e space t h e space f i n a l space a n d space i n i t i a l space t o t a l space e n e r g i e s comma space t h e n space left parenthesis straight E subscript straight f space – E subscript i right parenthesis space i s space colon space$

$T w o space c o a x i a l space d i s c s comma space h a v i n g space m o m e n t s space o f space i n e r t i a space space space straight I subscript 1 text and end text straight I subscript 1 over 2 space comma space a r e space r o t a t i n g space w i t h space r e s p e c t i v e space a n g u l a r space v e l o c i t i e s space omega subscript 1 a n d space omega subscript 1 over 2 space comma space a b o u t space t h e i r space c o m m o n space a x i s. space T h e y space a r e space b r o u g h t space i n space c o n t a c t space w i t h space e a c h space o t h e r space a n d space t h e r e a f t e r space t h e y space r o t a t e space w i t h space a space c o m m o n space a n g u l a r space v e l o c i t y. space I f space straight E subscript straight f space space a n d space E subscript i space a r e space t h e space f i n a l space a n d space i n i t i a l space t o t a l space e n e r g i e s comma space t h e n space left parenthesis straight E subscript straight f space – E subscript i right parenthesis space i s space colon space$

PhysicsMechanics

Mechanics

Physics

$T w o space p a r t i c l e s comma space o f space m a s s e s space M space a n d space 2 M comma space m o v i n g comma space a s space s h o w n comma space w i t h space s p e e d s space o f space 10 space m divided by s space a n d space 5 space m divided by s comma space c o l l i d e space e l a s t i c a l l y space a t space t h e space o r i g i n. space A f t e r space t h e space c o l l i s i o n comma space t h e y space m o v e space a l o n g space t h e space i n d i c a t e d space d i r e c t i o n s space w i t h space s p e e d s space v subscript 1 space a n d space v subscript 2 comma space r e s p e c t i v e l y T h e space v a l u e s space o f space v subscript 1 space a n d space v subscript 2 space a r e space n e a r l y space colon$

$T w o space p a r t i c l e s comma space o f space m a s s e s space M space a n d space 2 M comma space m o v i n g comma space a s space s h o w n comma space w i t h space s p e e d s space o f space 10 space m divided by s space a n d space 5 space m divided by s comma space c o l l i d e space e l a s t i c a l l y space a t space t h e space o r i g i n. space A f t e r space t h e space c o l l i s i o n comma space t h e y space m o v e space a l o n g space t h e space i n d i c a t e d space d i r e c t i o n s space w i t h space s p e e d s space v subscript 1 space a n d space v subscript 2 comma space r e s p e c t i v e l y T h e space v a l u e s space o f space v subscript 1 space a n d space v subscript 2 space a r e space n e a r l y space colon$

PhysicsMechanics

Mechanics

Physics

$A space b u l l e t space o f space m a s s space 20 space g space h a s space a n space i n i t i a l space s p e e d space o f space 1 space m s to the power of negative 1 end exponent comma space j u s t space b e f o r e space i t space s t a r t s space p e n e t r a t i n g space a space m u d space w a l l space o f space t h i c k n e s s space 20 space c m. I f space t h e space w a l l space o f f e r s space a space m e a n space r e s i s tan c e space o f space 2.5 cross times 10 to the power of negative 2 end exponent straight N comma space t h e space s p e e d space o f space t h e space b u l l e t space a f t e r space e m e r g i n g space f r o m space t h e space o t h e r space s i d e space o f space t h e space w a l l space i s space c l o s e space t o space colon$

$A space b u l l e t space o f space m a s s space 20 space g space h a s space a n space i n i t i a l space s p e e d space o f space 1 space m s to the power of negative 1 end exponent comma space j u s t space b e f o r e space i t space s t a r t s space p e n e t r a t i n g space a space m u d space w a l l space o f space t h i c k n e s s space 20 space c m. I f space t h e space w a l l space o f f e r s space a space m e a n space r e s i s tan c e space o f space 2.5 cross times 10 to the power of negative 2 end exponent straight N comma space t h e space s p e e d space o f space t h e space b u l l e t space a f t e r space e m e r g i n g space f r o m space t h e space o t h e r space s i d e space o f space t h e space w a l l space i s space c l o s e space t o space colon$

PhysicsMechanics

Mechanics

Physics

$A space t h i n space d i s c space o f space m a s s space M space a n d space r a d i u s space R space h a s space m a s s space p e r space u n i t space a r e a space sigma left parenthesis r right parenthesis equals k r squared space w h e r e space r space i s space t h e space d i s tan c e space f r o m space i t s space c e n t r e. I t s space m o m e n t space o f space i n e r t i a space a b o u t space a n space a x i s space g o i n g space t h r o u g h space i t s space c e n t r e space o f space m a s s space a n d space p e r p e n d i c u l a r space t o space i t s space p l a n e space i s space colon space$

$A space t h i n space d i s c space o f space m a s s space M space a n d space r a d i u s space R space h a s space m a s s space p e r space u n i t space a r e a space sigma left parenthesis r right parenthesis equals k r squared space w h e r e space r space i s space t h e space d i s tan c e space f r o m space i t s space c e n t r e. I t s space m o m e n t space o f space i n e r t i a space a b o u t space a n space a x i s space g o i n g space t h r o u g h space i t s space c e n t r e space o f space m a s s space a n d space p e r p e n d i c u l a r space t o space i t s space p l a n e space i s space colon space$

PhysicsMechanics

Mechanics

Physics

$A space u n i f o r m space r e c tan g u l a r space t h i n space s h e e t space A B C D space o f space m a s space M space h a s space l e n g t h space a space a n d space b r e a d t h space b comma space a s space s h o w n space i n space t h e space f i g u r e. space I f space t h e space s h a d e d space p o r t i o n H B G O space i s space c u t minus o f f comma space t h e space c o o r d i n a t e s space o f space t h e space c e n t r e space o f space m a s s space o f space t h e space r e m a i n i n g space p o r t i o n space w i l l space b e$

$A space u n i f o r m space r e c tan g u l a r space t h i n space s h e e t space A B C D space o f space m a s space M space h a s space l e n g t h space a space a n d space b r e a d t h space b comma space a s space s h o w n space i n space t h e space f i g u r e. space I f space t h e space s h a d e d space p o r t i o n H B G O space i s space c u t minus o f f comma space t h e space c o o r d i n a t e s space o f space t h e space c e n t r e space o f space m a s s space o f space t h e space r e m a i n i n g space p o r t i o n space w i l l space b e$

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