Physics

Mechanics

Easy

Question

$T w o space b a l l s space w i t h space m a s s e s space m subscript 1 equals 3 k g space a n d space m subscript 2 equals 5 k g space h a v e space i n i t i a l space v e l o c i t i e s space v subscript 1 equals v subscript 2 equals 5 m divided by s space i n space t h e space d i r e c t i o n s space s h o w n space i n space f i g u r e. space T h e y space c o l l i d e space a t space t h e space o r i g i n. space F i n d space t h e space v e l o c i t y space o f space t h e space C M space 3 s space b e f o r e t h e space c o l l i s i o n.$

Two balls with masses and have initial velocities in the directions shown in figure. They collide at the origin. Find the velocity of the CM 3s before the collision.

$- 1.5 \hat{i} + 2 \hat{ȷ} m / s$
$1.5 \hat{i} + 2 \hat{ȷ} m / s$
$- 1.5 \hat{i} - 2 \hat{ȷ} m / s$
$- 1.5 \hat{i} + 3 \hat{ȷ} m / s$

The correct answer is: $negative 1.5 i with hat on top plus 2 ȷ with hat on top m divided by s$

$I f space w e space t a k e space b a l l space " 1 " space a n d space b a l l space apostrophe 2 apostrophe space a s space s y s t e m comma space t h e r e space i s space n o space e x t e m a l space f o r c e space a c t i n g space o n space t h e space s y s t e m. space H e n c e space t h e space v e l o c i t y space o f space t h e space C M space o f space t h e space s y s t e m space w i l l space b e space c o n s tan t. space F r o m space g i v e n space f i g u r e comma space w e space c a 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 C M comma space w h i c h space w i l l space a l s o space b e space t h e space v e l o c i t y space o f C M space 3 space s space b e f o r e space t h e space c o l l i s i o n. space x space minus c o m p o n e n t space o f space v e l o c i t y space o f space C M v subscript text CM(x) end text end subscript equals fraction numerator m subscript 1 v subscript 1 x end subscript plus m subscript 2 v subscript 2 x end subscript over denominator m subscript 1 plus m subscript 2 a end fraction equals fraction numerator left parenthesis 3 right parenthesis open parentheses negative 5 cos 37 degree close parentheses plus left parenthesis 5 right parenthesis left parenthesis 0 right parenthesis over denominator 8 k g end fraction equals negative 1.5 m divided by s y text -component of velocity of end text C M text , end text v subscript C M left parenthesis y right parenthesis end subscript equals fraction numerator m subscript 1 v subscript 1 y end subscript plus m subscript 2 v subscript 2 y end subscript over denominator m subscript 1 plus m subscript 2 end fraction equals fraction numerator left parenthesis 3 right parenthesis open parentheses negative 5 sin 37 degree close parentheses plus left parenthesis 5 cross times 5 right parenthesis over denominator 8 k g end fraction equals plus 2 m divided by s therefore v with rightwards arrow on top subscript C M end subscript equals negative 1.5 i with hat on top plus 2 j with rightwards arrow on top m divided by s$

$A space u n i f o r m space m e t a l space p l a t e space o f space r a d i u s space a comma space f r o m space w h i c h space a space d i s k space o f space d i a m e t e r space a space h a s space b e e n space s t a m p e d space o u t space left parenthesis r e m o v e d right parenthesis space a s space s h o w n space i n space f i g u r e. space L o c a t e space t h e space c e n t r e space o f space m a s s space o f space t h e space r e m a i n i n g space p l a t e. space F i n d space t h e space p o s i t i o n space o f space c e n t r e space o f space m a s s space o f space t h e space p l a t e.$

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Mechanics

Physics

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

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Mechanics

Physics

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

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Mechanics

Physics

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

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Mechanics

Physics

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

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Mechanics

Physics

$T h e space f i g u r e space s h o w s space a space s q u a r e space m e t a l space p l a t e space o f space s i d e space l space f r o m space w h i c h space a space s q u a r e space p l a t e space o f space s i d e space a space h a s space b e e n space c u t space a s space s h o w n space i n space t h e space f i g u r e. space F i n d space t h e space r a t i o space left parenthesis a divided by l right parenthesis space s o space t h a t space t h e space c e n t r e space o f space m a s s space o f space t h e space r e m a i n i n g space L minus s h a p e d space p l a t e space c o i n c i d e s space w i t h space t h e space p o i n t space A.$

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Mechanics

Physics

$D e t e r m i n e space t h e space c o o r d i n a t e space o f space c e n t r e space o f space m a s s space o f space a space t h i n space h o m o g e n e o u s space p l a t e space h a v i n g space t h e space f o r m space o f space a space r e c tan g l e space w i t h space s i d e s space r space a n d space 2 r space f r o m space w h i c h space a space s e m i c i r c l e space w i t h space a space r a d i u s space r space i s space c u t space o u t.$

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Mechanics

Physics

$F i n d space x space a n d space y space c o o r d i n a t e s space o f space t h e space c e n t r e space o f space m a s s space o f space t h e space p l a t e space s h o w n space i n space f i g u r e space f r o m space w h i c h space a space s q u a r e space o f space s i d e space 2 m space i s space c u t space o u t. space$

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Mechanics

Physics

$A space r i n g space a n d space a space p a r t i c l e space o f space m a s s e s space 2 m space a n d space m space a r e space k e p t space a s space s h o w n space i n space t h e space f i g u r e comma space s u c h space t h a t space t h e space C M space o f space t h e space c o m b i n a t i o n space l i e s space a t space t h e space o r i g i n. space F i n d space x.$

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Mechanics

Physics

$F i g u r e space s h o w s space a space u n i f o r m space d i s c space o f space r a d i u s space R comma space f r o m space w h i c h space a space h o l e space o f space r a d i u s space R divided by 2 space h a s space b e e n space c u t space o u t space f r o m space l e f t o f space t h e space c e n t r e space a n d space i s space p l a c e d space o n space t h e space r i g h t space o f space t h e space c e n t r e space o f space t h e space d i s c. space F i n d space t h e space C M space o f space t h e space r e s u l t i n g space d i s c.$

$A space s q u a r e space l a m i n a space o f space s i d e space a space a n d space a space c i r c u l a r space l a m i n a space o f space d i a m e t e r space a space a r e space p l a c e d space t o u c h i n g space e a c h space o t h e r space a s space s h o w n space i n space f i g u r e. space F i n d space d i s tan c e space o f space t h e i r space c e n t r e space o f space m a s space f r o m space p o i n t space O comma space t h e space c e n t r e space o f space s q u a r e.$

PhysicsMechanics

Mechanics

Physics

$T h r e e space p a r t i c l e s space o f space m a s s e s space 1 k g comma space 2 k g space a n d space 3 k g space a r e space p l a c e d space a t space t h e space c o r n e r s space A comma space B space a n d space C space r e s p e c t i v e l y space o f space a n space e q u i l a t e r a l space t r i a n g l e space A B C space o f space e d g e space 1 m. space F i n d space t h e space d i s tan c e space o f space t h e i r space c e n t r e space o f space m a s s space f r o m space A.$

PhysicsMechanics

Mechanics

Physics

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

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