Physics

Mechanics

Easy

Question

A spherical hole is made in a solid sphere of radius R. The mass of the sphere before hollowing was M. The gravitational field at the centre of the hole due to the remaining mass is –

Zero
$\frac{GM}{8 R^{2}}$
$\frac{GM}{2 R^{2}}$
$\frac{GM}{R^{2}}$

The correct answer is: $fraction numerator GM over denominator 2 straight R squared end fraction$

By the principle of superposition of fields $straight E with rightwards arrow on top equals straight E with rightwards arrow on top subscript 1 plus stack straight E subscript 2 with rightwards arrow on top text Here, end text straight E with rightwards arrow on top equals text net field at the centre of hole due to entire mass end text straight E with rightwards arrow on top subscript 1 equals text field due to remaining mass end text text and end text stack straight E subscript 2 with rightwards arrow on top equals text field due to mass in hole end text equals 0 stack straight E subscript 1 with rightwards arrow on top equals straight E with rightwards arrow on top equals open parentheses GM over straight R cubed close parentheses times straight r text where end text straight r equals straight R over 2 straight E with rightwards arrow on top equals fraction numerator GM over denominator 2 straight R squared end fraction$

A mass m is placed in the cavity inside a uniform hollow sphere of mass M as shown in the figure. What is the gravitational force on the mass m?

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Mechanics

Physics

$T w o space l e a d space s p h e r e s space o f space s a m e space r a d i u s space a r e space i n space c o n t a c t space w i t h space e a c h space o t h e r. space T h e space g r a v i t a t i o n a l space f o r c e space o f space a t t r a c t i o n space b e t w e e n space t h e m space i s space F. space I f space t w o space l e a d space s p h e r e s space o f space d o u b l e space t h e space p r e v i o u s space r a d i u s space a r e space i n space c o n t a c t space w i t h space e a c h space o t h e r comma space t h e space g r a v i t a t i o n a l space f o r c e space o f space a t t r a c t i o n space b e t w e e n space t h e m space w i l l space b e$

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Mechanics

Physics

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

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Mechanics

Physics

$T w o space p a r t i c l e s space e a c h space o f space m a s s space ‘ m ’ space a r e space p l a c e d space a t space A space a n d space C space a r e space s u c h space A B equals B C equals L. space T h e space g r a v i t a t i o n a l space f o r c e space o n space t h e space t h i r d space p a r t i c l e space p l a c e d space a t space D space a t space a space d i s tan c e space L space o n space t h e space p e r p e n d i c u l a r space b i s e c t o r space o f space t h e space l i n e space A C space i s$

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Mechanics

Physics

$T h r e e space i d e n t i c a l space p a r t i c l e s space e a c h space o f space m a s s space “ m ” space a r e space a r r a n g e d space a t space t h e space c o r n e r s space o f space a n space e q u i l a t e r a l space t r i a n g l e space o f space s i d e space “ L ”. space I f space t h e y space a r e space t o space b e space i n space e q u i l i b r i u m comma space t h e space s p e e d space w i t h space w h i c h space t h e y space m u s t space r e v o l v e space u n d e r space t h e space i n f l u e n c e space o f space o n e space a n o t h e r ’ s space g r a v i t y space i n space a space c i r c u l a r space o r b i t space c i r c u m s c r i b i n g space t h e space t r i a n g l e space i s$

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Mechanics

Physics

$T w o space l e a d space b a l l s space o f space m a s s e s space m space a n d space 5 m space h a v i n g space r a d i i space R space a n d space 2 R space a r e space s e p a r a t e d space b y space 12 R. space I f space t h e y space a t t r a c t space e a c h space o t h e r space b y space g r a v i t a t i o n a l space f o r c e comma space t h e space d i s tan c e space c o v e r e d space b y space s m a l l space s p h e r e space b e f o r e space t h e y space t o u c h space e a c h space$

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Mechanics

Physics

$I f space t h e space d i s tan c e space b e t w e e n space t h e space s u n space a n d space t h e space e a r t h space i s space i n c r e a s e d space b y space t h r e e space t i m e s comma space t h e n space t h e space g r a v i t a t i o n a l space f o r c e space b e t w e e n space t h e space t w o space w i l l$

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Mechanics

Physics

$I f space t h e space m a s s space o f space o n e space p a r t i c l e space i s space i n c r e a s e d space b y space 50 percent sign space a n d space t h e space m a s s space o f space a n o t h e r space p a r t i c l e space i s space d e c r e a s e d space b y space 50 space percent sign comma space t h e space f o r c e space b e t w e e n space t h e m$

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Mechanics

Physics

$T w o space m a s s e s space ‘ M ’ space a n d space ‘ 4 M ’ space a r e space a t space a space d i s tan c e space ‘ r ’ space a p a r t space o n space t h e space l i n e space j o i n i n g space t h e m comma space ‘ P ’ space i s p o i n t space w h e r e space t h e space r e s u l tan t space g r a v i t a t i o n a l space f o r c e space i s space z e r o space left parenthesis s u c h space a space p o i n t space i s space c a l l e d space a s space n u l l space p o i n t right parenthesis. space T h e space d i s tan c e space o f space ‘ P ’ space f r o m space t h e space m a s s space ‘ M ’ space i s$

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Mechanics

Physics

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

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Mechanics

Physics

$G r a v i t a t i o n a l space f o r c e space b e t w e e n space t w o space p o i n t space m a s s e s space m space a n d space M space s e p a r a t e d space b y space a space d i s tan c e space r space i s space F. space N o w space i f space a space p o i n t space m a s s space 3 m space i s space p l a c e d space n e x t space t o space m comma space t h e space t o t a l space f o r c e space o n space M space w i l l space b e$

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Mechanics

Physics

$A space 3 space k g space m a s s space a n d space a space 4 space k g space m a s s space a r e space p l a c e d space o n space x space a n d space y space a x e s space a t space a space d i s tan c e space o f space 1 space m e t r e space f r o m space t h e space o r i g i n space a n d space a space 1 space k g space m a s s space i s space p l a c e d space a t space t h e space o r i g i n. space T h e n space t h e space r e s u l tan t space g r a v i t a t i o n a l space f o r c e space o n space 1 space k g space m a s s space i s$

PhysicsMechanics

Mechanics

Physics

$T h r e e space u n i f o r m space s p h e r e s space e a c h space o f space m a s s space m space a n d space d i a m e t e r space D space a r e space k e p t space i n space s u c h space a space w a y space t h a t space e a c h space t o u c h e s space t h e space o t h e r space t w o comma space t h e n space m a g n i t u d e space o f space t h e space g r a v i t a t i o n a l space f o r c e space o n space a n y space o n e space s p h e r e space d u e space t o space t h e space o t h e r space t w o space i s$

PhysicsMechanics

Mechanics

Physics

$G r a v i t a t i o n a l space f o r c e space b e t w e e n space t w o space p o i n t space m a s s e s space m space a n d space M space s e p a r a t e d space b y space a space d i s tan c e space r space i s space F. space N o w space i f space a space p o i n t space m a s s space 3 m space i s space p l a c e d space n e x t space t o space m comma space t h e space f o r c e space o n space M space d u e space t o space m space b e c o m e s$

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Mechanics

Physics

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

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