$A space r o c k e t space h a s space t o space b e space l a u n c h e d space f r o m space e a r t h space i n space s u c h space a space w a y space t h a t space i t space n e v e r space r e t u r n s. space I f space E space i s space t h e space m i n i m u m space e n e r g y space d e l i v e r e d space b y space t h e space r o c k e t space l a u n c h e r comma space w h a t space s h o u l d space b e space t h e space m i n i m u m space e n e r g y space t h a t space t h e space l a u n c h e r space s h o u l d space h a v e space i f space t h e space s a m e space r o c k e t space i s space t o space b e space l a u n c h e d space f r o m space t h e space s u r f a c e space o f space t h e space m o o n ? space A s s u m e space t h a t space t h e space d e n s i t y space o f space t h e space e a r t h space a n d space t h e space m o o n space a r e space e q u a l space a n d space t h a t space o f space e a r t h apostrophe s space v o l u m e space i s space 64 space t i m e s space t h e space v o l u m e space o f space t h e space m o o n.$

$F o u r space i d e n t i c a l space p a r t i c l e s space o f space m a s s space M space a r e space l o c a t e d space a t space t h e space c o r n e r s space o f space a space s q u a r e space o f s i d e space ‘ a ’. space W h a t space s h o u l d space b e space t h e i r space space s p e e d space i f space e a c h space o f space t h e m space r e v o l v e s space u n d e r space t h e space i n f l u e n c e space o f space o t h e r s space g r a v i t a t i o n a l space f i e l d space i n space a space c i r c u l a r space o r b i t space c i r c u m s c r i b i n g space t h e space s q u a r e space ? space$

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Physics

$C o n s i d e r space a space s p h e r i c a l space g a s e o u s space c l o u d space o f space m a s s space d e n s i t y space rho left parenthesis r right parenthesis space i n space f r e e space s p a c e space w h e r e space r space i s space t h e space r a d i a l space d i s tan c e space f r o m space i t s space c e n t e r. space T h e space g a s e o u s space c l o u d space i s space m a d e space o f space p a r t i c l e s space o f space e q u a l space m a s s space m space m o v i n g space i n space c i r c u l a r space o r b i t s space a b o u t space t h e space c o m m o n space c e n t e r space w i t h space t h e space s a m e k i n e t i c space e n e r g y space K. space T h e space f o r c e space a c t i n g space o n space t h e space p a r t i c l e space i s space t h e i r space m u t u a l space g r a v i t a t i o n a l space f o r c e. space I f space rho left parenthesis r right parenthesis space i s space c o n s tan t space i n space t i m e comma space t h e space p a r t i c l e n u m b e r space d e n s i t y space n left parenthesis r right parenthesis equals rho left parenthesis r right parenthesis divided by m space i s space u n i v e r s a l space g r a v i t a t i o n a l space c o n s tan t right square bracket space$

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Physics

$A space v e r y space l o n g space s o l e n o i d space o f space r e a d i u s space R space i s space c a r r y i n g space c u r r e n t space straight I left parenthesis straight t right parenthesis equals kte to the power of negative alpha straight t end exponent left parenthesis straight k greater than 0 right parenthesis space comma space a s space a space f u n c t i o n space o f space t i m e space left parenthesis t space greater or equal than space 0 right parenthesis. C o u n t e r space c l o c k w i s e space c u r r e n t space i s space t a k e n space t o space b e space p o s i t i v e. space A space c i r c u l a r space c o n d u c t i n g space c o i l space o f space r a d i u s space 2 R space i s space p l a c e d space i n space t h e space e q u a t o r i a l space p l a n e space o f space t h e space s o l e n o i d space a n d space c o n c e n t r i c space w i t h space t h e space s o l e n o i d. space T h e space c u r r e n t space i n d u c e d space i n space t h e space o u t e r space c o i l space i s c o r r e c t l y space d e p i c t e d comma space a s space a space f u n c t i o n space o f space t i m e comma space b y space colon space$

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Physics

$T h e space p o s i t i o n space o f space a space p a r t i c l e space a s space a space f u n c t i o n space o f space t i m e space t comma space i s space g i v e n space b y space space space x left parenthesis t right parenthesis equals a t plus b t squared minus c t cubed w h e r e space a comma space b space a n d space c space a r e space c o n s tan t s. space W h e n space t h e space p a r t i c l e space a t t a i n s space z e r o space a c c e l e r a t i o n comma space t h e n space i t s space v e l o c i t y space w i l l space b e space colon space space$

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Physics

$A space s o l i d space s p h e r e space o f space m a s s space ‘ M ’ space a n d space r a d i u s space ‘ a ’ space i s space s u r r o u n d e d space b y space a space u n i f o r m space c o n c e n t r i c space s p h e r i c a l space s h e l l space o f space t h i c k n e s s 2 a space a n d space m a s s space 2 M. space T h e space g r a v i t a t i o n a l space f i e l d space a t space d i s tan c e space ‘ 3 a ’ space f r o m space t h e space c e n t r e space w i l l space b e space$

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Mechanics

Physics

$A space p a r t i c l e space o f space m a s s space m space i s space m o v i n g space a l o n g space a space t r a j e c t o r y space g i v e n space b y space table attributes columnalign left columnspacing 1em rowspacing 4 pt end attributes row cell x equals x subscript 0 plus a cos invisible function application omega subscript 1 t end cell row cell y equals y subscript 0 plus b sin invisible function application omega subscript 2 t end cell end table T h e space t o r q u e comma space a c t i n g space o n space t h e space p a r t i c l e space a b o u t space t h e space o r i g i n space comma space a t space t space equals space 0 space i s space colon$

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Physics

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

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Mechanics

Physics

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

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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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Physics

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

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