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Physics

Mechanics

Easy

Question

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

$\frac{3 K}{π r^{2} m^{2} G}$
$\frac{K}{2 π r^{2} m^{2} G}$
$\frac{3 K}{π r^{2} m^{2} G}$
$\frac{K}{6 π r^{2} m^{2} G}$

The correct answer is: $fraction numerator straight K over denominator 2 pi straight r squared straight m squared straight G end fraction$

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

PhysicsMechanics

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

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A stone of mass of 16 kg is attached to a string 144 m long and is whirled in a horizontal circle. The maximum tension the string can withstand is 16 Newton. The maximum velocity of revolution that can be given to the stone without breaking it, will be

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A stone of mass of 16 kg is attached to a string 144 m long and is whirled in a horizontal circle. The maximum tension the string can withstand is 16 Newton. The maximum velocity of revolution that can be given to the stone without breaking it, will be

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$I f a p a r t i c l e o f m a s s 1 divided by 2 m v squared equals m g l i s m o v i n g i n a h o r i z o n t a l c i r c l e o f r a d i u s v equals square root of 2 g l end root w i t h a c e n t r i p e t a l f o r c e space space g equals 10 m divided by sec squared comma space t h e t o t a l e n e r g y i s$

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$A space m a s s space o f space 2 space k g space i s space w h i r l e d space i n space a space h o r i z o n t a l space c i r c l e space b y space m e a n s space o f space a space s t r i n g space a t space a n space i n i t i a l space s p e e d space o f space 5 r e v o l u t i o n s space p e r space m i n u t e. space K e e p i n g space t h e space r a d i u s space c o n s tan t space t h e space t e n s i o n space i n space t h e space s t r i n g space i s space d o u b l e d. space T h e space n e w space s p e e d space i s space n e a r l y space$

PhysicsMechanics

$A space m o t o r c y c l e space i s space g o i n g space o n space a n space o v e r b r i d g e space o f space r a d i u s space equals fraction numerator 2 pi over denominator T end fraction equals fraction numerator 2 pi over denominator 24 end fraction r a d divided by h r. space T h e space d r i v e r space m a i n t a i n s space a space c o n s tan t space s p e e d. space A s space t h e space m o t o r c y c l e space i s space a s c e n d i n g space o n space t h e space o v e r b r i d g e comma space t h e space n o r m a l space f o r c e space o n space i t space$

$A space m o t o r c y c l e space i s space g o i n g space o n space a n space o v e r b r i d g e space o f space r a d i u s space equals fraction numerator 2 pi over denominator T end fraction equals fraction numerator 2 pi over denominator 24 end fraction r a d divided by h r. space T h e space d r i v e r space m a i n t a i n s space a space c o n s tan t space s p e e d. space A s space t h e space m o t o r c y c l e space i s space a s c e n d i n g space o n space t h e space o v e r b r i d g e comma space t h e space n o r m a l space f o r c e space o n space i t space$

PhysicsMechanics

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$W h e n space a space p a r t i c l e space m o v e s space i n space a space u n i f o r m space c i r c u l a r space m o t i o n. space I t space h a s space$

$W h e n space a space p a r t i c l e space m o v e s space i n space a space u n i f o r m space c i r c u l a r space m o t i o n. space I t space h a s space$

PhysicsMechanics

$R a d i u s o f t h e c u r v e d r o a d o n n a t i o n a l h i g h w a y i s space pi squared m s to the power of negative 2 end exponent. W i d t h o f t h e r o a d i s equals v squared over r equals omega squared r T h e o u t e r e d g e o f t h e r o a d i s r a i s e d b y equals 4 pi squared n squared r equals 4 pi squared open parentheses 22 over 44 close parentheses squared cross times 1 space w i t h r e s p e c t t o i n n e r e d g e s o t h a t a c a r w i t h v e l o c i t y equals pi squared m divided by s squared space c a n p a s s s a f e o v e r i t. T h e v a l u e o f m g equals R s i n space theta i s$

$R a d i u s o f t h e c u r v e d r o a d o n n a t i o n a l h i g h w a y i s space pi squared m s to the power of negative 2 end exponent. W i d t h o f t h e r o a d i s equals v squared over r equals omega squared r T h e o u t e r e d g e o f t h e r o a d i s r a i s e d b y equals 4 pi squared n squared r equals 4 pi squared open parentheses 22 over 44 close parentheses squared cross times 1 space w i t h r e s p e c t t o i n n e r e d g e s o t h a t a c a r w i t h v e l o c i t y equals pi squared m divided by s squared space c a n p a s s s a f e o v e r i t. T h e v a l u e o f m g equals R s i n space theta i s$

PhysicsMechanics

$T h e space a v e r a g e space a c c e l e r a t i o n space v e c t o r space f o r space a space p a r t i c l e space h a v i n g space a space u n i f o r m space c i r c u l a r space m o t i o n space i s space$

$T h e space a v e r a g e space a c c e l e r a t i o n space v e c t o r space f o r space a space p a r t i c l e space h a v i n g space a space u n i f o r m space c i r c u l a r space m o t i o n space i s space$

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