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

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

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Mechanics

Physics

$A space b o d y space o f space m a s s space m subscript 1 space m o v i n g space w i t h space a n space u n k n o w n space v e l o c i t y space o f space space v subscript 1 space i with hat on top space space comma space u n d e r g o e s space a space c o l l i n e a r space c o l l i s i o n space w i t h space a space b o d y space o f m a s s space m subscript 2 space m o v i n g space w i t h space a space v e l o c i t y space v subscript 2 space i with hat on top space space. space A f t e r space c o l l i s i o n comma space m subscript 1 space a n d space m subscript 2 space m o v e space w i t h space v e l o c i t i e s space o f space v subscript 2 space i with hat on top space space a n d space v subscript 4 space i with hat on top comma r e s p e c t i v e l y space. space I f space m subscript 2 space equals space 0.5 space m subscript 1 space a n d space v subscript 3 space equals space 0.5 space v subscript 1 comma space t h e n space v subscript 1 space i s space minus space space$

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Mechanics

Physics

$A space p a r t i c l e space s t a r t s space f r o m space o r i g i n space O space f r o m space r e s t space a n d space m o v e s space w i t h space a space u n i f o r m space a c c e l e r a t i o n space a l o n g space t h e space p o s i t i v e space x minus a x i s. space I d e n t i f y space a l l space f i g u r e s space t h a t space c o r r e c t l y space r e p r e s e n t space t h e space m o t i o n space q u a l i t a t i v e l y. space left parenthesis a space equals space a c c e l e r a t i o n comma space v equals space v e l o c i t y comma space x equals space d i s p l a c e m e n t comma space t equals space t i m e space right parenthesis$

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Mechanics

Physics

$A space r e c tan g u l a r space s o l i d space b o x space o f space l e n g t h space 0.3 space m space i s space h e l d space h o r i z o n t a l l y comma space w i t h space o n e space o f space i t s space s i d e s space o n space t h e space e d g e space o f space a space p l a t f o r m space o f space h e i g h t space 5 space m. space W h e n space r e l e a s e d comma space i t space s l i p s space o f f space t h e space t a b l e space i n space a space v e r y space s h o r t space t i m e space tau equals 0.01 straight s space r e m a i n i n g space e s s e n t i a l l y space h o r i z o n t a l. T h e space a n g l e space b y space w h i c h space i t space w o u l d space r o t a t e space w h e n space i t space h i t s space t h e space g r o u n d space w i l l space b e space left parenthesis i n space r a d i a n s right parenthesis space c l o s e space t o$

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Mechanics

Physics

$A space S o l i d space s p h e r e space a n d space s o l i d space c y l i n d e r space o f space i d e n t i c a l space r a d i i space a p p r o a c h space a n space i n c l i n e space w i t h space t h e space s a m e space l i n e a r space v e l o c i t y space left parenthesis S e e space f i g u r e right parenthesis. space B o t h space r o l l space w i t h o u t space s l i p p i n g space a l l space t h r o u g h o u t. space T h e space t w o space c l i m b space m a x i m u m space h e i g h t s space h blank subscript s p h end subscript space a n d space h subscript c y l end subscript space o n space t h e space i n c l i n e. space T h e space r a t i o space straight h subscript sph over straight h subscript cy 1 end subscript space i s space g i v e n space b y$

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Mechanics

Physics

$A space t h i n space c i r c u l a r space p l a t e space o f space m a s s space M space a n d space r a d i u s space R space h a s space i t s space d e n s i t y space v a r y i n g space a s space rho left parenthesis straight r right parenthesis equals rho subscript 0 straight r straight rho left parenthesis straight r right parenthesis equals straight rho subscript 0 straight r space w i t h space straight rho subscript 0 space a s space c o n s tan t space a n d space r space i s space t h e space d i s tan c e space f r o m space i t s space c e n t e r. space T h e space m o m e n t space o f space i n e r t i a space o f space t h e space c i r c u l a r space p l a t e space a b o u t space a n space a x i s space p e r p e n d i c u l a r space t o space t h e space p l a t e space a n d space p a s sin g space t h r o u g h space i t s space e d g e space i s space I equals a M R squared space. space T h e space v a l u e space o f space t h e space c o e f f i c i e n t space a space i s$

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Mechanics

Physics

$F o u r space p a r t i c l e s space A comma space B comma space C space a n d space D space w i t h space m a s s e s space m subscript A equals m comma space m subscript B equals 2 space m comma space m subscript C equals 3 subscript m space a n d space m subscript D equals 4 m space a r e space a t space t h e space c o r n e r s space o f space a space s q u a r e. space T h e y space h a v e space a c c e l e r a t i o n s space o f space e q u a l space m a g n i t u d e space w i t h space d i r e c t i o n s space a s space s h o w n. space T h e space a c c e l e r a t i o n space o f space t h e space c e n t r e o f space m a s s space o f space t h e space p a r t i c l e s space i s$

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Mechanics

Physics

$A space m o v i n g space c o i l space g a l v a n o m e t e r space h a s space a space c o i l space w i t h space 175 space t u r n s space a n d space a r e a space 1 space c m squared. space I t space u s e s space a space t o r s i o n space b a n d space o f space t o r s i o n c o n s tan t space 10 to the power of negative 6 end exponent space N minus m divided by r a d. space T h e space c o i l space i s space p l a c e d space i n space a space m a g n e t i c space f i e l d space B space p a r a l l e l space t o space i t s space p l a n e. space T h e space c o i l d e f l e c t s space b y space 1 degree space space f o r space a space c u r r e n t space o f space 1 space m A. space T h e space v a l u e space o f space B space left parenthesis i n space T e s l a right parenthesis space i s space a p p r o x i m a t e l y space colon space$

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Mechanics

Physics

$A space s t a t i o n a r y space h o r i z o n t a l space d i s c space i s space f r e e space t o space r o t a t e space a b o u t space i t s space a x i s. space W h e n space a space t o r q u e space i s space a p p l i e d o n space i t comma space i t s space k i n e t i c space e n e r g y space a s space a space f u n c t i o n space o f space theta comma space w h e r e space theta space i s space t h e space a n g l e space b y space w h i c h space i s space h a s space r o t a t e d comma space i s space g i v e n space a s space straight k theta squared. space I f space i t s space m o m e n t space o f space i n e r t i a space i s space I space t h e n space t h e space a n g u l a r space a c c e l e r a t i o n space o f space t h e space d i s c space i s space space$

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Mechanics

Physics

$I n space t h e space f o r m u l a space X equals 5 space Y space Z squared comma space X space a n d space Z space h a v e space d i m e n s i o n s space o f space c a p a c i tan c e space a n d space m a g n e t i c space f i e l d comma space r e s p e c t i v e l y. space W h a t space a r e space t h e space d i m e n s i o n s space o f space Y space i n space S I space u n i t s ?$

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Mechanics

Physics

$I f space m space left parenthesis g plus pi n r squared right parenthesis space a n d space m left parenthesis g plus pi n r right parenthesis space r e p r e s e n t space r a d i a l space a n d space tan g e n t i a l space a c c e l e r a t i o n s comma space t h e space m o t i o n space o f space a space p a r t i c l e space w i l l space b e space u n i f o r m l y space c i r c u l a r space i f$

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Mechanics

Physics

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

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Mechanics

Physics

$W h e n space a space b o d y space m o v e s space i n space a space c i r c u l a r space p a t h comma space n o space w o r k space i s space d o n e space b y space t h e space f o r c e space sin c e comma$

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Mechanics

Physics

In uniform circular motion

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